--- /srv/rebuilderd/tmp/rebuilderdyF8uf6/inputs/macaulay2-common_1.26.06+ds-5_all.deb +++ /srv/rebuilderd/tmp/rebuilderdyF8uf6/out/macaulay2-common_1.26.06+ds-5_all.deb ├── file list │ @@ -1,3 +1,3 @@ │ -rw-r--r-- 0 0 0 4 2026-10-24 01:47:31.000000 debian-binary │ --rw-r--r-- 0 0 0 567932 2026-10-24 01:47:31.000000 control.tar.xz │ --rw-r--r-- 0 0 0 33668792 2026-10-24 01:47:31.000000 data.tar.xz │ +-rw-r--r-- 0 0 0 567856 2026-10-24 01:47:31.000000 control.tar.xz │ +-rw-r--r-- 0 0 0 33668584 2026-10-24 01:47:31.000000 data.tar.xz ├── control.tar.xz │ ├── control.tar │ │ ├── ./control │ │ │ @@ -1,13 +1,13 @@ │ │ │ Package: macaulay2-common │ │ │ Source: macaulay2 │ │ │ Version: 1.26.06+ds-5 │ │ │ Architecture: all │ │ │ Maintainer: Debian Math Team │ │ │ -Installed-Size: 324507 │ │ │ +Installed-Size: 324505 │ │ │ Depends: fonts-katex (>= 0.16.10+~cs6.1.0), libjs-katex (>= 0.16.10+~cs6.1.0) │ │ │ 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 │ │ │ @@ -3496,25 +3496,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 47188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15630 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/A1BrouwerDegrees/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40893 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___A__Infinity.out │ │ │ --rw-r--r-- 0 root (0) root (0) 918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 917 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/___Check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4348 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_a__Infinity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 56403 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_burke__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_display__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_extract__Blocks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1714 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_golod__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_is__Golod__A__Inf.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2183 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/example-output/_picture.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 7431 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_a__Infinity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 67690 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_burke__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9839 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_display__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10574 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_extract__Blocks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9459 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_golod__Betti.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_has__Minimal__Mult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AInfinity/html/_is__Golod__A__Inf.html │ │ │ @@ -3610,25 +3610,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7817 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjointIdeal/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 37097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjoint__Matrix.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1748 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1747 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_expected__Dimension.out │ │ │ -rw-r--r-- 0 root (0) root (0) 558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_linear__System__On__Rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1889 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_rational__Surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1596 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_slow__Adjunction__Calculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2944 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_special__Families__Of__Sommese__Vande__Ven.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 23 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 9411 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ --rw-r--r-- 0 root (0) root (0) 11522 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6813 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_expected__Dimension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_linear__System__On__Rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9773 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9934 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_rational__Surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9512 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_slow__Adjunction__Calculation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12677 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_special__Families__Of__Sommese__Vande__Ven.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/index.html │ │ │ @@ -3818,15 +3818,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3474 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_beilinson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_bgg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2077 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_cohomology__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Complex_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_direct__Image__Complex_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2382 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2381 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_pure__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_sym__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 678 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_tate__Resolution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/example-output/_universal__Extension.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 3963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/___Exterior.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4281 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/___Regularity.html │ │ │ @@ -3834,15 +3834,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6717 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_bgg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12329 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_cohomology__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9244 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Complex_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10236 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_direct__Image__Complex_lp__Module_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5853 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_projective__Product.html │ │ │ --rw-r--r-- 0 root (0) root (0) 14296 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 14295 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7006 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_sym__Ext.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7850 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_tate__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/_universal__Extension.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12502 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BGG/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BIBasis/ │ │ │ @@ -3867,18 +3867,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 77417 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3108 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BeginningMacaulay2/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 2927 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 29 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 5777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 5788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Benchmark/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 289778 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BernsteinSato/example-output/ │ │ │ @@ -4142,15 +4142,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10613 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Bertini/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 178971 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 7054 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out │ │ │ --rw-r--r-- 0 root (0) root (0) 7686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 7685 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2329 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp4.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp5.out │ │ │ -rw-r--r-- 0 root (0) root (0) 900 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_sp^_sp__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 747 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 845 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_sp_us_sp__Z__Z.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2414 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Character_spclass.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Equality_spchecks.out │ │ │ @@ -4182,15 +4182,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 595 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 62 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6359 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6298 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Action__On__Graded__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15599 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html │ │ │ --rw-r--r-- 0 root (0) root (0) 19435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 19434 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8815 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9421 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp5.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14589 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7168 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Decomposition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7784 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_sp^_sp__List.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5919 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/BettiCharacters/html/___Character_sp__Array.html │ │ │ @@ -4691,49 +4691,49 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12305 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4872 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ChainComplexOperations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 123136 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 4376 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___C__S__M.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1503 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Smooth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Check__Toric__Variety__Valid.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3442 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chern.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2404 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Class__In__Toric__Chow__Ring.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Comp__Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler.out │ │ │ --rw-r--r-- 0 root (0) root (0) 341 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Euler__Affine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 760 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ --rw-r--r-- 0 root (0) root (0) 624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Input__Is__Smooth.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 623 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Method.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1049 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Multi__Proj__Coord__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Output.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3327 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Segre.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1666 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Toric__Chow__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 570 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_is__Multi__Homogeneous.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/_probabilistic_spalgorithm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 665 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 55 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 23815 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 23816 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11361 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Toric__Variety__Valid.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chern.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9699 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Class__In__Toric__Chow__Ring.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10506 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18685 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6076 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6077 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6025 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6588 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6135 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6587 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Multi__Proj__Coord__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16520 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Output.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Segre.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8812 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Toric__Chow__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5032 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_bertini__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_configuring_sp__Bertini.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CharacteristicClasses/html/_is__Multi__Homogeneous.html │ │ │ @@ -5089,15 +5089,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 231378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__G__G__L.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1959 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___B__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Complete__Intersection__Resolutions.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4598 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4596 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/___Eisenbud__Shamash__Total.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2743 2026-10-24 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01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_make__T.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_matrix__Factorization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 10478 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_new__Ext.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1284 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_odd__Ext__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_regularity__Sequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_splittings.out │ │ │ -rw-r--r-- 0 root (0) root (0) 377 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_sum__Two__Monomials.out │ │ │ --rw-r--r-- 0 root (0) root (0) 449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/example-output/_two__Monomials.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 50 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___A__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5136 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Augmentation.html │ │ │ 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01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Search.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4885 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Standard__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/___Use__Special__Algorithms.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14844 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_compare__F__P__T.html │ │ │ --rw-r--r-- 0 root (0) root (0) 26150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 26152 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__Jumping__Exponent.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__F__P__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10037 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_is__Simple__Normal__Crossing.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20074 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/toc.html │ │ │ @@ -7722,15 +7722,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_moment__Graph.out │ │ │ -rw-r--r-- 0 root (0) root (0) 732 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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) 8062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_orbit__Closure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1060 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pullback_lp__Equivariant__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_pushforward.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_set__Indicator.out │ │ │ -rw-r--r-- 0 root (0) root (0) 437 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_trivial__K__Class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/example-output/_underlying__Graph_lp__Moment__Graph_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/ │ │ │ @@ -7773,15 +7773,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 8095 2026-10-24 01:47:31.000000 ./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) 8595 2026-10-24 01:47:31.000000 ./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) 9369 2026-10-24 01:47:31.000000 ./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) 8650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7658 2026-10-24 01:47:31.000000 ./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) 5902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_moment__Graph_lp__G__K__M__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7819 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_normal__Toric__Variety_lp__G__K__M__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 20015 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 20016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_projective__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7393 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pullback_lp__Equivariant__Map_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7948 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_pushforward.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7166 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_set__Indicator.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5918 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_trivial__K__Class.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/_underlying__Graph_lp__Moment__Graph_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/GKMVarieties/html/index.html │ │ │ @@ -9020,15 +9020,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4893 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_css__Lead__Term.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5004 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_diff__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 802 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_distraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_euler__Operators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_gkz.out │ │ │ -rw-r--r-- 0 root (0) root (0) 663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_is__Torus__Fixed.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_put__Weyl__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2252 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out │ │ │ @@ -9038,15 +9038,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6851 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Appell__F1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Canonical_sp__Series_sp__Tutorial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Theta__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/___Wto__T.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_create__Theta__Ring.html │ │ │ 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_gkz.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_indicial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_is__Torus__Fixed.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4094 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/HolonomicSystems/html/_nilsson__Support.html │ │ │ @@ -9247,18 +9247,18 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 466 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Frac__P_lp..._cm__Verbosity_eq_gt..._rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 707 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Fractions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1444 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/IntegralClosure/example-output/_ic__Map.out │ │ │ 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./usr/share/doc/Macaulay2/Jets/html/___Example_sp4.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___J__J.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Saturate.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/_jet.html │ │ │ @@ -9585,62 +9585,62 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Jets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 101880 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1949 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_all__Gradings.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2785 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2787 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_analyze__Strand.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_canonical__Homotopies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1036 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 3425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Table.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3427 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1006 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_carpet__Det.out │ │ │ --rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_compute__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6823 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_correspondence__Scroll.out │ │ │ -rw-r--r-- 0 root (0) root (0) 804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_cox__Matrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3.out │ │ │ --rw-r--r-- 0 root (0) root (0) 6871 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 6876 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_degenerate__K3__Betti__Tables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/example-output/_homotopy__Ranks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 720 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./usr/share/doc/Macaulay2/K3Carpets/html/___Scrolls.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_all__Gradings.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9712 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6739 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Carpet.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_canonical__Homotopies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12167 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9965 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10795 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10797 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6909 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6910 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 18363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_correspondence__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_cox__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9602 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15071 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15076 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_gorenstein__Double.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_hankel__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8285 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_homotopy__Ranks.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_irrelevant__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8194 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_product__Of__Projective__Spaces.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7292 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Equations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_relative__Resolution__Twists.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6025 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Scroll.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7697 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_scheme__In__Product.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_schreyer__Name.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6558 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/_small__Diagonal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 32029 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/K3Carpets/html/toc.html │ │ │ @@ -10094,35 +10094,35 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 60313 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 790 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/___Linear__Truncations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 231 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_comp__Min.out │ │ │ -rw-r--r-- 0 root (0) root (0) 283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_diagonal__Multidegrees.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Mins.out │ │ │ --rw-r--r-- 0 root (0) root (0) 701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_find__Region.out │ │ │ +-rw-r--r-- 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./usr/share/doc/Macaulay2/LinearTruncations/example-output/_multigraded__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1807 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_partial__Regularities.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_regularity__Bound.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/example-output/_support__Of__Tor.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 61 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6259 2026-10-24 01:47:31.000000 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8765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/_support__Of__Tor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 25981 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15960 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/LinearTruncations/html/toc.html │ │ │ @@ -10356,15 +10356,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1999 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Expression.out │ │ │ -rw-r--r-- 0 root (0) root (0) 788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Matrix_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1449 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ext^__Z__Z_lp__Module_cm__Module_rp.out │ │ 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(0) 339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Numbered__Vertical__List.out │ │ │ -rw-r--r-- 0 root (0) root (0) 430 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option.out │ │ │ -rw-r--r-- 0 root (0) root (0) 571 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Option__Table_sp_gt_gt_sp__Function.out │ │ │ @@ -10466,20 +10466,20 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Family_sp_us_st.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1401 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map.out │ │ │ -rw-r--r-- 0 root (0) root (0) 428 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Ring__Map_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_sh_qu_sp__Thing.out │ │ │ -rw-r--r-- 0 root (0) root (0) 866 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set_sp_st_st_sp__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.10.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1721 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.8.out │ │ │ -rw-r--r-- 0 root (0) root (0) 541 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.1.9.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Singular_sp__Book_sp1.2.13.out │ │ │ @@ -10611,15 +10611,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 234 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 154 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan.out │ │ │ -rw-r--r-- 0 root (0) root (0) 245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atan2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_atanh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 773 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_augmented_spassignment.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_autoload.out │ │ │ -rw-r--r-- 0 root (0) root (0) 137 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_base__Name.out │ │ │ @@ -10634,25 +10634,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 202 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_between.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_binomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_block__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_borel_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 725 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_break.out │ │ │ -rw-r--r-- 0 root (0) root (0) 996 2026-10-24 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_can__Use__Hilbert__Hint.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 591 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8939 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_capture.out │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_ceiling_lp__Number_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 128 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_center__String.out │ │ │ -rw-r--r-- 0 root (0) root (0) 946 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Base.out │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_change__Directory.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_char.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_characters.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1522 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_check.out │ │ │ -rw-r--r-- 0 root (0) root (0) 303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_class.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1307 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_clean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_code.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Monomial__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_codim_lp__Quotient__Ring_rp.out │ │ │ @@ -10668,22 +10668,22 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 288 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_column__Swap.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_columnate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_combine.out │ │ │ -rw-r--r-- 0 root (0) root (0) 198 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_command__Interpreter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comments.out │ │ │ -rw-r--r-- 0 root (0) root (0) 331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_common__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_commonest.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1539 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1551 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_comodule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 372 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compact__Matrix__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 251 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compare__Exchange.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compose.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2477 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compositions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 272 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_compress.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4286 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_sp__Groebner_spbases.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_computing_spsyzygies.out │ │ │ -rw-r--r-- 0 root (0) root (0) 103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1767 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_concatenating_spmatrices.out │ │ │ -rw-r--r-- 0 root (0) root (0) 392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conditional_spexecution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 239 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 478 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_conjugate_lp__Partition_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_constructing_spmaps_spbetween_spmodules.out │ │ │ @@ -10694,15 +10694,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 740 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_copy__File_lp__String_cm__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 112 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cos.out │ │ │ -rw-r--r-- 0 root (0) root (0) 113 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cosh.out │ │ │ -rw-r--r-- 0 root (0) root (0) 114 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cot.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_coth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover__Map_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cover_lp__Module_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 314 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cpu__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 243 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_create__Task.out │ │ │ -rw-r--r-- 0 root (0) root (0) 483 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_span_spideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 451 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_creating_spand_spwriting_spfiles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csc.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_csch.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1090 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current.out │ │ │ -rw-r--r-- 0 root (0) root (0) 84 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_current__Column__Number.out │ │ │ @@ -10801,15 +10801,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Exists.out │ │ │ -rw-r--r-- 0 root (0) root (0) 379 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Length.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 351 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Mode_lp__Z__Z_cm__String_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 82 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Readable.out │ │ │ --rw-r--r-- 0 root (0) root (0) 93 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 92 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Time.out │ │ │ -rw-r--r-- 0 root (0) root (0) 82 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_file__Writable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_fill__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 184 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_find__Heft.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_find__Program.out │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_find__Synonyms_lp__Symbol_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11761 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_fine_spcontrol_spof_spa_sp__Groebner_spbasis_spcomputation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_finish.out │ │ │ @@ -10914,15 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01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_installing_spaugmented_spassignment_spmethods.out │ │ │ --rw-r--r-- 0 root (0) root (0) 932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_instances.out │ │ │ -rw-r--r-- 0 root (0) root (0) 316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integers_spmodulo_spa_spprime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 315 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_integrate.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect.out │ │ │ -rw-r--r-- 0 root (0) root (0) 997 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Ideal_cm__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__R__Ri_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 189 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersect_lp__Set_cm__Set_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 190 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_intersection_spof_spideals.out │ │ │ @@ -10957,15 +10957,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Mutable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Open.out │ │ │ -rw-r--r-- 0 root (0) root (0) 290 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Output__File_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 338 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Polynomial__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Prime.out │ │ │ -rw-r--r-- 0 root (0) root (0) 150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Primitive.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 487 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 360 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Quotient__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ready_lp__File_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 115 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Real.out │ │ │ -rw-r--r-- 0 root (0) root (0) 230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Regular__File.out │ │ │ -rw-r--r-- 0 root (0) root (0) 223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 460 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Skew__Commutative.out │ │ │ @@ -11062,31 +11062,31 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 769 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 319 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 422 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Ring_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_matrix_lp__Vector_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 560 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_max.out │ │ │ 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./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Exponent.out │ │ │ -rw-r--r-- 0 root (0) root (0) 310 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_min__Position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 925 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Groebner__Basis_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2181 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingens_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 737 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mingle.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1805 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1804 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Betti.out │ │ │ -rw-r--r-- 0 root (0) root (0) 951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ideal_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 607 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimal__Presentation_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minimize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 504 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_minors_lp__Z__Z_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 318 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_mkdir.out │ │ │ @@ -11162,16 +11162,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 681 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pack.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_package.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packages.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_packing_spmonomials_spfor_spefficiency.out │ │ │ -rw-r--r-- 0 root (0) root (0) 131 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pad.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1059 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pairs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 144 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel__Apply.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8673 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8672 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parse.out │ │ │ -rw-r--r-- 0 root (0) root (0) 317 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parsing_spprecedence_cm_spin_spdetail.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_partitions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 652 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parts.out │ │ │ -rw-r--r-- 0 root (0) root (0) 277 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_path.out │ │ │ @@ -11180,15 +11180,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_peek_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 267 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permanents.out │ │ │ -rw-r--r-- 0 root (0) root (0) 874 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffian.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2202 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pfaffians.out │ │ │ -rw-r--r-- 0 root (0) root (0) 301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pi.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_pivots_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 13709 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 13710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_poincare.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2331 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polarize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polylog.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4055 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_polynomial_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 624 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_position.out │ │ │ -rw-r--r-- 0 root (0) root (0) 475 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_positions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 274 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_powermod.out │ │ │ -rw-r--r-- 0 root (0) root (0) 173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_precision.out │ │ │ @@ -11224,15 +11224,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 796 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_lp__Ring__Element_cm__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient__Remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3975 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_lp__Matrix_cm__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1400 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_quotient_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 94 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out │ │ │ -rw-r--r-- 0 root (0) root (0) 669 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Mutable__Matrix_lp__Z__Z_cm__Z__Z_cm__R__R_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__Subset.out │ │ │ -rw-r--r-- 0 root (0) root (0) 843 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__List_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1109 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Module_cm__Module_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Q__Q_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Type_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random_lp__Z__Z_cm__Ideal_rp.out │ │ │ @@ -11249,15 +11249,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 246 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_real__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 462 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_realpath.out │ │ │ -rw-r--r-- 0 root (0) root (0) 345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_recursion__Depth.out │ │ │ -rw-r--r-- 0 root (0) root (0) 729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reduce__Hilbert.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5963 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_reduced__Row__Echelon__Form.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1345 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 160 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regex__Quote.out │ │ │ --rw-r--r-- 0 root (0) root (0) 712 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 659 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1212 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 238 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularized__Beta.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_regularized__Gamma.out │ │ │ -rw-r--r-- 0 root (0) root (0) 225 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_relativize__Filename.out │ │ │ -rw-r--r-- 0 root (0) root (0) 631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remainder.out │ │ │ -rw-r--r-- 0 root (0) root (0) 713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remainder_sq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove.out │ │ │ @@ -11320,15 +11320,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 561 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_shuffle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 175 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sign.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2452 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_simple_sp__Groebner_spbasis_spcomputations_spover_spvarious_springs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 100 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sin.out │ │ │ -rw-r--r-- 0 root (0) root (0) 933 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_singular__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_size2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_smith__Normal__Form_lp__Matrix_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4508 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_solve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 891 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_some__Terms.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1085 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort__Columns.out │ │ │ -rw-r--r-- 0 root (0) root (0) 495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_sort_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 278 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_source_lp__Ring__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_span.out │ │ │ @@ -11431,15 +11431,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 220 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unique.out │ │ │ -rw-r--r-- 0 root (0) root (0) 268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unique__Permutations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 122 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unlock.out │ │ │ -rw-r--r-- 0 root (0) root (0) 196 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_unsequence.out │ │ │ -rw-r--r-- 0 root (0) root (0) 168 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_url__Encode.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_use_lp__Package_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 519 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_use_lp__Ring_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1173 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1174 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spfunctions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spfunctions_spwith_spoptional_spinputs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_sphooks.out │ │ │ -rw-r--r-- 0 root (0) root (0) 269 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_utf8.out │ │ │ -rw-r--r-- 0 root (0) root (0) 122 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_utf8substring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_valid_spnames.out │ │ │ -rw-r--r-- 0 root (0) root (0) 577 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_value_lp__Expression_rp.out │ │ │ @@ -11449,15 +11449,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_value_lp__Symbol_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 201 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_values.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_variables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 153 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Monoid_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vars_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1063 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_vector.out │ │ │ --rw-r--r-- 0 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./usr/share/doc/Macaulay2/MinimalPrimes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4222 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MinimalPrimes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Miura/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Miura/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 15856 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Miura/dump/rawdocumentation.dump │ │ │ @@ -14638,31 +14638,31 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39495 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 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16013 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10683 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4822 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplicitySequence/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplierIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiplierIdeals/dump/ │ │ │ @@ -14729,15 +14729,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 507 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/example-output/___Multiprojective__Variety_sp_eq_eq_sp__Multiprojective__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 390 2026-10-24 01:47:31.000000 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-rw-r--r-- 0 root (0) root (0) 755 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Edges.out │ │ │ -rw-r--r-- 0 root (0) root (0) 232 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_remove__Isomorphs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 203 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/example-output/_string__To__Graph.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 32 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6397 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Comparison_spof_sp__Graph6_spand_sp__Sparse6_spformats.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5908 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Checking_spfor_spisomorphic_spgraphs.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8749 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8750 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_add__Edges.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7835 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_are__Isomorphic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13355 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_build__Graph__Filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8002 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_count__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_filter__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9334 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Bipartite__Graphs.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Graphs.html │ │ │ @@ -15495,15 +15495,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 179808 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 431 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Dependent__Set.out │ │ │ -rw-r--r-- 0 root (0) root (0) 791 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op.out │ │ │ -rw-r--r-- 0 root (0) root (0) 553 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Diff__Op_sp__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2350 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 614 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_amult.out │ │ │ -rw-r--r-- 0 root (0) root (0) 931 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_coordinate__Change__Ops.out │ │ │ -rw-r--r-- 0 root (0) root (0) 354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 403 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_diff__Op_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_differential__Primary__Decomposition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 349 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_eliminating__Dual.out │ │ │ -rw-r--r-- 0 root (0) root (0) 270 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/example-output/_evaluate_lp__Diff__Op_cm__Abstract__Point_rp.out │ │ │ @@ -15536,15 +15536,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 75 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8726 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Dependent__Set.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9234 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Diff__Op_sp__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5207 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Sampler.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Hybrid_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8027 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Macaulay__Matrix_dq.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8016 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8017 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Tolerance_sp_lp__Noetherian__Operators_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6940 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_amult.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5516 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_colon.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8646 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_coordinate__Change__Ops.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6720 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NoetherianOperators/html/_diff__Op_lp__Matrix_rp.html │ │ │ @@ -15616,15 +15616,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 20057 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16625 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6404 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NonPrincipalTestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 609279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 3513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 3516 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Chow_spring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3270 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp^_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 908 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___Normal__Toric__Variety_sp_us_sp__Array.out │ │ │ -rw-r--r-- 0 root (0) root (0) 708 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/___O__O_sp__Toric__Divisor.out │ │ │ @@ -15688,17 +15688,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1271 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_lattice__Points_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1229 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Simplicial_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1568 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_make__Smooth_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 601 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_matrix_lp__Toric__Map_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 543 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_max_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1557 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1560 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_monomials_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2228 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_nef__Generators_lp__Normal__Toric__Variety_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1002 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Fan_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__List_cm__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Polyhedron_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 704 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_normal__Toric__Variety_lp__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1599 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_orbits_lp__Normal__Toric__Variety_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_picard__Group_lp__Normal__Toric__Variety_rp.out │ │ │ @@ -15726,15 +15726,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 881 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vector_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_vertices_lp__Toric__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weighted__Projective__Space_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 452 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/example-output/_weil__Divisor__Group_lp__Toric__Map_rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 68 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 13066 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 13069 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12164 2026-10-24 01:47:31.000000 ./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) 27816 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp^_st_st_sp__Z__Z.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9625 2026-10-24 01:47:31.000000 ./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) 13241 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Array.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Normal__Toric__Variety_sp_us_sp__Z__Z.html │ │ │ @@ -15803,17 +15803,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 10776 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Simplicial_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13861 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_make__Smooth_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_making_spnormal_sptoric_spvarieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11657 2026-10-24 01:47:31.000000 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11171 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_nef__Generators_lp__Normal__Toric__Variety_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 10532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 10534 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 19370 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__List_cm__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Matrix_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10007 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12299 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10951 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_orbits_lp__Normal__Toric__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12368 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_picard__Group_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -16123,40 +16123,40 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 40622 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 24531 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/NumericalCertification/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6381 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│ │ │ -rw-r--r-- 0 root (0) root (0) 7937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_extracting_spinformation_spfrom_spchain_spcomplexes.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OldChainComplexes/html/_free_spresolutions_spof_spmodules.html │ │ │ @@ -16657,17 +16657,17 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6580 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OpenMath/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5219 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OpenMath/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/OpenMath/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Oscillators/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 84707 2026-10-24 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./usr/share/doc/Macaulay2/Oscillators/html/_find__Real__Solutions.html │ │ │ @@ -16989,15 +16989,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 12822 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Parsing/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9410 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Parsing/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Parsing/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 135847 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/ │ │ │ --rw-r--r-- 0 root (0) root (0) 2841 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2842 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___A_spfamily_spof_sppaths_spon_spa_spcone.out │ │ │ -rw-r--r-- 0 root (0) root (0) 456 2026-10-24 01:47:31.000000 ./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 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Axis__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___C__Mon__Tensor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2634 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1530 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Path__Signatures.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___Polynomial_sppaths_spof_spdegree_spm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 11000 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/___The_spuniversal_spvariety_spand_sptoric_spcoordinates.out │ │ │ @@ -17024,15 +17024,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 886 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_tensor__Parametrization.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2023 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Algebra.out │ │ │ -rw-r--r-- 0 root (0) root (0) 387 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/example-output/_word__String.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 38 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/.Headline │ │ │ --rw-r--r-- 0 root (0) root (0) 11585 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 11586 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Array_sp_us_sp__N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Axis__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8469 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___C__Mon__Tensor.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4107 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Computing_sp__Path_sp__Varieties.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7635 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Polynomial__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21707 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___N__C__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15628 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/PathSignatures/html/___Path.html │ │ │ @@ -17971,15 +17971,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 162 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/example-output/_filter.out │ │ │ -rw-r--r-- 0 root (0) root (0) 465 2026-10-24 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./usr/share/doc/Macaulay2/Posets/example-output/_index__Labeling.out │ │ │ @@ -18100,15 +18100,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 6303 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_filter.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7577 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_filtration.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7126 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_flag__Chains.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_flag__Poset.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7011 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_flagf__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7014 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_flagh__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_gap__Convert__Poset.html │ │ │ --rw-r--r-- 0 root (0) root (0) 9269 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 9268 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6552 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_h__Polynomial.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6346 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_hasse__Diagram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5523 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_height_lp__Poset_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6940 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_hibi__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9862 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_hibi__Ring.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_incomparability__Graph.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7118 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Posets/html/_index__Labeling.html │ │ │ @@ -18650,15 +18650,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2633 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Example_sp__Type_sp[300c].out │ │ │ -rw-r--r-- 0 root (0) root (0) 2039 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.out │ │ │ -rw-r--r-- 0 root (0) root (0) 9657 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 13630 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3258 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1713 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Half_spcanonical_spdegree_sp20.out │ │ │ --rw-r--r-- 0 root (0) root (0) 21924 2026-10-24 01:47:31.000000 ./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) 21923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Noether-__Lefschetz_spexamples.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7051 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Pfaffians_spon_spquadrics.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1233 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Singularities_spof_splifting_spof_sptype_sp[300b].out │ │ │ -rw-r--r-- 0 root (0) root (0) 4091 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2380 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1645 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2176 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Type_sp[331]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.out │ │ │ @@ -18692,15 +18692,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9263 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 10050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Example_sp__Type_sp[300c].html │ │ │ -rw-r--r-- 0 root (0) root (0) 7030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spall_sppossible_spbetti_sptables_spfor_spquadratic_spcomponent_spof_spinverse_spsystem_spfor_spquartics_spin_sp4_spvariables.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16374 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp16_spbetti_sptables_sppossible_spfor_spquartic_spforms_spin_sp4_spvariables_cm_spand_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 20058 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sp__Betti_spstratum_spof_spa_spgiven_spquartic.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Finding_spthe_sppossible_spbetti_sptables_spfor_sppoints_spin_sp__P^3_spwith_spgiven_spgeometry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8895 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Half_spcanonical_spdegree_sp20.html │ │ │ --rw-r--r-- 0 root (0) root (0) 47772 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 47771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11186 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Noether-__Lefschetz_spexamples.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4325 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12642 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Pfaffians_spon_spquadrics.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8788 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Singularities_spof_splifting_spof_sptype_sp[300b].html │ │ │ -rw-r--r-- 0 root (0) root (0) 14894 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[000]_cm_sp__C__Y_spof_spdegree_sp20.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[210]_cm_sp__C__Y_spof_spdegree_sp18_spvia_splinkage.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Type_sp[310]_cm_sp__C__Y_spof_spdegree_sp17_spvia_splinkage.html │ │ │ @@ -18960,27 +18960,27 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 185 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/___Zero__Mean.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2643 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_disturb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1358 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_histogram.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1300 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_maximal__Entry.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1502 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_normalize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 5291 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Chain__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 435 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_random__Simplicial__Complex.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1143 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/example-output/_test__Time__For__L__L__Lon__Syzygies.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 44 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 5439 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Discrete.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___With__L__L__L.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/___Zero__Mean.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10101 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_disturb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7432 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_histogram.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7407 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_maximal__Entry.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7072 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_normalize.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14710 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Chain__Complex.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6729 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_random__Simplicial__Complex.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8245 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8266 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html │ │ │ -rw-r--r-- 0 root (0) root (0) 13046 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9662 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4959 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomComplexes/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 52982 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurves/example-output/ │ │ │ @@ -19045,33 +19045,33 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 86199 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 399 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Alexander__Probability.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Finding_sp__Extreme_sp__Examples.out │ │ │ --rw-r--r-- 0 root (0) root (0) 447 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 448 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/___Random__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 481 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__Chain__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 308 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_ideal__From__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_is__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Addition.out │ │ │ -rw-r--r-- 0 root (0) root (0) 906 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 564 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 479 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Elements__From__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 461 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 287 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 289 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial.out │ │ │ -rw-r--r-- 0 root (0) root (0) 384 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Monomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 359 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Pure__Binomial__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 733 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shellable__Ideal__Chain.out │ │ │ -rw-r--r-- 0 root (0) root (0) 752 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Shelling.out │ │ │ -rw-r--r-- 0 root (0) root (0) 426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Sparse__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 491 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Monomial__Ideal.out │ │ │ --rw-r--r-- 0 root (0) root (0) 8838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 8825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1627 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Toric__Edge__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 233 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_reg__Seq.out │ │ │ -rw-r--r-- 0 root (0) root (0) 350 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_square__Free.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 39 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 6207 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Alexander__Probability.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10488 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/___Finding_sp__Extreme_sp__Examples.html │ │ │ @@ -19080,28 +19080,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5977 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_is__Shelling.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7962 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Addition.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7486 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7888 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Binomial__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6753 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Edge__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_undeclare__Variable.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6744 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_value__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9003 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/_working_spwith_spgate_spmatrices.html │ │ │ --rw-r--r-- 0 root (0) root (0) 39799 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 39797 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23485 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7224 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLPexpressions/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 42792 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 3218 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/example-output/_sl2__Equivariant__Constant__Rank__Matrix.out │ │ │ @@ -20232,15 +20232,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1688 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Generation_spof_spformulas.out │ │ │ -rw-r--r-- 0 root (0) root (0) 902 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Grassmannian_spof_splines_spin_sp__P3.out │ │ │ -rw-r--r-- 0 root (0) root (0) 639 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Hilbert_sppolynomial_spand_sp__Todd_spclass_spof_spprojective_sp3-space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 803 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Lines_spon_spa_spquintic_spthreefold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 511 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__Riemann-__Roch_spformulas.out │ │ │ -rw-r--r-- 0 root (0) root (0) 569 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Example_spfrom_sp__Schubert_co_sp__The_spnumber_spof_spelliptic_spcubics_spon_spa_spsextic_sp4-fold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 247 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Hom_lp__Abstract__Sheaf_cm__Abstract__Sheaf_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1584 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Lines_spon_sphypersurfaces.out │ │ │ -rw-r--r-- 0 root (0) root (0) 256 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Abstract__Variety.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1332 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___O__O_sp_us_sp__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 913 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Quotient__Bundles.out │ │ │ -rw-r--r-- 0 root (0) root (0) 869 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spcurve.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1741 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spon_spa_spsurface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7382 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Riemann-__Roch_spwithout_spdenominators.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/example-output/___Ring_sp_us_sp__Chern__Class__Variable.out │ │ │ @@ -20352,15 +20352,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5827 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Examples_spfrom_sp__Schubert_cm_sptranslated.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Flag__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6592 2026-10-24 01:47:31.000000 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Abstract__Variety.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6575 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___O__O_sp_us_sp__Ring__Element.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4499 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Pull__Back.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6751 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Quotient__Bundles.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5376 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spcurve.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spon_spa_spsurface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11608 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Schubert2/html/___Riemann-__Roch_spwithout_spdenominators.html │ │ │ @@ -20538,15 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./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1282 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1608 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Group__Acting.out │ │ │ -rw-r--r-- 0 root (0) root (0) 565 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 701 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Odd__Or__Even.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___S__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1943 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Ring.out │ │ │ -rw-r--r-- 0 root (0) root (0) 954 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Ring__Element.out │ │ │ -rw-r--r-- 0 root (0) root (0) 719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/___Schur__Ring__Indexed__Variable__Table.out │ │ │ @@ -20561,15 +20561,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1749 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_dim_lp__List_cm__Schur__Ring__Element_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_e__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 640 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_h__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_internal__Product.out │ │ │ -rw-r--r-- 0 root (0) root (0) 978 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/SchurRings/example-output/_modification__Rule.out │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_numgens_lp__Schur__Ring_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 840 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_p__Variable.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3004 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_partitions_lp__Set_cm__Basic__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1932 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_plethysm.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_plethysm_lp__Basic__List_cm__Ring__Element_rp.out │ │ │ @@ -20599,15 +20599,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 1775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_to__Sp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 797 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/example-output/_to__Symm.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 69 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 10650 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Basis.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Class__Function.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8459 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___E__H__P__Variables.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8366 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SchurRings/html/___Eor__H.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8365 2026-10-24 01:47:31.000000 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./usr/share/doc/Macaulay2/SpechtModule/example-output/_inner__Product_lp__Z__Z_cm__Mutable__Matrix_cm__Mutable__Matrix_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 207 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_list__To__Tableau_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 993 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/example-output/_matrix__Representation.out │ │ │ @@ -21582,15 +21582,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5582 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_cycle__Decomposition_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_elementary__Symmetric__Polynomials_lp__Polynomial__Ring_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5252 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpechtModule/html/_entries_lp__Young__Tableau_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6545 2026-10-24 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2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize__Fano__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1966 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parametrize_lp__Hodge__Special__Fourfold_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 13809 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1488 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_special__Fourfold.out │ │ │ -rw-r--r-- 0 root (0) root (0) 413 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface.out │ │ │ -rw-r--r-- 0 root (0) root (0) 656 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_surface_lp__List_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2427 2026-10-24 01:47:31.000000 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│ │ @@ -21713,28 +21713,28 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5619 2026-10-24 01:47:31.000000 ./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) 8109 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Congruence__Of__Curves_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Cubic__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5784 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_map_lp__Gushel__Mukai__Fourfold_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_mirror__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6573 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_normal__Sheaf.html │ │ │ 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(0) root (0) 25215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10521 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6716 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_special__Fourfold_lp__String_cm__Z__Z_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7886 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__List_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9392 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_surface_lp__Multiprojective__Variety_cm__Multiprojective__Variety_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5826 2026-10-24 01:47:31.000000 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(0) root (0) 7138 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html │ │ │ -rw-r--r-- 0 root (0) root (0) 40566 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 35967 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 15426 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/SpectralSequences/ │ │ │ @@ -22558,15 +22558,15 @@ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 166859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 7476 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_action__On__Direct__Image.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1183 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson.out │ │ │ -rw-r--r-- 0 root (0) root (0) 926 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Bundle.out │ │ │ -rw-r--r-- 0 root (0) root (0) 632 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Contraction.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1692 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2581 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_bgg.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1129 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_coarse__Multigraded__Regularity.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2746 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_cohomology__Hash__Table.out │ │ │ -rw-r--r-- 0 root (0) root (0) 837 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_cohomology__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 46631 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_composed__Functions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 8706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_corner__Complex.out │ │ │ -rw-r--r-- 0 root (0) root (0) 6302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_direct__Image__Complex.out │ │ │ @@ -22600,15 +22600,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 3825 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/___Rings.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5425 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/___Sub__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 3987 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/___Tate__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27030 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_action__On__Direct__Image.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10364 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9213 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Bundle.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Contraction.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8993 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8994 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11253 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_bgg.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_coarse__Multigraded__Regularity.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12770 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_cohomology__Hash__Table.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9999 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_cohomology__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 69237 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_composed__Functions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5768 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_contraction__Data.html │ │ │ -rw-r--r-- 0 root (0) root (0) 23344 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TateOnProducts/html/_corner__Complex.html │ │ │ @@ -22749,25 +22749,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 806 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_compatible__Ideals.out │ │ │ -rw-r--r-- 0 root (0) root (0) 378 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_decompose__Fraction.out │ │ │ -rw-r--r-- 0 root (0) root (0) 606 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_descend__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 116 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_floor__Log.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1009 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1518 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 227 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Preimage.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1326 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1117 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Trace__On__Canonical__Module.out │ │ │ -rw-r--r-- 0 root (0) root (0) 488 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__Cohen__Macaulay.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1817 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1823 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Injective.out │ │ │ -rw-r--r-- 0 root (0) root (0) 637 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Pure.out │ │ │ -rw-r--r-- 0 root (0) root (0) 517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Rational.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1492 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_is__F__Regular.out │ │ │ -rw-r--r-- 0 root (0) root (0) 187 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_multiplicative__Order.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_parameter__Test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 216 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Element.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1513 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1514 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2756 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/example-output/_test__Module.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 590 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 40 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 4497 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Ascent__Count.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4556 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__C__M.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5301 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/___Assume__Domain.html │ │ │ @@ -22800,25 +22800,25 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 9322 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_compatible__Ideals.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7339 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_decompose__Fraction.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9505 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_descend__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5396 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_floor__Log.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10782 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12579 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Power.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5408 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Preimage.html │ │ │ --rw-r--r-- 0 root (0) root (0) 16914 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 16916 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8725 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Trace__On__Canonical__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8387 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15975 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15981 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10517 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Pure.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Rational.html │ │ │ -rw-r--r-- 0 root (0) root (0) 16563 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5995 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_multiplicative__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8061 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_parameter__Test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6473 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Element.html │ │ │ --rw-r--r-- 0 root (0) root (0) 15164 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 15165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html │ │ │ -rw-r--r-- 0 root (0) root (0) 17879 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Module.html │ │ │ -rw-r--r-- 0 root (0) root (0) 37686 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 34923 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12593 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TestIdeals/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Text/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Text/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 137618 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Text/dump/rawdocumentation.dump │ │ │ @@ -23020,30 +23020,30 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 27964 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 11775 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThinSincereQuivers/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 24955 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Lineage__Table.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1467 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1472 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Minimal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 7067 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/___Threaded__G__B.out │ │ │ --rw-r--r-- 0 root (0) root (0) 653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 615 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1034 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_minimize_lp__Lineage__Table_rp.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1155 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ --rw-r--r-- 0 root (0) root (0) 2087 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 920 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 2165 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb.out │ │ │ -rw-r--r-- 0 root (0) root (0) 328 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_tgb_lp..._cm__Verbose_eq_gt..._rp.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/ │ │ │ -rw-r--r-- 0 root (0) root (0) 605 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Certification │ │ │ -rw-r--r-- 0 root (0) root (0) 77 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/.Headline │ │ │ -rw-r--r-- 0 root (0) root (0) 8702 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Lineage__Table.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7200 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ --rw-r--r-- 0 root (0) root (0) 6899 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7205 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 6861 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7103 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_minimize_lp__Lineage__Table_rp.html │ │ │ --rw-r--r-- 0 root (0) root (0) 7297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ --rw-r--r-- 0 root (0) root (0) 12859 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 7062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 12937 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7621 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/_tgb_lp..._cm__Verbose_eq_gt..._rp.html │ │ │ -rw-r--r-- 0 root (0) root (0) 21706 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6872 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4771 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/ThreadedGB/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 67031 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump │ │ │ @@ -23382,15 +23382,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 7386 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/TriangularSets/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 166062 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 424 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/___Chirotope.out │ │ │ -rw-r--r-- 0 root (0) root 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./usr/share/doc/Macaulay2/Triangulations/example-output/_degree__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_delaunay__Subdivision.out │ │ │ -rw-r--r-- 0 root (0) root (0) 304 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_delaunay__Weights.out │ │ │ -rw-r--r-- 0 root (0) root (0) 383 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/example-output/_fine__Star__Triangulation.out │ │ │ @@ -23450,15 +23450,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_regular__Triangulation__Weights.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12280 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_secondary__Cone.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9250 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_some__Triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14609 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_triangulation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_vectors.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9663 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_volume__Vector.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9265 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/_wall__Circuits.html │ │ │ --rw-r--r-- 0 root (0) root (0) 39758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 39759 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/index.html │ │ │ -rw-r--r-- 0 root (0) root (0) 27693 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10498 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triangulations/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 49559 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 191 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/Triplets/example-output/___Betti1_lp__Triplet_rp.out │ │ │ @@ -24243,15 +24243,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 11283 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/master.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9346 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VectorGraphics/html/toc.html │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/ │ │ │ -rw-r--r-- 0 root (0) root (0) 226354 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/ │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Def__Param.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1215 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1213 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/___Smart__Lift.out │ │ │ -rw-r--r-- 0 root (0) root (0) 416 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Comparison__Theorem.out │ │ │ -rw-r--r-- 0 root (0) root (0) 406 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_check__Tangent__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 862 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_correct__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 433 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 774 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_cotangent__Cohomology2.out │ │ │ -rw-r--r-- 0 root (0) root (0) 562 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_ext__Matrix.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3316 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/example-output/_families.out │ │ │ @@ -24280,15 +24280,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5903 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7898 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Def__Param.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Degree__Bound.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5123 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Highest__Order.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9653 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Nested__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5248 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Polynomial__Check.html │ │ │ -rw-r--r-- 0 root (0) root (0) 6257 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Sanity__Check.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8261 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8259 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7148 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Verbose.html │ │ │ -rw-r--r-- 0 root (0) root (0) 4635 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/___Versal__Deformation__Results.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8028 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_check__Comparison__Theorem.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7831 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_check__Tangent__Space.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10134 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_correct__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 5625 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_correction__Matrix.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10071 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/VersalDeformations/html/_cotangent__Cohomology1.html │ │ │ @@ -24412,15 +24412,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 2302 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_flattening__Relations.out │ │ │ -rw-r--r-- 0 root (0) root (0) 4832 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Flat__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 110 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__From__Disk.out │ │ │ -rw-r--r-- 0 root (0) root (0) 990 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Range__Of__One__Parameter__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1616 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Smoothing__Family__With__Versal__Deformation.out │ │ │ -rw-r--r-- 0 root (0) root (0) 842 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_give1683__Format.out │ │ │ -rw-r--r-- 0 root (0) root (0) 612 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_hilbert__Burch__Matrices.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1293 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1294 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3279 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Unfolding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1617 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_prune__Family.out │ │ │ -rw-r--r-- 0 root (0) root (0) 15838 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_restricted__Unfolding.out │ │ │ -rw-r--r-- 0 root (0) root (0) 3363 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_satisfies__Degree__Condition1.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1050 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_smoothness__With__Reductions.out │ │ │ -rw-r--r-- 0 root (0) root (0) 968 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_to__Do__List.out │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/ │ │ │ @@ -24449,15 +24449,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 5708 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Range__Of__One__Parameter__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12022 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8651 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family__With__Versal__Deformation.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8793 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_give1683__Format.html │ │ │ -rw-r--r-- 0 root (0) root (0) 7419 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_hilbert__Burch__Matrices.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_improve__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 14619 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_is__Smoothing__Family.html │ │ │ --rw-r--r-- 0 root (0) root (0) 8149 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html │ │ │ +-rw-r--r-- 0 root (0) root (0) 8150 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12736 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Unfolding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 10223 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_prune__Family.html │ │ │ -rw-r--r-- 0 root (0) root (0) 22777 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_restricted__Unfolding.html │ │ │ -rw-r--r-- 0 root (0) root (0) 9758 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_satisfies__Degree__Condition1.html │ │ │ -rw-r--r-- 0 root (0) root (0) 8297 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_smoothness__With__Reductions.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12230 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_solving__Flattening__Relations.html │ │ │ -rw-r--r-- 0 root (0) root (0) 12188 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_test__Bound.html │ │ │ @@ -24480,15 +24480,15 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 620 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ceiling_lp__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 264 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clean__Support.out │ │ │ -rw-r--r-- 0 root (0) root (0) 470 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_clear__Cache.out │ │ │ -rw-r--r-- 0 root (0) root (0) 254 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Basic__List_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 335 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficient_lp__Ideal_cm__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 728 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_coefficients_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 2714 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1719 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1718 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_dualize.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1174 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_embed__As__Ideal.out │ │ │ -rw-r--r-- 0 root (0) root (0) 500 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_find__Element__Of__Degree.out │ │ │ -rw-r--r-- 0 root (0) root (0) 532 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_gbs.out │ │ │ -rw-r--r-- 0 root (0) root (0) 831 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Linear__Diophantine__Solution.out │ │ │ -rw-r--r-- 0 root (0) root (0) 501 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Count.out │ │ │ -rw-r--r-- 0 root (0) root (0) 357 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_get__Prime__Divisors.out │ │ │ -rw-r--r-- 0 root (0) root (0) 336 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ideal__Power.out │ │ │ @@ -24512,16 +24512,16 @@ │ │ │ -rw-r--r-- 0 root (0) root (0) 441 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Zero__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 667 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_map__To__Projective__Space.out │ │ │ -rw-r--r-- 0 root (0) root (0) 1342 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_non__Cartier__Locus.out │ │ │ -rw-r--r-- 0 root (0) root (0) 780 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_positive__Part.out │ │ │ -rw-r--r-- 0 root (0) root (0) 604 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_primes.out │ │ │ -rw-r--r-- 0 root (0) root (0) 765 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 846 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ramification__Divisor.out │ │ │ --rw-r--r-- 0 root (0) root (0) 4359 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ --rw-r--r-- 0 root (0) root (0) 1095 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 4355 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexify.out │ │ │ +-rw-r--r-- 0 root (0) root (0) 1097 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_reflexive__Power.out │ │ │ -rw-r--r-- 0 root (0) root (0) 249 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_ring_lp__Basic__Divisor_rp.out │ │ │ -rw-r--r-- 0 root (0) root (0) 375 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__Q__Weil__Divisor.out │ │ │ -rw-r--r-- 0 root (0) root (0) 458 2026-10-24 01:47:31.000000 ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_to__R__Weil__Divisor.out 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./usr/share/info/ThinSincereQuivers.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 8350 2026-10-24 01:47:31.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 8409 2026-10-24 01:47:31.000000 ./usr/share/info/ThreadedGB.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13976 2026-10-24 01:47:31.000000 ./usr/share/info/Topcom.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7724 2026-10-24 01:47:31.000000 ./usr/share/info/TorAlgebra.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 6967 2026-10-24 01:47:31.000000 ./usr/share/info/ToricHigherDirectImages.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 4013 2026-10-24 01:47:31.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 4015 2026-10-24 01:47:31.000000 ./usr/share/info/ToricInvariants.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9922 2026-10-24 01:47:31.000000 ./usr/share/info/ToricTopology.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 31419 2026-10-24 01:47:31.000000 ./usr/share/info/ToricVectorBundles.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 9577 2026-10-24 01:47:31.000000 ./usr/share/info/TriangularSets.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33244 2026-10-24 01:47:31.000000 ./usr/share/info/Triangulations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33246 2026-10-24 01:47:31.000000 ./usr/share/info/Triangulations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 7304 2026-10-24 01:47:31.000000 ./usr/share/info/Triplets.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 11010 2026-10-24 01:47:31.000000 ./usr/share/info/Tropical.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10699 2026-10-24 01:47:31.000000 ./usr/share/info/TropicalToric.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 5763 2026-10-24 01:47:31.000000 ./usr/share/info/Truncations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8693 2026-10-24 01:47:31.000000 ./usr/share/info/Units.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 4688 2026-10-24 01:47:31.000000 ./usr/share/info/VNumber.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8937 2026-10-24 01:47:31.000000 ./usr/share/info/Valuations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 44396 2026-10-24 01:47:31.000000 ./usr/share/info/Varieties.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19350 2026-10-24 01:47:31.000000 ./usr/share/info/VectorFields.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 51845 2026-10-24 01:47:31.000000 ./usr/share/info/VectorGraphics.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 41360 2026-10-24 01:47:31.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 41359 2026-10-24 01:47:31.000000 ./usr/share/info/VersalDeformations.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 13242 2026-10-24 01:47:31.000000 ./usr/share/info/VirtualResolutions.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10436 2026-10-24 01:47:31.000000 ./usr/share/info/Visualize.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33371 2026-10-24 01:47:31.000000 ./usr/share/info/WeierstrassSemigroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 38095 2026-10-24 01:47:31.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33368 2026-10-24 01:47:31.000000 ./usr/share/info/WeierstrassSemigroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 38073 2026-10-24 01:47:31.000000 ./usr/share/info/WeilDivisors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 10618 2026-10-24 01:47:31.000000 ./usr/share/info/WeylAlgebras.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 33219 2026-10-24 01:47:31.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ --rw-r--r-- 0 root (0) root (0) 14718 2026-10-24 01:47:31.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 33224 2026-10-24 01:47:31.000000 ./usr/share/info/WeylGroups.info.gz │ │ │ +-rw-r--r-- 0 root (0) root (0) 14716 2026-10-24 01:47:31.000000 ./usr/share/info/WhitneyStratifications.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 19544 2026-10-24 01:47:31.000000 ./usr/share/info/WittVectors.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 8865 2026-10-24 01:47:31.000000 ./usr/share/info/XML.info.gz │ │ │ -rw-r--r-- 0 root (0) root (0) 49515 2026-10-24 01:47:31.000000 ./usr/share/info/gfanInterface.info.gz │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/lintian/ │ │ │ drwxr-xr-x 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/lintian/overrides/ │ │ │ -rw-r--r-- 0 root (0) root (0) 10748 2026-10-23 21:15:50.000000 ./usr/share/lintian/overrides/macaulay2-common │ │ │ lrwxrwxrwx 0 root (0) root (0) 0 2026-10-24 01:47:31.000000 ./usr/share/Macaulay2/Style/katex/contrib/auto-render.min.js -> ../../../../javascript/katex/contrib/auto-render.js │ │ ├── ./usr/share/doc/Macaulay2/A1BrouwerDegrees/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ Z2V0VHJhY2U= │ │ │ #:len=2312 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIHRyYWNlIG92ZXIg │ │ │ JGskIGZvciBhbiBlbGVtZW50IGluIGEgZmluaXRlIGRpbWVuc2lvbmFsICRrJCAtYWxnZWJyYSIs │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aGFzTWluaW1hbE11bHQoUmluZyxJbmZpbml0ZU51bWJlcik= │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU1OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaGFzTWluaW1hbE11bHQsUmluZyxJbmZpbml0ZU51 │ │ ├── ./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) │ │ │ - -- 2.44098s elapsed │ │ │ + -- 1.33736s 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) │ │ │ - -- 2.39145s elapsed │ │ │ + -- 1.6045s elapsed │ │ │ │ │ │ 1 3 9 27 81 243 729 2187 │ │ │ o4 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 4 5 6 7 │ │ │ │ │ │ o4 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AInfinity/html/___Check.html │ │ │ @@ -95,28 +95,28 @@ │ │ │ 1 │ │ │ o2 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i3 : elapsedTime burkeResolution(M, 7, Check => false)
│ │ │ - -- 2.44098s elapsed
│ │ │ + -- 1.33736s 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)
│ │ │ - -- 2.39145s elapsed
│ │ │ + -- 1.6045s 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) │ │ │ │ - -- 2.44098s elapsed │ │ │ │ + -- 1.33736s 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) │ │ │ │ - -- 2.39145s elapsed │ │ │ │ + -- 1.6045s 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/AbstractSimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ ZmFjZXRzKEFic3RyYWN0U2ltcGxpY2lhbENvbXBsZXgp │ │ │ #:len=368 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmYWNldHMsQWJzdHJhY3RTaW1wbGljaWFsQ29tcGxl │ │ ├── ./usr/share/doc/Macaulay2/AbstractToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ QWJzdHJhY3RUb3JpY1ZhcmlldGllcw== │ │ │ #:len=379 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibGlua3MgYWJzdHJhY3Qgc2ltcGxpY2lh │ │ │ bCAobm9ybWFsKSB0b3JpYyB2YXJpZXRpZXMgdG8gU2NodWJlcnQyIiwgRGVzY3JpcHRpb24gPT4g │ │ ├── ./usr/share/doc/Macaulay2/AdjointIdeal/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ dHJhY2VNYXRyaXgoSWRlYWwsTWF0cml4KQ== │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE3MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VNYXRyaXgsSWRlYWwsTWF0cml4KSwidHJh │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ YWRqdW5jdGlvblByb2Nlc3MoSWRlYWwsWlop │ │ │ #:len=310 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjc4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGp1bmN0aW9uUHJvY2VzcyxJZGVhbCxaWiksImFk │ │ ├── ./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 │ │ │ - -- .0257838s elapsed │ │ │ + -- .0303774s elapsed │ │ │ │ │ │ 1 14 33 28 8 │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/example-output/_adjunction__Process.out │ │ │ @@ -87,30 +87,30 @@ │ │ │ o13 : BettiTally │ │ │ │ │ │ i14 : phi=map(P2,Pn,H); │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ i15 : elapsedTime betti(I'=trim ker phi) │ │ │ - -- .609185s elapsed │ │ │ + -- .532622s elapsed │ │ │ │ │ │ 0 1 │ │ │ o15 = total: 1 11 │ │ │ 0: 1 . │ │ │ 1: . 3 │ │ │ 2: . 8 │ │ │ │ │ │ o15 : BettiTally │ │ │ │ │ │ i16 : I'== I │ │ │ │ │ │ o16 = true │ │ │ │ │ │ i17 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 6.98144s elapsed │ │ │ + -- 5.7203s 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) │ │ │ - -- .0200526s elapsed │ │ │ + -- .0118019s 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) │ │ │ - -- .0565566s elapsed │ │ │ + -- .0737255s elapsed │ │ │ │ │ │ 0 1 │ │ │ o16 = total: 1 12 │ │ │ 0: 1 . │ │ │ 1: . 12 │ │ │ │ │ │ o16 : BettiTally │ │ │ │ │ │ i17 : I'== I │ │ │ │ │ │ o17 = true │ │ │ │ │ │ i18 : elapsedTime basePts=primaryDecomposition ideal H; │ │ │ - -- 1.96041s elapsed │ │ │ + -- 1.72537s elapsed │ │ │ │ │ │ i19 : tally apply(basePts,c->(dim c, degree c, betti c)) │ │ │ │ │ │ 0 1 │ │ │ o19 = Tally{(0, 34, total: 1 15) => 1} │ │ │ 0: 1 . │ │ │ 1: . . │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjoint__Matrix.html │ │ │ @@ -154,15 +154,15 @@ │ │ │ │ │ │ o9 = 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i10 : elapsedTime fI=res I
│ │ │ - -- .0257838s elapsed
│ │ │ + -- .0303774s elapsed
│ │ │  
│ │ │          1       14       33       28       8
│ │ │  o10 = Pn  <-- Pn   <-- Pn   <-- Pn   <-- Pn
│ │ │                                            
│ │ │        0       1        2        3        4
│ │ │  
│ │ │  o10 : Complex
│ │ │ ├── html2text {} │ │ │ │ @@ -54,15 +54,15 @@ │ │ │ │ 2: . 12 │ │ │ │ │ │ │ │ o8 : BettiTally │ │ │ │ i9 : c=codim I │ │ │ │ │ │ │ │ o9 = 4 │ │ │ │ i10 : elapsedTime fI=res I │ │ │ │ - -- .0257838s elapsed │ │ │ │ + -- .0303774s elapsed │ │ │ │ │ │ │ │ 1 14 33 28 8 │ │ │ │ o10 = Pn <-- Pn <-- Pn <-- Pn <-- Pn │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o10 : Complex │ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_adjunction__Process.html │ │ │ @@ -222,15 +222,15 @@ │ │ │ │ │ │ o14 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i15 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .609185s elapsed
│ │ │ + -- .532622s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o15 = total: 1 11
│ │ │            0: 1  .
│ │ │            1: .  3
│ │ │            2: .  8
│ │ │  
│ │ │ @@ -243,15 +243,15 @@
│ │ │  
│ │ │  o16 = true
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 6.98144s elapsed
│ │ │ + -- 5.7203s 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)
│ │ │ │ - -- .609185s elapsed
│ │ │ │ + -- .532622s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o15 = total: 1 11
│ │ │ │            0: 1  .
│ │ │ │            1: .  3
│ │ │ │            2: .  8
│ │ │ │  
│ │ │ │  o15 : BettiTally
│ │ │ │  i16 : I'== I
│ │ │ │  
│ │ │ │  o16 = true
│ │ │ │  i17 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 6.98144s elapsed
│ │ │ │ + -- 5.7203s elapsed
│ │ │ │  i18 : tally apply(basePts,c->(dim c, degree c, betti c))
│ │ │ │  
│ │ │ │                            0 1
│ │ │ │  o18 = Tally{(1, 1, total: 1 2) => 5}
│ │ │ │                         0: 1 2
│ │ │ │                            0 1
│ │ │ │              (1, 3, total: 1 3) => 8
│ │ ├── ./usr/share/doc/Macaulay2/AdjunctionForSurfaces/html/_parametrization.html
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o13 : BettiTally
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : elapsedTime sub(I,H)
│ │ │ - -- .0200526s elapsed
│ │ │ + -- .0118019s elapsed
│ │ │  
│ │ │  o14 = ideal (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
│ │ │  
│ │ │  o14 : Ideal of P2
│ │ │ │ │ │ │ │ │ │ │ │ @@ -215,15 +215,15 @@ │ │ │ │ │ │ o15 : RingMap P2 <-- Pn │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i16 : elapsedTime betti(I'=trim ker phi)
│ │ │ - -- .0565566s elapsed
│ │ │ + -- .0737255s elapsed
│ │ │  
│ │ │               0  1
│ │ │  o16 = total: 1 12
│ │ │            0: 1  .
│ │ │            1: . 12
│ │ │  
│ │ │  o16 : BettiTally
│ │ │ @@ -235,15 +235,15 @@ │ │ │ │ │ │ o17 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ - -- 1.96041s elapsed
│ │ │ + -- 1.72537s 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)
│ │ │ │ - -- .0200526s elapsed
│ │ │ │ + -- .0118019s 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)
│ │ │ │ - -- .0565566s elapsed
│ │ │ │ + -- .0737255s elapsed
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o16 = total: 1 12
│ │ │ │            0: 1  .
│ │ │ │            1: . 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ │ │  i17 : I'== I
│ │ │ │  
│ │ │ │  o17 = true
│ │ │ │  i18 : elapsedTime basePts=primaryDecomposition ideal H;
│ │ │ │ - -- 1.96041s elapsed
│ │ │ │ + -- 1.72537s 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/AlgebraicSplines/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c3RhbmxleVJlaXNuZXIoTGlzdCxMaXN0KQ==
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3RhbmxleVJlaXNuZXIsTGlzdCxMaXN0KSwic3Rh
│ │ ├── ./usr/share/doc/Macaulay2/AllMarkovBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  cmFuZG9tTWFya292KE1hdHJpeCk=
│ │ │  #:len=265
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTY5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21NYXJrb3YsTWF0cml4KSwicmFuZG9tTWFy
│ │ ├── ./usr/share/doc/Macaulay2/AnalyzeSheafOnP1/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  YW5hbHl6ZShNb2R1bGUp
│ │ │  #:len=251
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbmFseXplLE1vZHVsZSksImFuYWx5emUoTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/AssociativeAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2VxdWVuY2VUb1ZhcmlhYmxlU3ltYm9scw==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgsICJ1bmRvY3VtZW50ZWQiID0+IHRy
│ │ │  dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7InNlcXVlbmNl
│ │ ├── ./usr/share/doc/Macaulay2/BGG/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  dGF0ZVJlc29sdXRpb24=
│ │ │  #:len=2005
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluaXRlIHBpZWNlIG9mIHRoZSBUYXRl
│ │ │  IHJlc29sdXRpb24iLCAibGluZW51bSIgPT4gNzUzLCBJbnB1dHMgPT4ge1NQQU57VFR7Im0ifSwi
│ │ ├── ./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 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ + -- used 0.523032s (cpu); 0.379306s (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 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ + -- used 0.475544s (cpu); 0.39874s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o15 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BGG/html/_pure__Resolution.html
│ │ │ @@ -258,15 +258,15 @@
│ │ │                4      4
│ │ │  o13 : Matrix A  <-- A
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
i14 : time betti (F = pureResolution(M,{0,2,4}))
│ │ │ - -- used 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ + -- used 0.523032s (cpu); 0.379306s (thread); 0s (gc)
│ │ │  
│ │ │               0 1 2
│ │ │  o14 = total: 3 6 3
│ │ │            0: 3 . .
│ │ │            1: . 6 .
│ │ │            2: . . 3
│ │ │  
│ │ │ @@ -277,15 +277,15 @@
│ │ │          

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

│ │ │          

│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -192,27 +192,27 @@
│ │ │ │  o18 : ActionOnComplex
│ │ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │ │  
│ │ │ │  o19 = Complex with 6 actors
│ │ │ │  
│ │ │ │  o19 : ActionOnComplex
│ │ │ │  i20 : elapsedTime a1 = character A1
│ │ │ │ - -- 1.06712s elapsed
│ │ │ │ + -- .563934s elapsed
│ │ │ │  
│ │ │ │  o20 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │                   |                4    2         4    2
│ │ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │ │  
│ │ │ │  o20 : Character
│ │ │ │  i21 : elapsedTime a2 = character A2
│ │ │ │ - -- 36.3539s elapsed
│ │ │ │ + -- 21.3377s elapsed
│ │ │ │  
│ │ │ │  o21 = Character over kk
│ │ │ │  
│ │ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │ │                   |                4    2         4    2
│ │ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ │ @@ -319,15 +319,15 @@
│ │ │ │  i30 : M = Is2 / I2;
│ │ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │ │  
│ │ │ │  o31 = Module with 6 actors
│ │ │ │  
│ │ │ │  o31 : ActionOnGradedModule
│ │ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ │ - -- 16.4277s elapsed
│ │ │ │ + -- 9.6772s elapsed
│ │ │ │  
│ │ │ │  o32 = Character over kk
│ │ │ │  
│ │ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │ │  
│ │ │ │  o32 : Character
│ │ │ │  i33 : b/T
│ │ ├── ./usr/share/doc/Macaulay2/BinomialEdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=3
│ │ │  YmVp
│ │ │  #:len=336
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQmlub21pYWwgZWRnZSBpZGVhbCIsIERl
│ │ │  c2NyaXB0aW9uID0+ICgiYmVpIGlzIGEgc3lub255bSBmb3IgIixUT3tuZXcgRG9jdW1lbnRUYWcg
│ │ ├── ./usr/share/doc/Macaulay2/Binomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Ymlub21pYWxJc1ByaW1l
│ │ │  #:len=1312
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGVzdCBmb3IgcHJpbWVuZXNzIG9mIGEg
│ │ │  Ymlub21pYWwgaWRlYWwiLCAibGluZW51bSIgPT4gMTU4MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJ
│ │ ├── ./usr/share/doc/Macaulay2/BoijSoederberg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bWF0cml4KEJldHRpVGFsbHksWlop
│ │ │  #:len=291
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAyMCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF0cml4LEJldHRpVGFsbHksWlopLCJtYXRyaXgo
│ │ ├── ./usr/share/doc/Macaulay2/Book3264Examples/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  SW50ZXJzZWN0aW9uIFRoZW9yeSBTZWN0aW9uIDUuMg==
│ │ │  #:len=1578
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQmFzaWNzIG9mIHZlY3RvciBidW5kbGVz
│ │ │  IGFuZCBDaGVybiBjbGFzc2VzIiwgRGVzY3JpcHRpb24gPT4gKERJVntQQVJBe1RFWHsiSW4gU2No
│ │ ├── ./usr/share/doc/Macaulay2/BooleanGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Z2JCb29sZWFuKElkZWFsKQ==
│ │ │  #:len=1781
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZSBHcm9lYm5lciBCYXNpcyBm
│ │ │  b3IgSWRlYWxzIGluIEJvb2xlYW4gUG9seW5vbWlhbCBRdW90aWVudCBSaW5nIiwgImxpbmVudW0i
│ │ ├── ./usr/share/doc/Macaulay2/Brackets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  R0NBbGdlYnJh
│ │ │  #:len=1521
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAoRElWe1BBUkF7VEVYeyJBbiBvYmpl
│ │ │  Y3Qgb2YgY2xhc3MgR0NBbGdlYnJhIHJlcHJlc2VudHMgYSBHcmFzc21hbm4tQ2F5bGV5IGFsZ2Vi
│ │ ├── ./usr/share/doc/Macaulay2/Browse/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  QnJvd3Nl
│ │ │  #:len=397
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBtZXRob2QgZm9yIGJyb3dzaW5nIGFu
│ │ │  ZCBleGFtaW5pbmcgTWFjYXVsYXkyIGRhdGEgc3RydWN0dXJlcyIsIERlc2NyaXB0aW9uID0+ICgi
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNTeXp5Z3koTW9kdWxlLFpaKQ==
│ │ │  #:len=228
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc1N5enlneSxNb2R1bGUsWlopLCJpc1N5enlneShN
│ │ ├── ./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 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ + -- used 0.348453s (cpu); 0.220881s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of S
│ │ │  
│ │ │  i24 : betti res j
│ │ │  
│ │ │               0 1 2 3 4
│ │ │  o24 = total: 1 3 6 5 1
│ │ ├── ./usr/share/doc/Macaulay2/Bruns/html/_bruns.html
│ │ │ @@ -385,15 +385,15 @@
│ │ │  
│ │ │  o22 : BettiTally
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          

│ │ │                
i15 : time betti (F = pureResolution(11,4,{0,2,4}))
│ │ │ - -- used 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ + -- used 0.475544s (cpu); 0.39874s (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 0.450017s (cpu); 0.376863s (thread); 0s (gc)
│ │ │ │ + -- used 0.523032s (cpu); 0.379306s (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 0.468351s (cpu); 0.403732s (thread); 0s (gc)
│ │ │ │ + -- used 0.475544s (cpu); 0.39874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0 1 2
│ │ │ │  o15 = total: 3 6 3
│ │ │ │            0: 3 . .
│ │ │ │            1: . 6 .
│ │ │ │            2: . . 3
│ │ ├── ./usr/share/doc/Macaulay2/BIBasis/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  YmlCYXNpcyguLi4sdG9Hcm9lYm5lcj0+Li4uKQ==
│ │ │  #:len=240
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODMsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2JpQmFzaXMsdG9Hcm9lYm5lcl0sImJpQmFzaXMoLi4u
│ │ ├── ./usr/share/doc/Macaulay2/BeginningMacaulay2/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  QmVnaW5uaW5nTWFjYXVsYXky
│ │ │  #:len=29304
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTWF0aGVtYXRpY2lhbnMnIEludHJvZHVj
│ │ │  dGlvbiB0byAgTWFjYXVsYXkyIiwgRGVzY3JpcHRpb24gPT4gKERJVntQQVJBe1RFWHsiV2UgYXNz
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  QmVuY2htYXJr
│ │ │  #:len=347
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3RhbmRhcmQgTWFjYXVsYXkyIGJlbmNo
│ │ │  bWFya3MiLCBEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMgcGFja2FnZSBwcm92
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/example-output/_run__Benchmarks.out
│ │ │ @@ -1,10 +1,10 @@
│ │ │  -- -*- M2-comint -*- hash: 1330545576567
│ │ │  
│ │ │  i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Sat Oct 24 09:36:46 UTC 2026
│ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2100.028  
│ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds
│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .150504 seconds
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Benchmark/html/_run__Benchmarks.html
│ │ │ @@ -80,19 +80,19 @@
│ │ │          

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

│ │ │          

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

│ │ │                
i1 : runBenchmarks "res39"
│ │ │ --- beginning computation Sat Oct 24 03:20:06 UTC 2026
│ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250  
│ │ │ +-- beginning computation Sat Oct 24 09:36:46 UTC 2026
│ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2100.028  
│ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds

│ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .150504 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 Sat Oct 24 03:20:06 UTC 2026
│ │ │ │ --- Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1
│ │ │ │ -(2026-10-15) x86_64 GNU/Linux
│ │ │ │ --- AMD EPYC 7702P 64-Core Processor  AuthenticAMD  cpu MHz 1996.250
│ │ │ │ +-- beginning computation Sat Oct 24 09:36:46 UTC 2026
│ │ │ │ +-- Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │ │ +-- Intel Xeon Processor (Skylake, IBRS)  GenuineIntel  cpu MHz 2100.028
│ │ │ │  -- Macaulay2 1.26.06, compiled with gcc 16.2.0
│ │ │ │ --- res39: res of a generic 3 by 9 matrix over ZZ/101: .30315 seconds
│ │ │ │ +-- res39: res of a generic 3 by 9 matrix over ZZ/101: .150504 seconds
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_u_n_B_e_n_c_h_m_a_r_k_s is a _c_o_m_m_a_n_d.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Benchmark.m2:319:0.
│ │ ├── ./usr/share/doc/Macaulay2/BernsteinSato/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RGxvY2FsaXplQWxsKElkZWFsLFJpbmdFbGVtZW50KQ==
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/Bertini/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  U3RlcHNGb3JJbmNyZWFzZQ==
│ │ │  #:len=268
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzM3Mywgc3ltYm9sIERvY3VtZW50VGFn
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│ │ │ @@ -515,15 +515,15 @@
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│ │ │  MFwiIn0sIiwgIixTUEFOeyJPcHRpb24gdG8gY2hhbmdlIGRpcmVjdG9yeSBmb3IgZmlsZSBzdG9y
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│ │ │  YmVydGluaVVzZXJIb21vdG9weSIsImJlcnRpbmlVc2VySG9tb3RvcHkiLCJCZXJ0aW5pIn0sIEtl
│ │ │ @@ -1100,15 +1100,15 @@
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│ │ │  cHkoLi4uLFZlcmJvc2U9Pi4uLikiLCJCZXJ0aW5pIn0sIlZlcmJvc2UifSxUVHsiID0+ICJ9LFRU
│ │ │  eyIuLi4ifSwiLCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwiZmFsc2UifSwiLCAiLFNQQU57Ik9w
│ │ │  dGlvbiB0byBzaWxlbmNlIGFkZGl0aW9uYWwgb3V0cHV0In19fSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiYmVydGluaVBhcmFtZXRlckhvbW90b3B5IiwiYmVy
│ │ │  dGluaVBhcmFtZXRlckhvbW90b3B5IiwiQmVydGluaSJ9LCBLZXkgPT4gYmVydGluaVBhcmFtZXRl
│ │ │ @@ -2449,15 +2449,15 @@
│ │ │  YWw9Pi4uLikiLCJCZXJ0aW5pIn0sIlJhbmRvbVJlYWwifSxUVHsiID0+ICJ9LFRUeyIuLi4ifSwi
│ │ │  LCAiLFNQQU57ImRlZmF1bHQgdmFsdWUgIiwie30ifSwiLCAiLFNQQU57ImFuIG9wdGlvbiB3aGlj
│ │ │  aCBkZXNpZ25hdGVzIHN5bWJvbHMvc3RyaW5ncy92YXJpYWJsZXMgdGhhdCB3aWxsIGJlIHNldCB0
│ │ │  byBiZSBhIHJhbmRvbSByZWFsIG51bWJlciBvciByYW5kb20gY29tcGxleCBudW1iZXIifX0sU1BB
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│ │ │  dG9yeV0sImJlcnRpbmlaZXJvRGltU29sdmUoLi4uLFRvcERpcmVjdG9yeT0+Li4uKSIsIkJlcnRp
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│ │ │ -dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0yMzY2OS0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
│ │ │ +dWx0IHZhbHVlICIsIlwiL3RtcC9NMi0zMDEwOS0wLzBcIiJ9LCIsICIsU1BBTnsiT3B0aW9uIHRv
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│ │ │  bmlaZXJvRGltU29sdmUoLi4uLFVzZVJlZ2VuZXJhdGlvbj0+Li4uKSIsIkJlcnRpbmkifSwiVXNl
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│ │ │  bHVlICIsIi0xIn0sIiwgIixTUEFOe319LFNQQU57VE8ye25ldyBEb2N1bWVudFRhZyBmcm9tIHtb
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│ │ │  Ym9zZT0+Li4uKSIsIkJlcnRpbmkifSwiVmVyYm9zZSJ9LFRUeyIgPT4gIn0sVFR7Ii4uLiJ9LCIs
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Parameter__Homotopy.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │              
  • HomVariableGroup => ..., default value {}, an option to group variables and use multihomogeneous homotopies
    • 
│ │ │              
  • M2Precision (missing documentation)
│ │ │   => ..., default value 53, 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30109-0/0", 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/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30109-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S, a _l_i_s_t, a list whose entries are lists of solutions for each
│ │ │ │              target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__User__Homotopy.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │   => ..., default value 53, 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │              
  • RandomReal (missing documentation)
│ │ │   => ..., default value {}, 
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30109-0/0", 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/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30109-0/0", Option to
│ │ │ │              change directory for file storage.
│ │ │ │            o _V_e_r_b_o_s_e => ..., default value false, Option to silence additional
│ │ │ │              output
│ │ │ │      * Outputs:
│ │ │ │            o S0, a _l_i_s_t, a list of solutions to the target system
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This method calls Bertini to track a user-defined homotopy. The user needs to
│ │ ├── ./usr/share/doc/Macaulay2/Bertini/html/_bertini__Zero__Dim__Solve.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │   => ..., default value "main_data", 
│ │ │              
  • NameSolutionsFile (missing documentation)
│ │ │   => ..., default value "raw_solutions", 
    • 
│ │ │              
  • OutputStyle (missing documentation)
│ │ │   => ..., default value "OutPoints", 
    • 
│ │ │              
  • RandomComplex => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │              
  • RandomReal => ..., default value {}, an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
    • 
│ │ │ -            
  • TopDirectory => ..., default value "/tmp/M2-23669-0/0", Option to change directory for file storage.
    • 
│ │ │ +            
  • TopDirectory => ..., default value "/tmp/M2-30109-0/0", 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/M2-23669-0/0", Option to
│ │ │ │ +          o _T_o_p_D_i_r_e_c_t_o_r_y => ..., default value "/tmp/M2-30109-0/0", 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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  YWN0aW9uKE1vZHVsZSxMaXN0KQ==
│ │ │  #:len=290
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgzNywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYWN0aW9uLE1vZHVsZSxMaXN0KSwiYWN0aW9uKE1v
│ │ ├── ./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
│ │ │ - -- .449182s elapsed
│ │ │ + -- .243752s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp2.out
│ │ │ @@ -100,15 +100,15 @@
│ │ │  i6 : A=action(RI,S6)
│ │ │  
│ │ │  o6 = Complex with 11 actors
│ │ │  
│ │ │  o6 : ActionOnComplex
│ │ │  
│ │ │  i7 : elapsedTime c=character A
│ │ │ - -- 1.06265s elapsed
│ │ │ + -- .389533s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/example-output/___Betti__Characters_sp__Example_sp3.out
│ │ │ @@ -187,28 +187,28 @@
│ │ │  i19 : A2 = action(RI2,G,Sub=>false)
│ │ │  
│ │ │  o19 = Complex with 6 actors
│ │ │  
│ │ │  o19 : ActionOnComplex
│ │ │  
│ │ │  i20 : elapsedTime a1 = character A1
│ │ │ - -- 1.06712s elapsed
│ │ │ + -- .563934s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │        (2, {13})  |  1   1  1  1            1                  1
│ │ │  
│ │ │  o20 : Character
│ │ │  
│ │ │  i21 : elapsedTime a2 = character A2
│ │ │ - -- 36.3539s elapsed
│ │ │ + -- 21.3377s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -308,15 +308,15 @@
│ │ │  i31 : B = action(M,G,Sub=>false)
│ │ │  
│ │ │  o31 = Module with 6 actors
│ │ │  
│ │ │  o31 : ActionOnGradedModule
│ │ │  
│ │ │  i32 : elapsedTime b = character(B,21)
│ │ │ - -- 16.4277s elapsed
│ │ │ + -- 9.6772s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp1.html
│ │ │ @@ -167,15 +167,15 @@
│ │ │  
│ │ │  o8 : ActionOnComplex
    • 
│ │ │              
    
│ │ │                
    i9 : elapsedTime c = character A
│ │ │ - -- .449182s elapsed
│ │ │ + -- .243752s elapsed
│ │ │  
│ │ │  o9 = Character over QQ
│ │ │        
│ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ │ ├── html2text {}
│ │ │ │ @@ -91,15 +91,15 @@
│ │ │ │  o7 : List
│ │ │ │  i8 : A = action(RI,S7)
│ │ │ │  
│ │ │ │  o8 = Complex with 15 actors
│ │ │ │  
│ │ │ │  o8 : ActionOnComplex
│ │ │ │  i9 : elapsedTime c = character A
│ │ │ │ - -- .449182s elapsed
│ │ │ │ + -- .243752s elapsed
│ │ │ │  
│ │ │ │  o9 = Character over QQ
│ │ │ │  
│ │ │ │       (0, {0})  |  1   1   1   1   1  1   1   1   1   1   1   1   1   1   1
│ │ │ │       (1, {2})  |  0  -1   1  -1   0  0   0  -1   2   0   2   2   2   6  14
│ │ │ │       (2, {3})  |  0   1   0   0  -1  1  -1  -1  -1  -1  -1   1  -1   5  35
│ │ │ │       (3, {4})  |  0  -1   0   0   1  1   1  -1  -1   1  -1  -1  -1  -5  35
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp2.html
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o6 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime c=character A
│ │ │ - -- 1.06265s elapsed
│ │ │ + -- .389533s elapsed
│ │ │  
│ │ │  o7 = Character over QQ
│ │ │        
│ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ │ ├── html2text {}
│ │ │ │ @@ -113,15 +113,15 @@
│ │ │ │  o5 : List
│ │ │ │  i6 : A=action(RI,S6)
│ │ │ │  
│ │ │ │  o6 = Complex with 11 actors
│ │ │ │  
│ │ │ │  o6 : ActionOnComplex
│ │ │ │  i7 : elapsedTime c=character A
│ │ │ │ - -- 1.06265s elapsed
│ │ │ │ + -- .389533s elapsed
│ │ │ │  
│ │ │ │  o7 = Character over QQ
│ │ │ │  
│ │ │ │        (0, {0})  |   1  1   1   1   1   1  1   1   1   1    1
│ │ │ │        (1, {5})  |   0  1   0   2   0   1  3   0   2   4    6
│ │ │ │        (1, {7})  |   0  0   0   0   0   1  3   0   4  16   60
│ │ │ │        (1, {9})  |   0  0   0   0   2   2  2   0   4   8   20
│ │ ├── ./usr/share/doc/Macaulay2/BettiCharacters/html/___Betti__Characters_sp__Example_sp3.html
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o19 : ActionOnComplex
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime a1 = character A1
│ │ │ - -- 1.06712s elapsed
│ │ │ + -- .563934s elapsed
│ │ │  
│ │ │  o20 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │                   |                4    2         4    2
│ │ │         (1, {8})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │        (2, {11})  |  1   1  1  1            1                  1
│ │ │ @@ -331,15 +331,15 @@
│ │ │  
│ │ │  o20 : Character
    
│ │ │              
    
│ │ │                
    i21 : elapsedTime a2 = character A2
│ │ │ - -- 36.3539s elapsed
│ │ │ + -- 21.3377s elapsed
│ │ │  
│ │ │  o21 = Character over kk
│ │ │         
│ │ │         (0, {0})  |  1   1  1  1            1                  1
│ │ │        (1, {16})  |  6   2  0  0           -1                 -1
│ │ │                   |                4    2         4    2
│ │ │        (2, {19})  |  3  -1  0  1  a  + a  + a  - a  - a  - a - 1
│ │ │ @@ -483,15 +483,15 @@
│ │ │  
│ │ │  o31 : ActionOnGradedModule
    
│ │ │              
    
│ │ │                
    i32 : elapsedTime b = character(B,21)
│ │ │ - -- 16.4277s elapsed
│ │ │ + -- 9.6772s elapsed
│ │ │  
│ │ │  o32 = Character over kk
│ │ │         
│ │ │        (0, {21})  |  1  1  1  1  1  1
│ │ │  
│ │ │  o32 : Character
    
│ │ │              
    
│ │ │                
    i23 : time j=bruns F.dd_3;
│ │ │ - -- used 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ + -- used 0.348453s (cpu); 0.220881s (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 0.402268s (cpu); 0.246249s (thread); 0s (gc)
│ │ │ │ + -- used 0.348453s (cpu); 0.220881s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c3ViY29tcGxleChDZWxsQ29tcGxleCxMaXN0KQ==
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU2OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ViY29tcGxleCxDZWxsQ29tcGxleCxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  RUtSZXNvbHV0aW9uKE1vbm9taWFsSWRlYWwp
│ │ │  #:len=292
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoRUtSZXNvbHV0aW9uLE1vbm9taWFsSWRlYWwpLCJF
│ │ ├── ./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 0.296573s (cpu); 0.217064s (thread); 0s (gc)
│ │ │ + -- used 0.297594s (cpu); 0.222289s (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 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
│ │ │ + -- used 0.09219s (cpu); 0.0916783s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
│ │ │ + -- used 0.112858s (cpu); 0.116184s (thread); 0s (gc)
│ │ │  
│ │ │  i10 : betti source m
│ │ │  
│ │ │               0  1  2   3   4   5   6   7
│ │ │  o10 = total: 1 19 80 181 312 484 447 156
│ │ │            0: 1  3  3   1   .   .   .   .
│ │ │            1: .  .  1   3   3   .   .   .
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_minimize_lp__Chain__Complex_rp.html
│ │ │ @@ -186,15 +186,15 @@
│ │ │          
    
│ │ │            
    Now we minimize the result. The free summand we added to the end maps to zero, and thus is part of the minimization.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -306,15 +306,15 @@
│ │ │  
│ │ │  o21 : A
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -160,15 +160,15 @@
│ │ │ │  
│ │ │ │                2              2
│ │ │ │  o14 = ideal (x x  - x x x , x x )
│ │ │ │                0 3    1 2 4   2 5
│ │ │ │  
│ │ │ │  o14 : Ideal of R
│ │ │ │  i15 : time csmK=CSM(A,K)
│ │ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ │ + -- used 0.825791s (cpu); 0.316258s (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 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ │ + -- used 0.205415s (cpu); 0.0917261s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          2 2     2         2    2            2
│ │ │ │  o22 = 7h h  + 5h h  + 4h h  + h  + 3h h  + h
│ │ │ │          1 2     1 2     1 2    1     1 2    2
│ │ │ │  
│ │ │ │  o22 : A
│ │ │ │  In the case where the ambient space is a toric variety which is not a product
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Check__Smooth.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │  
│ │ │  o2 : NormalToricVariety
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i13 : time m = minimize (E[1]);
│ │ │ - -- used 0.296573s (cpu); 0.217064s (thread); 0s (gc)
    
│ │ │ + -- used 0.297594s (cpu); 0.222289s (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 0.296573s (cpu); 0.217064s (thread); 0s (gc)
│ │ │ │ + -- used 0.297594s (cpu); 0.222289s (thread); 0s (gc)
│ │ │ │  i14 : isQuasiIsomorphism m
│ │ │ │  
│ │ │ │  o14 = true
│ │ │ │  i15 : E[1] == source m
│ │ │ │  
│ │ │ │  o15 = true
│ │ │ │  i16 : E' = target m
│ │ ├── ./usr/share/doc/Macaulay2/ChainComplexExtras/html/_resolution__Of__Chain__Complex.html
│ │ │ @@ -134,21 +134,21 @@
│ │ │                  		
│ │ │                
    i7 : C = chainComplex for i from min C+1 to max C list map(mods_(i-1),mods_i,substitute(matrix C.dd_i,S));
    
│ │ │              
    
│ │ │                
    i8 : time m = resolutionOfChainComplex C;
│ │ │ - -- used 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
    
│ │ │ + -- used 0.09219s (cpu); 0.0916783s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i9 : time n = cartanEilenbergResolution C;
│ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
    
│ │ │ + -- used 0.112858s (cpu); 0.116184s (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 0.0995083s (cpu); 0.0987657s (thread); 0s (gc)
│ │ │ │ + -- used 0.09219s (cpu); 0.0916783s (thread); 0s (gc)
│ │ │ │  i9 : time n = cartanEilenbergResolution C;
│ │ │ │ - -- used 0.113261s (cpu); 0.1149s (thread); 0s (gc)
│ │ │ │ + -- used 0.112858s (cpu); 0.116184s (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/ChainComplexOperations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c3ltMihDb21wbGV4KQ==
│ │ │  #:len=259
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW0yLENvbXBsZXgpLCJzeW0yKENvbXBsZXgpIiwi
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  TXVsdGlQcm9qQ29vcmRSaW5n
│ │ │  #:len=2206
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQSBxdWljayB3YXkgdG8gYnVpbGQgdGhl
│ │ │  IGNvb3JkaW5hdGUgcmluZyBvZiBhIHByb2R1Y3Qgb2YgcHJvamVjdGl2ZSBzcGFjZXMiLCAibGlu
│ │ ├── ./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 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ + -- used 0.825791s (cpu); 0.316258s (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 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ + -- used 0.205415s (cpu); 0.0917261s (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 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ + -- used 0.248158s (cpu); 0.167897s (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 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ + -- used 0.356711s (cpu); 0.281838s (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 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ + -- used 0.57897s (cpu); 0.32109s (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 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ + -- used 2.01269s (cpu); 1.77491s (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 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ + -- used 0.260334s (cpu); 0.190679s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │  o10 : ------
│ │ │           4
│ │ │          h
│ │ │           1
│ │ │  
│ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ + -- used 0.0923611s (cpu); 0.0923676s (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 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ + -- used 0.0680213s (cpu); 0.0426126s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
│ │ │  
│ │ │  i5 : time Euler I
│ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ + -- used 0.251398s (cpu); 0.145082s (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 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ + -- used 0.100727s (cpu); 0.0743991s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
│ │ │  
│ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ + -- used 0.186888s (cpu); 0.0858216s (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 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ + -- used 0.0615714s (cpu); 0.0540657s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/example-output/___Inds__Of__Smooth.out
│ │ │ @@ -7,29 +7,29 @@
│ │ │  o1 : PolynomialRing
│ │ │  
│ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ + -- used 3.87257s (cpu); 1.09014s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │  o3 : ----------
│ │ │          3   3
│ │ │        (h , h )
│ │ │          1   2
│ │ │  
│ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ + -- used 3.97584s (cpu); 1.06847s (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 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ + -- used 0.799563s (cpu); 0.418812s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o3 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ + -- used 0.0566005s (cpu); 0.0336231s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │  o4 : ------
│ │ │          5
│ │ │         h
│ │ │          1
│ │ │  
│ │ │  i5 : time Chern I
│ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ + -- used 0.0546206s (cpu); 0.0323409s (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 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ + -- used 2.19398s (cpu); 0.93049s (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 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ + -- used 0.568612s (cpu); 0.211517s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o4 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___C__S__M.html
│ │ │ @@ -239,15 +239,15 @@
│ │ │  
│ │ │  o14 : Ideal of R
    
│ │ │              
    
│ │ │                
    i15 : time csmK=CSM(A,K)
│ │ │ - -- used 0.399939s (cpu); 0.310473s (thread); 0s (gc)
│ │ │ + -- used 0.825791s (cpu); 0.316258s (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 0.182686s (cpu); 0.107681s (thread); 0s (gc)
│ │ │ + -- used 0.205415s (cpu); 0.0917261s (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 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ + -- used 0.248158s (cpu); 0.167897s (thread); 0s (gc)
│ │ │  
│ │ │         7      6      5      4      3      2
│ │ │  o3 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │         7      7      7      7      7      7     7
│ │ │  
│ │ │                                                  ZZ[x ..x ]
│ │ │                                                      0   7
│ │ │ @@ -93,15 +93,15 @@
│ │ │       (x x x x x x x x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x , - x  + x )
│ │ │         0 1 2 3 4 5 6 7     0    1     0    2     0    3     0    4     0    5     0    6     0    7
    
│ │ │              
    
│ │ │                
    i4 : time CSM(U,CheckSmooth=>false)
│ │ │ - -- used 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ + -- used 0.356711s (cpu); 0.281838s (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 0.272841s (cpu); 0.176546s (thread); 0s (gc)
│ │ │ │ + -- used 0.248158s (cpu); 0.167897s (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 0.414662s (cpu); 0.322186s (thread); 0s (gc)
│ │ │ │ + -- used 0.356711s (cpu); 0.281838s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         7      6      5      4      3      2
│ │ │ │  o4 = 8x  + 28x  + 56x  + 70x  + 56x  + 28x  + 8x  + 1
│ │ │ │         7      7      7      7      7      7     7
│ │ │ │  
│ │ │ │                                                  ZZ[x ..x ]
│ │ │ │                                                      0   7
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Comp__Method.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time CSM(I,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ + -- used 0.57897s (cpu); 0.32109s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o5 = 6h  + 14h  + 14h  + 10h
│ │ │         1      1      1      1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -114,15 +114,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i6 : time CSM(I,CompMethod=>PnResidual)
│ │ │ - -- used 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ + -- used 2.01269s (cpu); 1.77491s (thread); 0s (gc)
│ │ │  
│ │ │         5      4      3      2
│ │ │  o6 = 6H  + 14H  + 14H  + 10H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o6 : -----
│ │ │          6
│ │ │ @@ -147,15 +147,15 @@
│ │ │  
│ │ │  o9 : Ideal of S
    
│ │ │              
    
│ │ │                
    i10 : time CSM(K,CompMethod=>ProjectiveDegree)
│ │ │ - -- used 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ + -- used 0.260334s (cpu); 0.190679s (thread); 0s (gc)
│ │ │  
│ │ │          3     2
│ │ │  o10 = 3h  + 5h
│ │ │          1     1
│ │ │  
│ │ │        ZZ[h ]
│ │ │            1
│ │ │ @@ -164,15 +164,15 @@
│ │ │          h
│ │ │           1
    
│ │ │              
    
│ │ │                
    i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ + -- used 0.0923611s (cpu); 0.0923676s (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 0.553623s (cpu); 0.383064s (thread); 0s (gc)
│ │ │ │ + -- used 0.57897s (cpu); 0.32109s (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 2.31313s (cpu); 1.85451s (thread); 0s (gc)
│ │ │ │ + -- used 2.01269s (cpu); 1.77491s (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 0.302112s (cpu); 0.205175s (thread); 0s (gc)
│ │ │ │ + -- used 0.260334s (cpu); 0.190679s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o10 = 3h  + 5h
│ │ │ │          1     1
│ │ │ │  
│ │ │ │        ZZ[h ]
│ │ │ │            1
│ │ │ │  o10 : ------
│ │ │ │           4
│ │ │ │          h
│ │ │ │           1
│ │ │ │  i11 : time CSM(K,CompMethod=>PnResidual)
│ │ │ │ - -- used 0.0879468s (cpu); 0.087955s (thread); 0s (gc)
│ │ │ │ + -- used 0.0923611s (cpu); 0.0923676s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          3     2
│ │ │ │  o11 = 3H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o11 : -----
│ │ │ │           4
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler.html
│ │ │ @@ -130,23 +130,23 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : time Euler(I,InputIsSmooth=>true)
│ │ │ - -- used 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ + -- used 0.0680213s (cpu); 0.0426126s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 4
    
│ │ │              
    
│ │ │                
    i5 : time Euler I
│ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ + -- used 0.251398s (cpu); 0.145082s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 4
    
│ │ │              
    
│ │ │                
    i6 : EulerIHash=Euler(I,Output=>HashForm);
    
│ │ │ @@ -194,23 +194,23 @@
│ │ │          
    
│ │ │            
    Note that the ideal J above is a complete intersection, thus we may change the method option which may speed computation in some cases. We may also note that the ideal generated by the first 2 generators of I defines a smooth scheme and input this information into the method. This may also improve computation speed.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Euler(J,Method=>DirectCompleteInt)
│ │ │ - -- used 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ + -- used 0.100727s (cpu); 0.0743991s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 2
    
│ │ │              
    
│ │ │                
    i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ + -- used 0.186888s (cpu); 0.0858216s (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 0.0376966s (cpu); 0.0369951s (thread); 0s (gc)
│ │ │ │ + -- used 0.0680213s (cpu); 0.0426126s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 4
│ │ │ │  i5 : time Euler I
│ │ │ │ - -- used 0.257354s (cpu); 0.164445s (thread); 0s (gc)
│ │ │ │ + -- used 0.251398s (cpu); 0.145082s (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 0.0792323s (cpu); 0.0754648s (thread); 0s (gc)
│ │ │ │ + -- used 0.100727s (cpu); 0.0743991s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 2
│ │ │ │  i11 : time Euler(J,Method=>DirectCompleteInt,IndsOfSmooth=>{0,1})
│ │ │ │ - -- used 0.195562s (cpu); 0.103705s (thread); 0s (gc)
│ │ │ │ + -- used 0.186888s (cpu); 0.0858216s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = 2
│ │ │ │  Now consider an example in \PP^2 \times \PP^2.
│ │ │ │  i12 : R=MultiProjCoordRing({2,2})
│ │ │ │  
│ │ │ │  o12 = R
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Euler__Affine.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │              
    
│ │ │                
    i4 : time EulerAffine I
│ │ │ - -- used 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ + -- used 0.0615714s (cpu); 0.0540657s (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 0.047108s (cpu); 0.0467349s (thread); 0s (gc)
│ │ │ │ + -- used 0.0615714s (cpu); 0.0540657s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 2
│ │ │ │  Observe that the algorithm is a probabilistic algorithm and may give a wrong
│ │ │ │  answer with a small but nonzero probability. Read more under _p_r_o_b_a_b_i_l_i_s_t_i_c
│ │ │ │  _a_l_g_o_r_i_t_h_m.
│ │ │ │  ********** WWaayyss ttoo uussee EEuulleerrAAffffiinnee:: **********
│ │ │ │      * EulerAffine(Ideal)
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Inds__Of__Smooth.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ + -- used 3.87257s (cpu); 1.09014s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ @@ -92,15 +92,15 @@
│ │ │        (h , h )
│ │ │          1   2
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ + -- used 3.97584s (cpu); 1.06847s (thread); 0s (gc)
│ │ │  
│ │ │         2 2     2         2
│ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │         1 2     1 2     1 2
│ │ │  
│ │ │       ZZ[h ..h ]
│ │ │           1   2
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,28 +16,28 @@
│ │ │ │  o1 = R
│ │ │ │  
│ │ │ │  o1 : PolynomialRing
│ │ │ │  i2 : I=ideal(R_0*R_1*R_3-R_0^2*R_3,random({0,1},R),random({1,2},R));
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : time CSM(I,Method=>DirectCompletInt)
│ │ │ │ - -- used 1.72196s (cpu); 1.16792s (thread); 0s (gc)
│ │ │ │ + -- used 3.87257s (cpu); 1.09014s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o3 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ │ │  o3 : ----------
│ │ │ │          3   3
│ │ │ │        (h , h )
│ │ │ │          1   2
│ │ │ │  i4 : time CSM(I,Method=>DirectCompletInt,IndsOfSmooth=>{1,2})
│ │ │ │ - -- used 1.40622s (cpu); 1.05667s (thread); 0s (gc)
│ │ │ │ + -- used 3.97584s (cpu); 1.06847s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         2 2     2         2
│ │ │ │  o4 = 2h h  + 2h h  + 5h h
│ │ │ │         1 2     1 2     1 2
│ │ │ │  
│ │ │ │       ZZ[h ..h ]
│ │ │ │           1   2
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Input__Is__Smooth.html
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ + -- used 0.799563s (cpu); 0.418812s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o3 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -88,15 +88,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ + -- used 0.0566005s (cpu); 0.0336231s (thread); 0s (gc)
│ │ │  
│ │ │         3
│ │ │  o4 = 4h
│ │ │         1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -110,15 +110,15 @@
│ │ │          
    
│ │ │            
    Note that one could, equivalently, use the command Chern instead in this case.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -438,26 +438,26 @@
│ │ │  
│ │ │  o26 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -182,15 +182,15 @@
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        0, 0, 0, -1, -1}}
│ │ │ │  
│ │ │ │  o19 : List
│ │ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ │ - -- 3.09141s elapsed
│ │ │ │ + -- 2.763s 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)
│ │ │ │ - -- .3809s elapsed
│ │ │ │ + -- .475756s 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))
│ │ │ │ - -- 11.5175s elapsed
│ │ │ │ + -- 7.99828s 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)
│ │ │ │ - -- .397928s elapsed
│ │ │ │ + -- .434887s 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))
│ │ │ │ - -- .553978s elapsed
│ │ │ │ - -- .554009s elapsed
│ │ │ │ + -- .374195s elapsed
│ │ │ │ + -- .374217s 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/CoincidentRootLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  Y29tcGxleHJhbmsoUmluZ0VsZW1lbnQp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjb21wbGV4cmFuayxSaW5nRWxlbWVudCksImNvbXBs
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  RWlzZW5idWRTaGFtYXNoVG90YWwoTW9kdWxlKQ==
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTE2OSwgc3ltYm9sIERvY3VtZW50VGFn
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│ │ ├── ./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 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ + -- used 6.58715s (cpu); 5.06523s (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 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ + -- used 0.185324s (cpu); 0.11575s (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 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ + -- used 0.990714s (cpu); 0.775146s (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 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ + -- used 1.03035s (cpu); 0.834039s (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 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ + -- used 0.642372s (cpu); 0.396618s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ + -- used 0.15689s (cpu); 0.110412s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ + -- used 3.023e-06s (cpu); 2.62e-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 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ + -- used 0.924677s (cpu); 0.628534s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ + -- used 0.548809s (cpu); 0.381602s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ + -- used 0.119118s (cpu); 0.108486s (thread); 0s (gc)
│ │ │  4
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/___Eisenbud__Shamash.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o6 = 10
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : time Chern I
│ │ │ - -- used 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ + -- used 0.0546206s (cpu); 0.0323409s (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 0.618754s (cpu); 0.466376s (thread); 0s (gc)
│ │ │ │ + -- used 0.799563s (cpu); 0.418812s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o3 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ │ │  o3 : ------
│ │ │ │          5
│ │ │ │         h
│ │ │ │          1
│ │ │ │  i4 : time CSM(I,InputIsSmooth=>true)
│ │ │ │ - -- used 0.033987s (cpu); 0.0334213s (thread); 0s (gc)
│ │ │ │ + -- used 0.0566005s (cpu); 0.0336231s (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 0.0322268s (cpu); 0.0317709s (thread); 0s (gc)
│ │ │ │ + -- used 0.0546206s (cpu); 0.0323409s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3
│ │ │ │  o5 = 4h
│ │ │ │         1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/CharacteristicClasses/html/___Method.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
    
│ │ │              
    
│ │ │                
    i3 : time CSM I
│ │ │ - -- used 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ + -- used 2.19398s (cpu); 0.93049s (thread); 0s (gc)
│ │ │  
│ │ │          5      4     3
│ │ │  o3 = 12h  + 10h  + 6h
│ │ │          1      1     1
│ │ │  
│ │ │       ZZ[h ]
│ │ │           1
│ │ │ @@ -92,15 +92,15 @@
│ │ │         h
│ │ │          1
    
│ │ │              
    
│ │ │                
    i4 : time CSM(I,Method=>DirectCompleteInt)
│ │ │ - -- used 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ + -- used 0.568612s (cpu); 0.211517s (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 1.19911s (cpu); 0.877209s (thread); 0s (gc)
│ │ │ │ + -- used 2.19398s (cpu); 0.93049s (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 0.310562s (cpu); 0.233341s (thread); 0s (gc)
│ │ │ │ + -- used 0.568612s (cpu); 0.211517s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          5      4     3
│ │ │ │  o4 = 12h  + 10h  + 6h
│ │ │ │          1      1     1
│ │ │ │  
│ │ │ │       ZZ[h ]
│ │ │ │           1
│ │ ├── ./usr/share/doc/Macaulay2/Chordal/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
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│ │ │  #:file=rawdocumentation-dcba-8.db
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│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  UmluZ01hcCBDaG9yZGFsTmV0
│ │ │  #:len=1424
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│ │ ├── ./usr/share/doc/Macaulay2/Classic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
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│ │ ├── ./usr/share/doc/Macaulay2/CodingTheory/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  VmFuaXNoaW5nSWRlYWw=
│ │ │  #:len=1237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmFuaXNoaW5nIGlkZWFsIG9mIGFuIGV2
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│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  Y29ob21DYWxnKE5vcm1hbFRvcmljVmFyaWV0eSk=
│ │ │  #:len=2221
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibG9jYWxseSBzdGFzaGVkIGNvaG9tb2xv
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│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/example-output/___Cohom__Calg.out
│ │ │ @@ -184,15 +184,15 @@
│ │ │        {0, -1, 0, 0, 0, -1}, {0, 0, -1, 0, 0, -1}, {0, 0, 0, -1, 0, -1}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, -1, -1}}
│ │ │  
│ │ │  o19 : List
│ │ │  
│ │ │  i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.09141s elapsed
│ │ │ + -- 2.763s 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)
│ │ │ - -- .3809s elapsed
│ │ │ + -- .475756s 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))
│ │ │ - -- 11.5175s elapsed
│ │ │ + -- 7.99828s 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)
│ │ │ - -- .397928s elapsed
│ │ │ + -- .434887s 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))
│ │ │ - -- .553978s elapsed
│ │ │ - -- .554009s elapsed
│ │ │ + -- .374195s elapsed
│ │ │ + -- .374217s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
│ │ │  
│ │ │  i29 : assert(cohomvec1 == cohomvec2)
│ │ ├── ./usr/share/doc/Macaulay2/CohomCalg/html/index.html
│ │ │ @@ -314,15 +314,15 @@
│ │ │  
│ │ │  o19 : List
    
│ │ │              
    
│ │ │                
    i20 : elapsedTime hvecs = cohomCalg(X, D2)
│ │ │ - -- 3.09141s elapsed
│ │ │ + -- 2.763s elapsed
│ │ │  
│ │ │  o20 = {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0},
│ │ │        -----------------------------------------------------------------------
│ │ │        {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 0, 0, 0}, {0,
│ │ │        -----------------------------------------------------------------------
│ │ │        0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}, {0, 0,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -404,25 +404,25 @@
│ │ │  
│ │ │  o22 : List
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 + X_8)
│ │ │ - -- .3809s elapsed
│ │ │ + -- .475756s 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))
│ │ │ - -- 11.5175s elapsed
│ │ │ + -- 7.99828s elapsed
│ │ │  
│ │ │  o24 = {1, 0, 0, 0, 0}
│ │ │  
│ │ │  o24 : List
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime cohomvec1 = cohomCalg(X_3 + X_7 - X_8)
│ │ │ - -- .397928s elapsed
│ │ │ + -- .434887s 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))
│ │ │ - -- .553978s elapsed
│ │ │ - -- .554009s elapsed
│ │ │ + -- .374195s elapsed
│ │ │ + -- .374217s elapsed
│ │ │  
│ │ │  o28 = {0, 0, 0, 0, 0}
│ │ │  
│ │ │  o28 : List
    
│ │ │              
    
│ │ │                
    i7 : time G = EisenbudShamash(ff,F,len)
│ │ │ - -- used 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ + -- used 6.58715s (cpu); 5.06523s (thread); 0s (gc)
│ │ │  
│ │ │       /    S   \1     /    S   \5     /    S   \12     /    S   \20     /    S   \28     /    S   \36     /    S   \44     /    S   \52     /    S   \60     /    S   \68     /    S   \76
│ │ │  o7 = |--------|  <-- |--------|  <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|   <-- |--------|
│ │ │       |  2   3 |      |  2   3 |      |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |       |  2   3 |
│ │ │       |(x , x )|      |(x , x )|      |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|       |(x , x )|
│ │ │       \  0   1 /      \  0   1 /      \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /       \  0   1 /
│ │ │                                                                                                                                                                                
│ │ │ @@ -300,28 +300,28 @@
│ │ │  
│ │ │  o19 : QuotientRing
    
│ │ │              
    
│ │ │                
    i20 : FF = time Shamash(R1,F,4)
│ │ │ - -- used 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ + -- used 0.185324s (cpu); 0.11575s (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 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ + -- used 0.990714s (cpu); 0.775146s (thread); 0s (gc)
│ │ │  
│ │ │        / R\1     / R\6     / R\18     / R\38     / R\66
│ │ │  o21 = |--|  <-- |--|  <-- |--|   <-- |--|   <-- |--|
│ │ │        | 3|      | 3|      | 3|       | 3|       | 3|
│ │ │        \c /      \c /      \c /       \c /       \c /
│ │ │                                                   
│ │ │        0         1         2          3          4
│ │ │ @@ -333,15 +333,15 @@
│ │ │          
    
│ │ │            
    The function also deals correctly with complexes F where min F is not 0:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i22 : GG = time EisenbudShamash(R1,F[2],4)
│ │ │ - -- used 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ + -- used 1.03035s (cpu); 0.834039s (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 6.68392s (cpu); 5.01853s (thread); 0s (gc)
│ │ │ │ + -- used 6.58715s (cpu); 5.06523s (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 0.232858s (cpu); 0.156146s (thread); 0s (gc)
│ │ │ │ + -- used 0.185324s (cpu); 0.11575s (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 0.980622s (cpu); 0.754248s (thread); 0s (gc)
│ │ │ │ + -- used 0.990714s (cpu); 0.775146s (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 0.994666s (cpu); 0.734783s (thread); 0s (gc)
│ │ │ │ + -- used 1.03035s (cpu); 0.834039s (thread); 0s (gc)
│ │ │ │  
│ │ │ │          1       6       18       38       66
│ │ │ │  o22 = R1  <-- R1  <-- R1   <-- R1   <-- R1
│ │ │ │  
│ │ │ │        -2      -1      0        1        2
│ │ │ │  
│ │ │ │  o22 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_sum__Two__Monomials.html
│ │ │ @@ -84,23 +84,23 @@
│ │ │  
│ │ │  o1 = 0
│ │ │              
    
│ │ │                
    i2 : sumTwoMonomials(2,3)
│ │ │ - -- used 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ + -- used 0.642372s (cpu); 0.396618s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │  
│ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ + -- used 0.15689s (cpu); 0.110412s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │  
│ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ + -- used 3.023e-06s (cpu); 2.62e-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 0.547511s (cpu); 0.41022s (thread); 0s (gc)
│ │ │ │ + -- used 0.642372s (cpu); 0.396618s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 3}
│ │ │ │  
│ │ │ │ - -- used 0.106844s (cpu); 0.10641s (thread); 0s (gc)
│ │ │ │ + -- used 0.15689s (cpu); 0.110412s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 1}
│ │ │ │  
│ │ │ │ - -- used 4.058e-06s (cpu); 3.486e-06s (thread); 0s (gc)
│ │ │ │ + -- used 3.023e-06s (cpu); 2.62e-06s (thread); 0s (gc)
│ │ │ │  4
│ │ │ │  Tally{}
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_w_o_M_o_n_o_m_i_a_l_s -- tally the sequences of BRanks for certain examples
│ │ │ │  ********** WWaayyss ttoo uussee ssuummTTwwooMMoonnoommiiaallss:: **********
│ │ │ │      * sumTwoMonomials(ZZ,ZZ)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/CompleteIntersectionResolutions/html/_two__Monomials.html
│ │ │ @@ -88,25 +88,25 @@
│ │ │  
│ │ │  o1 = 0
    
│ │ │              
    
│ │ │                
    i2 : twoMonomials(2,3)
│ │ │ - -- used 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ + -- used 0.924677s (cpu); 0.628534s (thread); 0s (gc)
│ │ │  2
│ │ │  Tally{{{1, 1}} => 2        }
│ │ │        {{2, 2}, {1, 2}} => 4
│ │ │  
│ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ + -- used 0.548809s (cpu); 0.381602s (thread); 0s (gc)
│ │ │  3
│ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │        {{3, 3}, {2, 3}} => 1
│ │ │  
│ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ + -- used 0.119118s (cpu); 0.108486s (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 1.04089s (cpu); 0.839487s (thread); 0s (gc)
│ │ │ │ + -- used 0.924677s (cpu); 0.628534s (thread); 0s (gc)
│ │ │ │  2
│ │ │ │  Tally{{{1, 1}} => 2        }
│ │ │ │        {{2, 2}, {1, 2}} => 4
│ │ │ │  
│ │ │ │ - -- used 0.552679s (cpu); 0.395293s (thread); 0s (gc)
│ │ │ │ + -- used 0.548809s (cpu); 0.381602s (thread); 0s (gc)
│ │ │ │  3
│ │ │ │  Tally{{{2, 2}, {1, 2}} => 2}
│ │ │ │        {{3, 3}, {2, 3}} => 1
│ │ │ │  
│ │ │ │ - -- used 0.10407s (cpu); 0.103794s (thread); 0s (gc)
│ │ │ │ + -- used 0.119118s (cpu); 0.108486s (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/Complexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  cHJ1bmVDb21wbGV4KC4uLixQcnVuaW5nTWFwPT4uLi4p
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQ4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcnVuZUNvbXBsZXgsUHJ1bmluZ01hcF0sInBydW5l
│ │ ├── ./usr/share/doc/Macaulay2/ConformalBlocks/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsRGl2aXNvck0wbmJhcg==
│ │ │  #:len=1172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY2xhc3Mgb2YgdGhl
│ │ │  IGNhbm9uaWNhbCBkaXZpc29yIG9uIHRoZSBtb2R1bGkgc3BhY2Ugb2Ygc3RhYmxlIG4tcG9pbnRl
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=39
│ │ │  aXNFcHNpbG9uRmFjdG9yaXplZChNYXRyaXgsUmluZ0VsZW1lbnQp
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0Vwc2lsb25GYWN0b3JpemVkLE1hdHJpeCxSaW5n
│ │ ├── ./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 = pfaffianSystem I;
│ │ │ - -- 4.39641s elapsed
│ │ │ + -- 2.51969s elapsed
│ │ │  
│ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.46882s elapsed
│ │ │ + -- 3.44069s 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);
│ │ │ - -- .606974s elapsed
│ │ │ + -- .445284s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
│ │ │  
│ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.30033s elapsed
│ │ │ + -- .97325s elapsed
│ │ │  
│ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .857943s elapsed
│ │ │ + -- .712307s 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);
│ │ │ - -- .393587s elapsed
│ │ │ + -- .292111s elapsed
│ │ │  
│ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- 1.06124s elapsed
│ │ │ + -- .604195s 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 = pfaffianSystem I;
│ │ │ - -- .32932s elapsed
│ │ │ + -- .1947s elapsed
│ │ │  
│ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .168768s elapsed
│ │ │ + -- .150569s elapsed
│ │ │  
│ │ │  i9 : netList A
│ │ │  
│ │ │       +--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o9 = || 0                                                       1                                               0                                                                     0                                                                      ||
│ │ │       || 0                                                       -1/x                                            1/x                                                                   y/x                                                                    ||
│ │ │       || -1/2xy                                                  -1/y                                            (-x-3y+1)/2xy                                                         (-x-y+1)/2x                                                            ||
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Cosmological_spcorrelator_spfor_spthe_sp2-site_spchain.html
│ │ │ @@ -123,21 +123,21 @@
│ │ │          
    
│ │ │            
    Then, we compute the system in connection form and verify that it meets the integrability conditions.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -82,15 +82,15 @@
│ │ │ │                2                 2
│ │ │ │       - a*c + e         - b*c + f
│ │ │ │       ----------*v, x + ----------*v)
│ │ │ │        d*e - a*f         d*e - a*f
│ │ │ │  
│ │ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i6 : time phi^** q
│ │ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ │ + -- used 0.324175s (cpu); 0.184591s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                  e        d        c        b        a
│ │ │ │  o6 = ideal (u - -*v, t - -*v, z - -*v, y - -*v, x - -*v)
│ │ │ │                  f        f        f        f        f
│ │ │ │  
│ │ │ │  o6 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
│ │ │ │  i7 : oo == p
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Segre__Class.html
│ │ │ @@ -139,59 +139,59 @@
│ │ │                x x  - 2x x x x  + x x  - 2x x x x  - 2x x x x  + 4x x x x  + x x  + 4x x x x  - 2x x x x  - 2x x x x  - 2x x x x  + x x
│ │ │                 3 4     2 3 4 5    2 5     1 3 4 6     1 2 5 6     0 3 5 6    1 6     1 2 4 7     0 3 4 7     0 2 5 7     0 1 6 7    0 7
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
    
│ │ │                
    i9 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- 4.39641s elapsed
    
│ │ │ + -- 2.51969s elapsed
│ │ │              
    
│ │ │                
    i10 : elapsedTime assert isIntegrable A
│ │ │ - -- 5.46882s elapsed
    
│ │ │ + -- 3.44069s elapsed
│ │ │              
    
│ │ │                
    i11 : netList A
│ │ │  
│ │ │        +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -172,30 +172,30 @@
│ │ │              
    		
│ │ │                
    i13 : B = {1_D,dx,dy,dx*dy};
    
│ │ │              
    
│ │ │                
    i14 : elapsedTime g = gaugeMatrix(I, B);
│ │ │ - -- .606974s elapsed
│ │ │ + -- .445284s elapsed
│ │ │  
│ │ │                4      4
│ │ │  o14 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ - -- 1.30033s elapsed
    
│ │ │ + -- .97325s elapsed
│ │ │              
    
│ │ │                
    i16 : elapsedTime assert isIntegrable A1
│ │ │ - -- .857943s elapsed
    
│ │ │ + -- .712307s elapsed
│ │ │              
    
│ │ │                
    i17 : netList A1
│ │ │  
│ │ │        +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │ @@ -232,21 +232,21 @@
│ │ │                4      4
│ │ │  o18 : Matrix F  <-- F
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime A2 = gaugeTransform(changeEps, A1);
│ │ │ - -- .393587s elapsed
    
│ │ │ + -- .292111s elapsed
│ │ │              
    
│ │ │                
    i20 : elapsedTime assert isIntegrable A2
│ │ │ - -- 1.06124s elapsed
    
│ │ │ + -- .604195s 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 = pfaffianSystem I;
│ │ │ │ - -- 4.39641s elapsed
│ │ │ │ + -- 2.51969s elapsed
│ │ │ │  i10 : elapsedTime assert isIntegrable A
│ │ │ │ - -- 5.46882s elapsed
│ │ │ │ + -- 3.44069s 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);
│ │ │ │ - -- .606974s elapsed
│ │ │ │ + -- .445284s elapsed
│ │ │ │  
│ │ │ │                4      4
│ │ │ │  o14 : Matrix F  <-- F
│ │ │ │  i15 : elapsedTime A1 = gaugeTransform(g, A);
│ │ │ │ - -- 1.30033s elapsed
│ │ │ │ + -- .97325s elapsed
│ │ │ │  i16 : elapsedTime assert isIntegrable A1
│ │ │ │ - -- .857943s elapsed
│ │ │ │ + -- .712307s 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);
│ │ │ │ - -- .393587s elapsed
│ │ │ │ + -- .292111s elapsed
│ │ │ │  i20 : elapsedTime assert isIntegrable A2
│ │ │ │ - -- 1.06124s elapsed
│ │ │ │ + -- .604195s elapsed
│ │ │ │  i21 : netList A2
│ │ │ │  
│ │ │ │        +------------------------------------------------------------------------
│ │ │ │  -------------------+
│ │ │ │  o21 = || ϵ/(x+z) 2zϵ/(x2-z2) 0       0                      |
│ │ │ │  |
│ │ │ │        || 0       ϵ/(x-z)     0       ϵ/(x+y)                |
│ │ ├── ./usr/share/doc/Macaulay2/ConnectionMatrices/html/___Massless_spone-loop_sptriangle_sp__Feynman_spdiagram.html
│ │ │ @@ -105,21 +105,21 @@
│ │ │          
    
│ │ │            
    Finally, we can compute the connection matrices.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : elapsedTime A = pfaffianSystem I;
│ │ │ - -- .32932s elapsed
    
│ │ │ + -- .1947s elapsed
│ │ │              
    
│ │ │                
    i8 : elapsedTime assert isIntegrable A
│ │ │ - -- .168768s elapsed
    
│ │ │ + -- .150569s 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 = pfaffianSystem I;
│ │ │ │ - -- .32932s elapsed
│ │ │ │ + -- .1947s elapsed
│ │ │ │  i8 : elapsedTime assert isIntegrable A
│ │ │ │ - -- .168768s elapsed
│ │ │ │ + -- .150569s elapsed
│ │ │ │  i9 : netList A
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -----------------+
│ │ │ │  o9 = || 0                                                       1
│ │ ├── ./usr/share/doc/Macaulay2/ConvexInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  bUNvbnZleEh1bGxGYWNlcyguLi4sdG9GaWxlPT4uLi4p
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzU3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttQ29udmV4SHVsbEZhY2VzLHRvRmlsZV0sIm1Db252
│ │ ├── ./usr/share/doc/Macaulay2/ConwayPolynomials/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  Y29ud2F5UG9seW5vbWlhbA==
│ │ │  #:len=1659
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvdmlkZSBhIENvbndheSBwb2x5bm9t
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│ │ ├── ./usr/share/doc/Macaulay2/CorrespondenceScrolls/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=52
│ │ │  cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ==
│ │ │  #:len=367
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│ │ ├── ./usr/share/doc/Macaulay2/CotangentSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  TGFiZWxz
│ │ │  #:len=183
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│ │ ├── ./usr/share/doc/Macaulay2/CpMackeyFunctors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  bWFrZUJ1cm5zaWRlTWFja2V5RnVuY3RvcihaWik=
│ │ │  #:len=345
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/Cremona/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY3JpYmUoUmF0aW9uYWxNYXAp
│ │ │  #:len=1096
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGVzY3JpYmUgYSByYXRpb25hbCBtYXAi
│ │ │  LCAibGluZW51bSIgPT4gOTgzLCBJbnB1dHMgPT4ge1NQQU57VFR7InBoaSJ9LCIsICIsU1BBTnsi
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/example-output/___Chern__Schwartz__Mac__Pherson.out
│ │ │ @@ -13,27 +13,27 @@
│ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │  
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
│ │ │  
│ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ + -- used 1.09149s (cpu); 0.850049s (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 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ + -- used 1.33551s (cpu); 0.841573s (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 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ + -- used 0.119657s (cpu); 0.119659s (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 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ + -- used 0.0708737s (cpu); 0.0192794s (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 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ + -- used 0.00459699s (cpu); 0.00459555s (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 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ + -- used 0.0466587s (cpu); 0.0466628s (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 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ + -- used 0.0298369s (cpu); 0.0298355s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projectiveDegrees phi
│ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ + -- used 0.464463s (cpu); 0.3923s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ + -- used 0.0590164s (cpu); 0.0590235s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ + -- used 0.00218726s (cpu); 0.00219061s (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 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ + -- used 0.482322s (cpu); 0.399664s (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 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ + -- used 0.00898651s (cpu); 0.00898696s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time degreeMap psi
│ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ + -- used 0.149425s (cpu); 0.149431s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
│ │ │  
│ │ │  i11 : time projectiveDegrees psi
│ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ + -- used 4.92152s (cpu); 4.44912s (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 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ + -- used 0.00225792s (cpu); 0.00226151s (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 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ + -- used 0.0545885s (cpu); 0.0545928s (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 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ + -- used 0.498961s (cpu); 0.423857s (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 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ + -- used 0.280506s (cpu); 0.261093s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : time degrees phi
│ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ + -- used 0.0291894s (cpu); 0.0170991s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
│ │ │  
│ │ │  i17 : time describe phi
│ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ + -- used 0.00322826s (cpu); 0.00323261s (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 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ + -- used 0.00976952s (cpu); 0.00977571s (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 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ + -- used 0.00962338s (cpu); 0.00962906s (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 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ + -- used 1.14177s (cpu); 0.850921s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
│ │ │  
│ │ │  i22 : time degree f
│ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ + -- used 1.4283e-05s (cpu); 1.3411e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
│ │ │  
│ │ │  i23 : time describe f
│ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ + -- used 0.00139563s (cpu); 0.00140042s (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 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ + -- used 0.484962s (cpu); 0.20826s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
│ │ │  
│ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ + -- used 0.0670834s (cpu); 0.0144547s (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 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ + -- used 0.0679067s (cpu); 0.057141s (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 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ + -- used 0.0544036s (cpu); 0.039766s (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 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ + -- used 0.324175s (cpu); 0.184591s (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 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ + -- used 0.844421s (cpu); 0.515648s (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 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ + -- used 0.396164s (cpu); 0.283943s (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 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ + -- used 0.0388774s (cpu); 0.0210279s (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 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ + -- used 0.260856s (cpu); 0.141402s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -98,22 +98,22 @@
│ │ │  
│ │ │  i8 : o4 == o6 and o5 == o7
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +o9 =   ZZ
│ │ │ + ------[x ..x ]
│ │ │ + 100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
│ │ │  
│ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ + -- used 0.0628304s (cpu); 0.0628336s (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 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ + -- used 0.351452s (cpu); 0.233532s (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 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ + -- used 0.370848s (cpu); 0.261296s (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 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ + -- used 1.40722s (cpu); 0.840178s (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 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ + -- used 0.000419229s (cpu); 0.000414949s (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 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ + -- used 0.123002s (cpu); 0.123008s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
│ │ │  
│ │ │  i6 : time rationalMap psi
│ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ + -- used 0.411917s (cpu); 0.340317s (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 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ + -- used 0.0418652s (cpu); 0.0418673s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ + -- used 2.20275s (cpu); 1.56914s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3
│ │ │  
│ │ │  i16 : time T2 = T * T
│ │ │ - -- used 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ + -- used 2.2041e-05s (cpu); 2.1092e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │        defining forms: given by a function
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │  
│ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ + -- used 3.3949s (cpu); 2.47518s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
│ │ │  
│ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │  
│ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │ @@ -169,15 +169,15 @@
│ │ │  i20 : T q
│ │ │  
│ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │  
│ │ │  o20 : List
│ │ │  
│ │ │  i21 : time f = rationalMap T
│ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ + -- used 2.93262s (cpu); 2.13759s (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 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ + -- used 0.371153s (cpu); 0.237742s (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 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ + -- used 0.227965s (cpu); 0.161642s (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 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ + -- used 1.98633s (cpu); 1.63356s (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 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ + -- used 2.70756s (cpu); 2.26027s (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 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ + -- used 0.0475906s (cpu); 0.0475915s (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 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ + -- used 1.25265s (cpu); 0.675682s (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 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
│ │ │ + -- used 0.000765688s (cpu); 0.000760851s (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 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
│ │ │ + -- used 0.0280952s (cpu); 0.0172637s (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 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
│ │ │ + -- used 0.0942155s (cpu); 0.041582s (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 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ + -- used 0.00351996s (cpu); 0.00351584s (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 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ + -- used 0.245702s (cpu); 0.133291s (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 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ + -- used 0.0739223s (cpu); 0.073923s (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 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ + -- used 0.255206s (cpu); 0.173658s (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 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ + -- used 0.0599719s (cpu); 0.0599182s (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 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ + -- used 0.0206066s (cpu); 0.0206054s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ + -- used 0.0373591s (cpu); 0.0163001s (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 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ + -- used 2.13185s (cpu); 1.85982s (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 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ + -- used 3.3215s (cpu); 2.28109s (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 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ + -- used 0.0181605s (cpu); 0.0180953s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
│ │ │  
│ │ │  i3 : time kernel(phi,2)
│ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ + -- used 0.903042s (cpu); 0.434661s (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 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ + -- used 0.00491444s (cpu); 0.00491219s (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 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ + -- used 0.48613s (cpu); 0.353755s (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 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ + -- used 0.226295s (cpu); 0.136901s (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 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ + -- used 0.0924105s (cpu); 0.0924147s (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 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ + -- used 0.0297214s (cpu); 0.0140827s (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 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ + -- used 0.0720352s (cpu); 0.0173512s (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 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ + -- used 6.3139e-05s (cpu); 5.5545e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ + -- used 2.3956e-05s (cpu); 2.3756e-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 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ + -- used 0.10024s (cpu); 0.10024s (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 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ + -- used 0.0285427s (cpu); 0.0285429s (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 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ + -- used 1.1704s (cpu); 0.553279s (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 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ + -- used 0.992458s (cpu); 0.911908s (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 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ + -- used 0.082983s (cpu); 0.0829819s (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 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ + -- used 0.0180094s (cpu); 0.0180098s (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 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ + -- used 0.0650044s (cpu); 0.0650038s (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 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ + -- used 0.00767562s (cpu); 0.00767575s (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 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ + -- used 0.0243386s (cpu); 0.0243395s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │  
│ │ │  i5 : time describe phi'
│ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ + -- used 0.00594743s (cpu); 0.0059516s (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 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ + -- used 0.00507663s (cpu); 0.00508411s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = rational map defined by forms of degree 2
│ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │       target variety: PP^3
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Chern__Schwartz__Mac__Pherson.html
│ │ │ @@ -102,30 +102,30 @@
│ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │                          0   4
    
│ │ │              
    
│ │ │                
    i3 : time ChernSchwartzMacPherson C
│ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ + -- used 1.09149s (cpu); 0.850049s (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 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ + -- used 1.33551s (cpu); 0.841573s (thread); 0s (gc)
│ │ │  
│ │ │         4     3     2
│ │ │  o4 = 3H  + 5H  + 3H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o4 : -----
│ │ │          5
│ │ │ @@ -172,30 +172,30 @@
│ │ │  o8 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i9 : time ChernClass G
│ │ │ - -- used 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ + -- used 0.119657s (cpu); 0.119659s (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 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ + -- used 0.0708737s (cpu); 0.0192794s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6      5      4     3
│ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o10 : -----
│ │ │          10
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,26 +39,26 @@
│ │ │ │                 2                           2
│ │ │ │  o2 = ideal (- x  + x x , - x x  + x x , - x  + x x )
│ │ │ │                 1    0 2     1 2    0 3     2    1 3
│ │ │ │  
│ │ │ │  o2 : Ideal of GF 78125[x ..x ]
│ │ │ │                          0   4
│ │ │ │  i3 : time ChernSchwartzMacPherson C
│ │ │ │ - -- used 1.26956s (cpu); 0.894031s (thread); 0s (gc)
│ │ │ │ + -- used 1.09149s (cpu); 0.850049s (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 1.46902s (cpu); 1.00309s (thread); 0s (gc)
│ │ │ │ + -- used 1.33551s (cpu); 0.841573s (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 0.110298s (cpu); 0.110305s (thread); 0s (gc)
│ │ │ │ + -- used 0.119657s (cpu); 0.119659s (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 0.0113359s (cpu); 0.0107235s (thread); 0s (gc)
│ │ │ │ + -- used 0.0708737s (cpu); 0.0192794s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           9      8      7      6      5      4     3
│ │ │ │  o10 = 10H  + 30H  + 60H  + 75H  + 57H  + 25H  + 5H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o10 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Euler__Characteristic.html
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o1 : Ideal of ------[p   ..p   , p   , p   , p   , p   , p   , p   , p   , p   ]
│ │ │                190181  0,1   0,2   1,2   0,3   1,3   2,3   0,4   1,4   2,4   3,4
    
│ │ │              
    
│ │ │                
    i2 : time EulerCharacteristic I
│ │ │ - -- used 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ + -- used 0.484962s (cpu); 0.20826s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 10
    
│ │ │              
    
│ │ │                
    i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ + -- used 0.0670834s (cpu); 0.0144547s (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 0.590242s (cpu); 0.241339s (thread); 0s (gc)
│ │ │ │ + -- used 0.484962s (cpu); 0.20826s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 10
│ │ │ │  i3 : time EulerCharacteristic(I,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0127473s (cpu); 0.0122967s (thread); 0s (gc)
│ │ │ │ + -- used 0.0670834s (cpu); 0.0144547s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = 10
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  No test is made to see if the projective variety is smooth.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_u_l_e_r_(_P_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y_) -- topological Euler characteristic of a
│ │ │ │        (smooth) projective variety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp!.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i4 : time phi! ;
│ │ │ - -- used 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ + -- used 0.0679067s (cpu); 0.057141s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
    
│ │ │              
    
│ │ │                
    i5 : describe phi
│ │ │ @@ -132,15 +132,15 @@
│ │ │       target variety: PP^5
│ │ │       coefficient ring: QQ
    
│ │ │              
    
│ │ │                
    i9 : time phi! ;
│ │ │ - -- used 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ + -- used 0.0544036s (cpu); 0.039766s (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 0.0572395s (cpu); 0.0568639s (thread); 0s (gc)
│ │ │ │ + -- used 0.0679067s (cpu); 0.057141s (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 0.0401527s (cpu); 0.03977s (thread); 0s (gc)
│ │ │ │ + -- used 0.0544036s (cpu); 0.039766s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
│ │ │ │  i10 : describe phi
│ │ │ │  
│ │ │ │  o10 = rational map defined by forms of degree 2
│ │ │ │        source variety: PP^4
│ │ │ │        target variety: PP^5
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/___Rational__Map_sp^_st_st_sp__Ideal.html
│ │ │ @@ -158,15 +158,15 @@
│ │ │  
│ │ │  o5 : Ideal of frac(QQ[a..f])[x, y, z, t, u, v]
    
│ │ │              
    
│ │ │                
    i6 : time phi^** q
│ │ │ - -- used 0.368428s (cpu); 0.207459s (thread); 0s (gc)
│ │ │ + -- used 0.324175s (cpu); 0.184591s (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 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ + -- used 0.844421s (cpu); 0.515648s (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 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ + -- used 0.396164s (cpu); 0.283943s (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 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ + -- used 0.0388774s (cpu); 0.0210279s (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 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ + -- used 0.260856s (cpu); 0.141402s (thread); 0s (gc)
│ │ │  
│ │ │            7        6       5      4      3
│ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │  
│ │ │       ZZ[H]
│ │ │  o7 : -----
│ │ │          8
│ │ │ @@ -208,25 +208,25 @@
│ │ │          
    
│ │ │          
    The method also accepts as input a ring map phi representing a rational map $\Phi:X\dashrightarrow Y$ between projective varieties. In this case, the method returns the push-forward to the Chow ring of the ambient projective space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it basically computes SegreClass ideal matrix phi. In the next example, we compute the Segre class of the base locus of a birational map $\mathbb{G}(1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i9 : use ZZ/100003[x_0..x_6]
│ │ │  
│ │ │ -o9     ZZ
│ │ │ -= ------[x ..x ]
│ │ │ -  100003  0   6
│ │ │ +o9 =   ZZ
│ │ │ + ------[x ..x ]
│ │ │ + 100003  0   6
│ │ │  
│ │ │  o9 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},{x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ + -- used 0.0628304s (cpu); 0.0628336s (thread); 0s (gc)
│ │ │  
│ │ │                                                          ZZ
│ │ │                                                        ------[y ..y ]
│ │ │                                                        100003  0   9                                                ZZ              2                              2
│ │ │  o10 = map (----------------------------------------------------------------------------------------------------, ------[x ..x ], {y  - y y  - y y , y y  - y y , y  - y y  - y y , y y  + y y  - y y , y y  - y y , y y  - y y  - y y , y y  - y y  - y y })
│ │ │             (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )  100003  0   6     3    0 5    1 6   3 4    1 7   4    2 7    0 9   2 5    3 5    1 8   4 5    1 9   4 8    2 9    3 9   7 8    4 9    6 9
│ │ │               5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
│ │ │ @@ -238,15 +238,15 @@
│ │ │                (y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y , y y  - y y  + y y )     100003  0   6
│ │ │                  5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i11 : time SegreClass phi
│ │ │ - -- used 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ + -- used 0.351452s (cpu); 0.233532s (thread); 0s (gc)
│ │ │  
│ │ │           9      8      7      6     5
│ │ │  o11 = 23H  - 42H  + 36H  - 22H  + 9H
│ │ │  
│ │ │        ZZ[H]
│ │ │  o11 : -----
│ │ │          10
│ │ │ @@ -272,30 +272,30 @@
│ │ │                   5 7    4 8    2 9   5 6    3 8    1 9   4 6    3 7    0 9   2 6    1 7    0 8   2 3    1 4    0 5
    
│ │ │              
    
│ │ │                
    i13 : -- Segre class of B in G(1,4)
│ │ │        time SegreClass B
│ │ │ - -- used 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ + -- used 0.370848s (cpu); 0.261296s (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 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ + -- used 1.40722s (cpu); 0.840178s (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 0.885027s (cpu); 0.51192s (thread); 0s (gc)
│ │ │ │ + -- used 0.844421s (cpu); 0.515648s (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 0.568535s (cpu); 0.343331s (thread); 0s (gc)
│ │ │ │ + -- used 0.396164s (cpu); 0.283943s (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 0.0216529s (cpu); 0.0213527s (thread); 0s (gc)
│ │ │ │ + -- used 0.0388774s (cpu); 0.0210279s (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 0.0973043s (cpu); 0.0970462s (thread); 0s (gc)
│ │ │ │ + -- used 0.260856s (cpu); 0.141402s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            7        6       5      4      3
│ │ │ │  o7 = 2816H  - 1056H  + 324H  - 78H  + 12H
│ │ │ │  
│ │ │ │       ZZ[H]
│ │ │ │  o7 : -----
│ │ │ │          8
│ │ │ │ @@ -134,22 +134,22 @@
│ │ │ │  method returns the push-forward to the Chow ring of the ambient projective
│ │ │ │  space of $X$ of the Segre class of the base locus of $\Phi$ in $X$, i.e., it
│ │ │ │  basically computes SegreClass ideal matrix phi. In the next example, we compute
│ │ │ │  the Segre class of the base locus of a birational map $\mathbb{G}
│ │ │ │  (1,4)\subset\mathbb{P}^9 \dashrightarrow \mathbb{P}^6$.
│ │ │ │  i9 : use ZZ/100003[x_0..x_6]
│ │ │ │  
│ │ │ │ -o9     ZZ
│ │ │ │ -= ------[x ..x ]
│ │ │ │ -  100003  0   6
│ │ │ │ +o9 =   ZZ
│ │ │ │ + ------[x ..x ]
│ │ │ │ + 100003  0   6
│ │ │ │  
│ │ │ │  o9 : PolynomialRing
│ │ │ │  i10 : time phi = inverseMap toMap(minors(2,matrix{{x_0,x_1,x_3,x_4,x_5},
│ │ │ │  {x_1,x_2,x_4,x_5,x_6}}),Dominant=>2)
│ │ │ │ - -- used 0.237104s (cpu); 0.125625s (thread); 0s (gc)
│ │ │ │ + -- used 0.0628304s (cpu); 0.0628336s (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 0.385145s (cpu); 0.250766s (thread); 0s (gc)
│ │ │ │ + -- used 0.351452s (cpu); 0.233532s (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 0.405977s (cpu); 0.267353s (thread); 0s (gc)
│ │ │ │ + -- used 0.370848s (cpu); 0.261296s (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 1.63119s (cpu); 0.951264s (thread); 0s (gc)
│ │ │ │ + -- used 1.40722s (cpu); 0.840178s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             9       8       7      6     5
│ │ │ │  o14 = 2764H  - 984H  + 294H  - 67H  + 9H
│ │ │ │  
│ │ │ │        ZZ[H]
│ │ │ │  o14 : -----
│ │ │ │          10
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_abstract__Rational__Map.html
│ │ │ @@ -106,15 +106,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : time psi = abstractRationalMap(P4,P5,f)
│ │ │ - -- used 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ + -- used 0.000419229s (cpu); 0.000414949s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: given by a function
│ │ │ @@ -124,23 +124,23 @@
│ │ │            
    
│ │ │          
    Now we compute first the degree of the forms defining the abstract map psi and then the corresponding concrete rational map.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
    
│ │ │                
    i5 : time projectiveDegrees(psi,3)
│ │ │ - -- used 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ + -- used 0.123002s (cpu); 0.123008s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    
│ │ │              
    
│ │ │                
    i6 : time rationalMap psi
│ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ + -- used 0.411917s (cpu); 0.340317s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │       source: Proj(QQ[t , t , t , t , t ])
│ │ │                        0   1   2   3   4
│ │ │       target: Proj(QQ[u , u , u , u , u , u ])
│ │ │                        0   1   2   3   4   5
│ │ │       defining forms: {
│ │ │ @@ -238,15 +238,15 @@
│ │ │  o13 : Ideal of -----[x ..x ]
│ │ │                 65521  0   3
    
│ │ │              
    
│ │ │                
    i14 : time T = abstractRationalMap(I,"OADP")
│ │ │ - -- used 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ + -- used 0.0418652s (cpu); 0.0418673s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -258,26 +258,26 @@
│ │ │            
    
│ │ │          
    The degree of the forms defining the abstract map T can be obtained by the following command:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time projectiveDegrees(T,2)
│ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ + -- used 2.20275s (cpu); 1.56914s (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 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ + -- used 2.2041e-05s (cpu); 2.1092e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(-----[x , x , x , x ])
│ │ │                     65521  0   1   2   3
│ │ │                       ZZ
│ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ @@ -286,15 +286,15 @@
│ │ │  
│ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
    
│ │ │              
    
│ │ │                
    i17 : time projectiveDegrees(T2,2)
│ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ + -- used 3.3949s (cpu); 2.47518s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = 1
    
│ │ │              
    
│ │ │          
    We verify that the composition of T with itself leaves a random point fixed:
    
│ │ │          
│ │ │ @@ -327,15 +327,15 @@
│ │ │            
│ │ │          
    
│ │ │          
    We now compute the concrete rational map corresponding to T:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i21 : time f = rationalMap T
│ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ + -- used 2.93262s (cpu); 2.13759s (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 0.000476825s (cpu); 0.000472506s (thread); 0s (gc)
│ │ │ │ + -- used 0.000419229s (cpu); 0.000414949s (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 0.1194s (cpu); 0.119423s (thread); 0s (gc)
│ │ │ │ + -- used 0.123002s (cpu); 0.123008s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 2
│ │ │ │  i6 : time rationalMap psi
│ │ │ │ - -- used 0.532769s (cpu); 0.401923s (thread); 0s (gc)
│ │ │ │ + -- used 0.411917s (cpu); 0.340317s (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 0.0383176s (cpu); 0.0383202s (thread); 0s (gc)
│ │ │ │ + -- used 0.0418652s (cpu); 0.0418673s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o14 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  The degree of the forms defining the abstract map T can be obtained by the
│ │ │ │  following command:
│ │ │ │  i15 : time projectiveDegrees(T,2)
│ │ │ │ - -- used 2.18663s (cpu); 1.5847s (thread); 0s (gc)
│ │ │ │ + -- used 2.20275s (cpu); 1.56914s (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 3.0007e-05s (cpu); 2.9705e-05s (thread); 0s (gc)
│ │ │ │ + -- used 2.2041e-05s (cpu); 2.1092e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │        defining forms: given by a function
│ │ │ │  
│ │ │ │  o16 : AbstractRationalMap (rational map from PP^3 to PP^3)
│ │ │ │  i17 : time projectiveDegrees(T2,2)
│ │ │ │ - -- used 3.67928s (cpu); 2.6019s (thread); 0s (gc)
│ │ │ │ + -- used 3.3949s (cpu); 2.47518s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 1
│ │ │ │  We verify that the composition of T with itself leaves a random point fixed:
│ │ │ │  i18 : p = apply(3,i->random(ZZ/65521))|{1}
│ │ │ │  
│ │ │ │  o18 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │ @@ -193,15 +193,15 @@
│ │ │ │  i20 : T q
│ │ │ │  
│ │ │ │  o20 = {-6648, -23396, -12311, 1}
│ │ │ │  
│ │ │ │  o20 : List
│ │ │ │  We now compute the concrete rational map corresponding to T:
│ │ │ │  i21 : time f = rationalMap T
│ │ │ │ - -- used 3.12656s (cpu); 2.217s (thread); 0s (gc)
│ │ │ │ + -- used 2.93262s (cpu); 2.13759s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = -- rational map --
│ │ │ │                       ZZ
│ │ │ │        source: Proj(-----[x , x , x , x ])
│ │ │ │                     65521  0   1   2   3
│ │ │ │                       ZZ
│ │ │ │        target: Proj(-----[x , x , x , x ])
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_approximate__Inverse__Map.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- approximateInverseMap: step 4 of 10
│ │ │  -- approximateInverseMap: step 5 of 10
│ │ │  -- approximateInverseMap: step 6 of 10
│ │ │  -- approximateInverseMap: step 7 of 10
│ │ │  -- approximateInverseMap: step 8 of 10
│ │ │  -- approximateInverseMap: step 9 of 10
│ │ │  -- approximateInverseMap: step 10 of 10
│ │ │ - -- used 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ + -- used 0.371153s (cpu); 0.237742s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = -- rational map --
│ │ │                    ZZ
│ │ │       source: Proj(--[t , t , t , t , t , t , t , t , t ])
│ │ │                    97  0   1   2   3   4   5   6   7   8
│ │ │                                  ZZ
│ │ │       target: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -205,15 +205,15 @@
│ │ │            
    
│ │ │                
    i5 : time psi' = approximateInverseMap(phi,CodimBsInv=>5);
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ + -- used 0.227965s (cpu); 0.161642s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : RationalMap (quadratic rational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i6 : assert(psi == psi')
    
│ │ │ @@ -300,15 +300,15 @@
│ │ │            
    
│ │ │                
    i8 : -- without the option 'CodimBsInv=>4', it takes about triple time 
│ │ │       time psi=approximateInverseMap(phi,CodimBsInv=>4)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │ - -- used 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ + -- used 1.98633s (cpu); 1.63356s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = -- rational map --
│ │ │                                  ZZ
│ │ │       source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │                                  97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │               {
│ │ │                                  2
│ │ │ @@ -372,15 +372,15 @@
│ │ │              
    		
│ │ │                
    i10 : -- in this case we can remedy enabling the option Certify
│ │ │        time psi = approximateInverseMap(phi,CodimBsInv=>4,Certify=>true)
│ │ │  -- approximateInverseMap: step 1 of 3
│ │ │  -- approximateInverseMap: step 2 of 3
│ │ │  -- approximateInverseMap: step 3 of 3
│ │ │  Certify: output certified!
│ │ │ - -- used 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ + -- used 2.70756s (cpu); 2.26027s (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 0.559174s (cpu); 0.374887s (thread); 0s (gc)
│ │ │ │ + -- used 0.371153s (cpu); 0.237742s (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 0.367939s (cpu); 0.280942s (thread); 0s (gc)
│ │ │ │ + -- used 0.227965s (cpu); 0.161642s (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 2.7613s (cpu); 1.89303s (thread); 0s (gc)
│ │ │ │ + -- used 1.98633s (cpu); 1.63356s (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 3.89555s (cpu); 2.92238s (thread); 0s (gc)
│ │ │ │ + -- used 2.70756s (cpu); 2.26027s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = -- rational map --
│ │ │ │                                   ZZ
│ │ │ │        source: subvariety of Proj(--[x , x , x , x , x , x , x , x , x , x , x  , x  ]) defined by
│ │ │ │                                   97  0   1   2   3   4   5   6   7   8   9   10   11
│ │ │ │                {
│ │ │ │                                   2
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_degree__Map.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 : RingMap ringP8 <-- ringP14
    
│ │ │              
    
│ │ │                
    i5 : time degreeMap phi
│ │ │ - -- used 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ + -- used 0.0475906s (cpu); 0.0475915s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 1
    
│ │ │              
    
│ │ │                
    i6 : -- Compose phi:P^8--->P^14 with a linear projection P^14--->P^8 from a general subspace of P^14 
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o6 : RingMap ringP8 <-- ringP8
    
│ │ │              
    
│ │ │                
    i7 : time degreeMap phi'
│ │ │ - -- used 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ + -- used 1.25265s (cpu); 0.675682s (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 0.0497248s (cpu); 0.0497258s (thread); 0s (gc)
│ │ │ │ + -- used 0.0475906s (cpu); 0.0475915s (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 1.46278s (cpu); 0.843714s (thread); 0s (gc)
│ │ │ │ + -- used 1.25265s (cpu); 0.675682s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = 14
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map
│ │ │ │      * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s -- projective degrees of a rational map between
│ │ │ │        projective varieties
│ │ │ │  ********** WWaayyss ttoo uussee ddeeggrreeeeMMaapp:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_force__Image.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o3 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i4 : time forceImage(Phi,ideal 0_(target Phi))
│ │ │ - -- used 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
    
│ │ │ + -- used 0.000765688s (cpu); 0.000760851s (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 0.000667762s (cpu); 0.00065607s (thread); 0s (gc)
│ │ │ │ + -- used 0.000765688s (cpu); 0.000760851s (thread); 0s (gc)
│ │ │ │  i5 : Phi;
│ │ │ │  
│ │ │ │  o5 : RationalMap (cubic dominant rational map from PP^6 to 6-dimensional
│ │ │ │  subvariety of PP^9)
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  If the declaration is false, nonsensical answers may result.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_graph.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time (p1,p2) = graph phi;
│ │ │ - -- used 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
    
│ │ │ + -- used 0.0280952s (cpu); 0.0172637s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i4 : p1
│ │ │  
│ │ │  o4 = -- rational map --
│ │ │ @@ -277,15 +277,15 @@
│ │ │            
    
│ │ │          
    When the source of the rational map is a multi-projective variety, the method returns all the projections.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -98,15 +98,15 @@
│ │ │ │  
│ │ │ │                        w w  - w w  + w w
│ │ │ │                         2 4    1 5    0 6
│ │ │ │                       }
│ │ │ │  
│ │ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
│ │ │ │  i2 : time psi = inverseMap phi
│ │ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ │ + -- used 0.0739223s (cpu); 0.073923s (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 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ │ + -- used 0.255206s (cpu); 0.173658s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = map (QQ[w ..w  ], QQ[w ..w  ], {- w w   + w w   + w  w   - w  w   - w w  ,
│ │ │ │  - w w   + w w   + w  w   - w  w   - w w  , - w w   + w w   + w  w   - w  w   -
│ │ │ │  w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   + w  w   -
│ │ │ │  w  w   - w w  , - w w   - w  w   + w  w   - w  w   - w w  , - w w   - w  w   +
│ │ │ │  w  w   - w  w   - w w  , w  w   - w  w   + w  w   - w  w   - w w  , - w  w   +
│ │ │ │  w  w   - w  w   + w  w   - w  w  , - w  w   + w  w   - w  w   + w  w   - w  w
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse_lp__Rational__Map_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o2 : RationalMap (rational map from PP^4 to PP^4)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : time g = graph p2;
│ │ │ - -- used 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
    
│ │ │ + -- used 0.0942155s (cpu); 0.041582s (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 0.0166208s (cpu); 0.0163389s (thread); 0s (gc)
│ │ │ │ + -- used 0.0280952s (cpu); 0.0172637s (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 0.0319997s (cpu); 0.0316715s (thread); 0s (gc)
│ │ │ │ + -- used 0.0942155s (cpu); 0.041582s (thread); 0s (gc)
│ │ │ │  i10 : g_0;
│ │ │ │  
│ │ │ │  o10 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ │ │  of PP^4 x PP^5 x PP^5 to PP^4)
│ │ │ │  i11 : g_1;
│ │ │ │  
│ │ │ │  o11 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_ideal_lp__Rational__Map_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4)
│ │ │              
    
│ │ │                
    i3 : time ideal phi
│ │ │ - -- used 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ + -- used 0.00351996s (cpu); 0.00351584s (thread); 0s (gc)
│ │ │  
│ │ │               2                                     2                      
│ │ │  o3 = ideal (x  - x x , x x  - x x  + x x , x x  - x  + x x , x x  - x x  +
│ │ │               4    3 5   2 4    3 4    1 5   2 3    3    1 4   1 2    1 3  
│ │ │       ------------------------------------------------------------------------
│ │ │              2
│ │ │       x x , x  - x x )
│ │ │ @@ -200,15 +200,15 @@
│ │ │  
│ │ │  o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4)
    
│ │ │              
    
│ │ │                
    i6 : time ideal phi'
│ │ │ - -- used 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ + -- used 0.245702s (cpu); 0.133291s (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 0.00352782s (cpu); 0.00352276s (thread); 0s (gc)
│ │ │ │ + -- used 0.00351996s (cpu); 0.00351584s (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 0.281847s (cpu); 0.150706s (thread); 0s (gc)
│ │ │ │ + -- used 0.245702s (cpu); 0.133291s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = ideal 1
│ │ │ │  
│ │ │ │  
│ │ │ │  QQ[x ..x , y ..y ]
│ │ │ │  
│ │ │ │  0   5   0   4
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_inverse__Map.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic Cremona transformation of PP^20)
    
│ │ │              
    
│ │ │                
    i2 : time psi = inverseMap phi
│ │ │ - -- used 0.0833126s (cpu); 0.0833315s (thread); 0s (gc)
│ │ │ + -- used 0.0739223s (cpu); 0.073923s (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: {
│ │ │ @@ -255,15 +255,15 @@
│ │ │  o4 : RingMap QQ[w ..w  ] <-- QQ[w ..w  ]
│ │ │                   0   26          0   26
    
│ │ │              
    
│ │ │                
    i5 : time psi = inverseMap phi
│ │ │ - -- used 0.300391s (cpu); 0.208058s (thread); 0s (gc)
│ │ │ + -- used 0.255206s (cpu); 0.173658s (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 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ + -- used 0.0599719s (cpu); 0.0599182s (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 0.0578597s (cpu); 0.0578645s (thread); 0s (gc)
│ │ │ │ + -- used 0.0599719s (cpu); 0.0599182s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │       source: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       target: Proj(QQ[x , x , x , x , x ])
│ │ │ │                        0   1   2   3   4
│ │ │ │       defining forms: {
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Birational.html
│ │ │ @@ -128,24 +128,24 @@
│ │ │  
│ │ │  o2 : RationalMap (quadratic dominant rational map from PP^4 to hypersurface in PP^5)
    
│ │ │              
    
│ │ │                
    i3 : time isBirational phi
│ │ │ - -- used 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ + -- used 0.0206066s (cpu); 0.0206054s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │                
    i4 : time isBirational(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ + -- used 0.0373591s (cpu); 0.0163001s (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 0.0202901s (cpu); 0.0202894s (thread); 0s (gc)
│ │ │ │ + -- used 0.0206066s (cpu); 0.0206054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : time isBirational(phi,Certify=>true)
│ │ │ │  Certify: output certified!
│ │ │ │ - -- used 0.0145752s (cpu); 0.014156s (thread); 0s (gc)
│ │ │ │ + -- used 0.0373591s (cpu); 0.0163001s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_D_o_m_i_n_a_n_t -- whether a rational map is dominant
│ │ │ │  ********** WWaayyss ttoo uussee iissBBiirraattiioonnaall:: **********
│ │ │ │      * isBirational(RationalMap)
│ │ │ │      * isBirational(RingMap)
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_is__Dominant.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │  o2 : RationalMap (rational map from PP^8 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 3.33576s (cpu); 2.59272s (thread); 0s (gc)
│ │ │ + -- used 2.13185s (cpu); 1.85982s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : P7 = ZZ/101[x_0..x_7];
    │ │ │ @@ -120,15 +120,15 @@ │ │ │ o6 : RationalMap (cubic rational map from PP^7 to PP^7) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDominant(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 4.05735s (cpu); 2.55688s (thread); 0s (gc)
│ │ │ + -- used 3.3215s (cpu); 2.28109s (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 3.33576s (cpu); 2.59272s (thread); 0s (gc) │ │ │ │ + -- used 2.13185s (cpu); 1.85982s (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 4.05735s (cpu); 2.55688s (thread); 0s (gc) │ │ │ │ + -- used 3.3215s (cpu); 2.28109s (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 │ │ │ @@ -93,26 +93,26 @@ │ │ │ o1 : RingMap QQ[x ..x ] <-- QQ[y ..y ] │ │ │ 0 8 0 11 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time kernel(phi,1)
│ │ │ - -- used 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ + -- used 0.0181605s (cpu); 0.0180953s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = ideal ()
│ │ │  
│ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │                    0   11
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time kernel(phi,2)
│ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ + -- used 0.903042s (cpu); 0.434661s (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 0.0180149s (cpu); 0.0180127s (thread); 0s (gc)
│ │ │ │ + -- used 0.0181605s (cpu); 0.0180953s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = ideal ()
│ │ │ │  
│ │ │ │  o2 : Ideal of QQ[y ..y  ]
│ │ │ │                    0   11
│ │ │ │  i3 : time kernel(phi,2)
│ │ │ │ - -- used 1.06451s (cpu); 0.555512s (thread); 0s (gc)
│ │ │ │ + -- used 0.903042s (cpu); 0.434661s (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
│ │ │ @@ -109,15 +109,15 @@
│ │ │  o2 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time parametrize L
│ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ + -- used 0.00491444s (cpu); 0.00491219s (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 ])
│ │ │ @@ -205,15 +205,15 @@
│ │ │  o4 : Ideal of --------[x ..x ]
│ │ │                10000019  0   9
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time parametrize Q
│ │ │ - -- used 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ + -- used 0.48613s (cpu); 0.353755s (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 {}
│ │ │ │ @@ -39,15 +39,15 @@
│ │ │ │       - 849671x  + 3034137x )
│ │ │ │                8           9
│ │ │ │  
│ │ │ │                   ZZ
│ │ │ │  o2 : Ideal of --------[x ..x ]
│ │ │ │                10000019  0   9
│ │ │ │  i3 : time parametrize L
│ │ │ │ - -- used 0.00443973s (cpu); 0.00443525s (thread); 0s (gc)
│ │ │ │ + -- used 0.00491444s (cpu); 0.00491219s (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 ])
│ │ │ │ @@ -135,15 +135,15 @@
│ │ │ │       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 0.76754s (cpu); 0.460282s (thread); 0s (gc)
│ │ │ │ + -- used 0.48613s (cpu); 0.353755s (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
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic rational map from 8-dimensional subvariety of PP^11 to PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time p = point source f
│ │ │ - -- used 0.298623s (cpu); 0.198159s (thread); 0s (gc)
│ │ │ + -- used 0.226295s (cpu); 0.136901s (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  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -104,15 +104,15 @@
│ │ │                (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 0.176944s (cpu); 0.176946s (thread); 0s (gc)
│ │ │ + -- used 0.0924105s (cpu); 0.0924147s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ _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 0.298623s (cpu); 0.198159s (thread); 0s (gc) │ │ │ │ + -- used 0.226295s (cpu); 0.136901s (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 0.176944s (cpu); 0.176946s (thread); 0s (gc) │ │ │ │ + -- used 0.0924105s (cpu); 0.0924147s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t -- pick a random K rational point on the scheme X │ │ │ │ defined by I │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * point(PolynomialRing) │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_projective__Degrees.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ 0 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time projectiveDegrees(phi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.0278401s (cpu); 0.0272526s (thread); 0s (gc)
│ │ │ + -- used 0.0297214s (cpu); 0.0140827s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = {1, 2, 4, 4, 2}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -122,15 +122,15 @@ │ │ │ 2 3 1 4 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projectiveDegrees(psi,Certify=>true)
│ │ │  Certify: output certified!
│ │ │ - -- used 0.019353s (cpu); 0.0187436s (thread); 0s (gc)
│ │ │ + -- used 0.0720352s (cpu); 0.0173512s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {2, 4, 4, 2, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -148,25 +148,25 @@ │ │ │ o6 : RingMap ------[x ..x ] <-- ------[x ..x ] │ │ │ 300007 0 6 300007 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time projectiveDegrees phi
│ │ │ - -- used 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc)
│ │ │ + -- used 6.3139e-05s (cpu); 5.5545e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {1, 2, 4, 8, 8, 4, 1}
│ │ │  
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time projectiveDegrees(phi,NumDegrees=>1)
│ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc)
│ │ │ + -- used 2.3956e-05s (cpu); 2.3756e-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 0.0278401s (cpu); 0.0272526s (thread); 0s (gc) │ │ │ │ + -- used 0.0297214s (cpu); 0.0140827s (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 0.019353s (cpu); 0.0187436s (thread); 0s (gc) │ │ │ │ + -- used 0.0720352s (cpu); 0.0173512s (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 5.2489e-05s (cpu); 4.7148e-05s (thread); 0s (gc) │ │ │ │ + -- used 6.3139e-05s (cpu); 5.5545e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {1, 2, 4, 8, 8, 4, 1} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : time projectiveDegrees(phi,NumDegrees=>1) │ │ │ │ - -- used 2.626e-05s (cpu); 2.6219e-05s (thread); 0s (gc) │ │ │ │ + -- used 2.3956e-05s (cpu); 2.3756e-05s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = {4, 1} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ Another way to use this method is by passing an integer i as second argument. │ │ │ │ However, this is equivalent to first projectiveDegrees(phi,NumDegrees=>i) and │ │ │ │ generally it is not faster. │ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ideal_cm__Z__Z_cm__Z__Z_rp.html │ │ │ @@ -93,15 +93,15 @@ │ │ │ o2 : Ideal of -----[x ..x ] │ │ │ 33331 0 6 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time phi = rationalMap(V,3,2)
│ │ │ - -- used 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ + -- used 0.10024s (cpu); 0.10024s (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 0.117734s (cpu); 0.117709s (thread); 0s (gc)
│ │ │ │ + -- used 0.10024s (cpu); 0.10024s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = -- rational map --
│ │ │ │                      ZZ
│ │ │ │       source: Proj(-----[x , x , x , x , x , x , x ])
│ │ │ │                    33331  0   1   2   3   4   5   6
│ │ │ │                      ZZ
│ │ │ │       target: Proj(-----[y , y , y , y , y , y , y , y , y , y , y  , y  , y  ,
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_rational__Map_lp__Ring_cm__Tally_rp.html
│ │ │ @@ -116,15 +116,15 @@
│ │ │              
│ │ │                
    i5 : D = new Tally from {H => 2,C => 1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time phi = rationalMap D
│ │ │ - -- used 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ + -- used 0.0285427s (cpu); 0.0285429s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = -- rational map --
│ │ │                                    ZZ
│ │ │       source: subvariety of Proj(-----[x , x , x , x , x , x ]) defined by
│ │ │                                  65521  0   1   2   3   4   5
│ │ │               {
│ │ │                   2                  2
│ │ │ @@ -224,15 +224,15 @@
│ │ │  
│ │ │  o6 : RationalMap (cubic rational map from surface in PP^5 to PP^20)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time ? image(phi,"F4")
│ │ │ - -- used 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ + -- used 1.1704s (cpu); 0.553279s (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 0.029123s (cpu); 0.029123s (thread); 0s (gc)
│ │ │ │ + -- used 0.0285427s (cpu); 0.0285429s (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 1.23167s (cpu); 0.77726s (thread); 0s (gc)
│ │ │ │ + -- used 1.1704s (cpu); 0.553279s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = surface of degree 38 and sectional genus 20 in PP^20 cut out by 153
│ │ │ │       hypersurfaces of degree 2
│ │ │ │  See also the package _W_e_i_l_D_i_v_i_s_o_r_s, which provides general tools for working
│ │ │ │  with divisors.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_a_t_i_o_n_a_l_M_a_p -- makes a rational map
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cremona__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
│ │ │ - -- used 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ + -- used 0.992458s (cpu); 0.911908s (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 1.3736s (cpu); 1.13714s (thread); 0s (gc)
│ │ │ │ + -- used 0.992458s (cpu); 0.911908s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = (rational map defined by forms of degree 3,
│ │ │ │        source variety: PP^3
│ │ │ │        target variety: PP^3
│ │ │ │        dominance: true
│ │ │ │        birationality: true
│ │ │ │        projective degrees: {1, 3, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Cubic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialCubicTransformation 9
│ │ │ - -- used 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ + -- used 0.082983s (cpu); 0.0829819s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6
│ │ │       target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ + -- used 0.0180094s (cpu); 0.0180098s (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 0.102301s (cpu); 0.102302s (thread); 0s (gc)
│ │ │ │ + -- used 0.082983s (cpu); 0.0829819s (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 0.0245428s (cpu); 0.0245442s (thread); 0s (gc)
│ │ │ │ + -- used 0.0180094s (cpu); 0.0180098s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 3
│ │ │ │       source variety: PP^6
│ │ │ │       target variety: complete intersection of type (2,2,2) in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 3, 9, 17, 21, 16, 8}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_special__Quadratic__Transformation.html
│ │ │ @@ -75,15 +75,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    The field K is required to be large enough.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
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│ │ │            
│ │ │            
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│ │ │            
│ │ │          
    
│ │ │                
    i1 : time specialQuadraticTransformation 4
│ │ │ - -- used 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ + -- used 0.0650044s (cpu); 0.0650038s (thread); 0s (gc)
│ │ │  
│ │ │  o1 = -- rational map --
│ │ │       source: Proj(QQ[x , x , x , x , x , x , x , x , x ])
│ │ │                        0   1   2   3   4   5   6   7   8
│ │ │       target: subvariety of Proj(QQ[y , y , y , y , y , y , y , y , y , y ]) defined by
│ │ │                                      0   1   2   3   4   5   6   7   8   9
│ │ │               {
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o1 : RationalMap (quadratic birational map from PP^8 to hypersurface in PP^9)
    
│ │ │              
    
│ │ │                
    i2 : time describe oo
│ │ │ - -- used 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ + -- used 0.00767562s (cpu); 0.00767575s (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 0.0879444s (cpu); 0.0879457s (thread); 0s (gc)
│ │ │ │ + -- used 0.0650044s (cpu); 0.0650038s (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 0.00847425s (cpu); 0.00847698s (thread); 0s (gc)
│ │ │ │ + -- used 0.00767562s (cpu); 0.00767575s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = rational map defined by forms of degree 2
│ │ │ │       source variety: PP^8
│ │ │ │       target variety: hypersurface of degree 3 in PP^9
│ │ │ │       dominance: true
│ │ │ │       birationality: true
│ │ │ │       projective degrees: {1, 2, 4, 8, 16, 21, 17, 9, 3}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/_to__External__String_lp__Rational__Map_rp.html
│ │ │ @@ -93,23 +93,23 @@
│ │ │  
│ │ │  o3 = 6927
    
│ │ │              
    
│ │ │                
    i4 : time phi' = value str;
│ │ │ - -- used 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ + -- used 0.0243386s (cpu); 0.0243395s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
    
│ │ │              
    
│ │ │                
    i5 : time describe phi'
│ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ + -- used 0.00594743s (cpu); 0.0059516s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = rational map defined by forms of degree 3
│ │ │       source variety: PP^3
│ │ │       target variety: smooth quadric hypersurface in PP^4
│ │ │       dominance: true
│ │ │       birationality: true (the inverse map is already calculated)
│ │ │       projective degrees: {1, 3, 4, 2}
│ │ │ @@ -118,15 +118,15 @@
│ │ │       degree base locus: 5
│ │ │       coefficient ring: ZZ/33331
    
│ │ │              
    
│ │ │                
    i6 : time describe inverse phi'
│ │ │ - -- used 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ + -- used 0.00507663s (cpu); 0.00508411s (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 0.0227997s (cpu); 0.022798s (thread); 0s (gc)
│ │ │ │ + -- used 0.0243386s (cpu); 0.0243395s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : RationalMap (cubic birational map from PP^3 to hypersurface in PP^4)
│ │ │ │  i5 : time describe phi'
│ │ │ │ - -- used 0.00554153s (cpu); 0.00554197s (thread); 0s (gc)
│ │ │ │ + -- used 0.00594743s (cpu); 0.0059516s (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 0.00455456s (cpu); 0.00455575s (thread); 0s (gc)
│ │ │ │ + -- used 0.00507663s (cpu); 0.00508411s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = rational map defined by forms of degree 2
│ │ │ │       source variety: smooth quadric hypersurface in PP^4
│ │ │ │       target variety: PP^3
│ │ │ │       dominance: true
│ │ │ │       birationality: true (the inverse map is already calculated)
│ │ │ │       projective degrees: {2, 4, 3, 1}
│ │ ├── ./usr/share/doc/Macaulay2/Cremona/html/index.html
│ │ │ @@ -63,29 +63,29 @@
│ │ │              
    		
│ │ │                
    i1 : ZZ/300007[t_0..t_6];
    
│ │ │              
    
│ │ │                
    i2 : time phi = toMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ + -- used 0.00459699s (cpu); 0.00459555s (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 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ + -- used 0.0466587s (cpu); 0.0466628s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = ideal (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x 
│ │ │               6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4
│ │ │       ------------------------------------------------------------------------
│ │ │       - x x  + x x , x x  - x x  + x x )
│ │ │          1 6    0 8   2 4    1 5    0 7
│ │ │  
│ │ │ @@ -93,43 +93,43 @@
│ │ │  o3 : Ideal of ------[x ..x ]
│ │ │                300007  0   9
    
│ │ │              
    
│ │ │                
    i4 : time degreeMap phi
│ │ │ - -- used 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ + -- used 0.0298369s (cpu); 0.0298355s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    
│ │ │              
    
│ │ │                
    i5 : time projectiveDegrees phi
│ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ + -- used 0.464463s (cpu); 0.3923s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ + -- used 0.0590164s (cpu); 0.0590235s (thread); 0s (gc)
│ │ │  
│ │ │  o6 = {5}
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ + -- used 0.00218726s (cpu); 0.00219061s (thread); 0s (gc)
│ │ │  
│ │ │                                                                         ZZ
│ │ │                                                                       ------[x ..x ]
│ │ │              ZZ                                                       300007  0   9                                                  3                2    2                2        2                      2                  2    2                 2                       3                2    2                2                                 2                           2    2                                  2        2                      2                  2                        2                         2    2                 2                       3                2    2
│ │ │  o7 = map (------[t ..t ], ----------------------------------------------------------------------------------------------------, {- t  + 2t t t  - t t  - t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t , - t t  + t t t  + t t t  - t t t  - t t  + t t t , - t t t  + t t  + t t  - t t t  - t t t  + t t t , - t t  + t t  + t t t  - t t t  - t t  + t t t , - t t  + t t t  + t t t  - t t  - t t t  + t t t , - t t  + t t  + t t t  - t t  - t t t  + t t t , - t  + 2t t t  - t t  - t t  + t t t })
│ │ │            300007  0   6   (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )      2     1 2 3    0 3    1 4    0 2 4     2 3    1 3    1 2 4    0 3 4    1 5    0 2 5     2 3    2 4    1 3 4    0 4    1 2 5    0 3 5     3     2 3 4    1 4    2 5    1 3 5     2 4    1 3 4    1 2 5    0 3 5    1 6    0 2 6     2 3 4    1 4    2 5    0 4 5    1 2 6    0 3 6     3 4    2 4    2 3 5    1 4 5    2 6    1 3 6     2 4    2 3 5    1 4 5    0 5    1 3 6    0 4 6     3 4    3 5    2 4 5    1 5    2 3 6    1 4 6     4     3 4 5    2 5    3 6    2 4 6
│ │ │                              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -141,15 +141,15 @@
│ │ │               300007  0   6      (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )
│ │ │                                    6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i8 : time psi = inverseMap phi
│ │ │ - -- used 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ + -- used 0.482322s (cpu); 0.399664s (thread); 0s (gc)
│ │ │  
│ │ │                                                         ZZ
│ │ │                                                       ------[x ..x ]
│ │ │                                                       300007  0   9                                                ZZ              3                2               2    2                        2                          2     2        2                               2                                   2               2             2                       3                                                 2                 2    2                                  2    2                 2                                                 3                         2      2    2      2                                              2
│ │ │  o8 = map (----------------------------------------------------------------------------------------------------, ------[t ..t ], {x  - 2x x x  + x x  - x x x  + x x  + x x  + x x x  - x x x  + x x  - 2x x x  - x x x  - 2x x , x x  - x x  - x x x  + x x x  + x x x  + x x  - 2x x x  - x x x  + x x x , x x  - x x x  + x x  - x x x  + x x  - x x x  - x x x , x  - x x x  + x x x  + x x x  - 2x x x  - x x x , x x  - x x x  + x x  + x x  - x x x  - x x x  - x x x , x x  - x x  - x x x  + x x  + x x x  + x x x  - 2x x x  - x x x  + x x x , x  - 2x x x  - x x x  + x x  + x x  + x x  + x x  + x x x  - 2x x x  - x x x  - x x x  - 2x x })
│ │ │            (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )  300007  0   6     2     1 2 3    0 3    1 2 5    0 5    1 6    0 2 6    0 4 6    1 7     0 2 7    0 4 7     0 9   2 3    1 3    1 2 6    0 3 6    0 5 6    1 8     0 2 8    0 4 8    0 1 9   2 3    1 3 6    0 6    0 3 8    1 9    0 2 9    0 4 9   3    1 3 8    0 6 8    1 2 9     0 3 9    0 5 9   3 6    2 3 8    0 8    2 9    1 3 9    0 6 9    0 7 9   3 6    3 8    2 6 8    1 8    2 3 9    2 5 9     1 6 9    1 7 9    0 8 9   6     3 6 8    5 6 8    2 8    4 8    3 9    5 9    2 6 9     4 6 9    2 7 9    4 7 9     0 9
│ │ │              6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
│ │ │ @@ -161,44 +161,44 @@
│ │ │               (x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x , x x  - x x  + x x )     300007  0   6
│ │ │                 6 7    5 8    4 9   3 7    2 8    1 9   3 5    2 6    0 9   3 4    1 6    0 8   2 4    1 5    0 7
    
│ │ │              
    
│ │ │                
    i9 : time isInverseMap(phi,psi)
│ │ │ - -- used 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ + -- used 0.00898651s (cpu); 0.00898696s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time degreeMap psi
│ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ + -- used 0.149425s (cpu); 0.149431s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 1
    
│ │ │              
    
│ │ │                
    i11 : time projectiveDegrees psi
│ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ + -- used 4.92152s (cpu); 4.44912s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │          
    We repeat the example using the type RationalMap and using deterministic methods.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -53,15 +53,15 @@
│ │ │ │        2
│ │ │ │  o6 = y T
│ │ │ │          1,2
│ │ │ │  
│ │ │ │  o6 : R[T   ..T   ]
│ │ │ │          1,1   1,3
│ │ │ │  i7 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s
│ │ │ │ +Finding easy relations           :  -- used 0.0273023s (cpu); 0.0156337s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = H
│ │ │ │  
│ │ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
│ │ │ │  i8 : homologyClass(KR,z1*z2)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Module.html
│ │ │ @@ -134,15 +134,15 @@
│ │ │  
│ │ │  o5 : Complex
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time phi = rationalMap minors(3,matrix{{t_0..t_4},{t_1..t_5},{t_2..t_6}})
│ │ │ - -- used 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ + -- used 0.00225792s (cpu); 0.00226151s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                       ZZ
│ │ │        target: Proj(------[x , x , x , x , x , x , x , x , x , x ])
│ │ │ @@ -247,15 +247,15 @@
│ │ │  
│ │ │  o12 : RationalMap (cubic rational map from PP^6 to PP^9)
    
│ │ │              
    
│ │ │                
    i13 : time phi = rationalMap(phi,Dominant=>2)
│ │ │ - -- used 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ + -- used 0.0545885s (cpu); 0.0545928s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = -- rational map --
│ │ │                       ZZ
│ │ │        source: Proj(------[t , t , t , t , t , t , t ])
│ │ │                     300007  0   1   2   3   4   5   6
│ │ │                                     ZZ
│ │ │        target: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │ @@ -320,15 +320,15 @@
│ │ │  
│ │ │  o13 : RationalMap (cubic rational map from PP^6 to 6-dimensional subvariety of PP^9)
    
│ │ │              
    
│ │ │                
    i14 : time phi^(-1)
│ │ │ - -- used 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ + -- used 0.498961s (cpu); 0.423857s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = -- rational map --
│ │ │                                     ZZ
│ │ │        source: subvariety of Proj(------[x , x , x , x , x , x , x , x , x , x ]) defined by
│ │ │                                   300007  0   1   2   3   4   5   6   7   8   9
│ │ │                {
│ │ │                 x x  - x x  + x x ,
│ │ │ @@ -381,49 +381,49 @@
│ │ │  
│ │ │  o14 : RationalMap (cubic birational map from 6-dimensional subvariety of PP^9 to PP^6)
    
│ │ │              
    
│ │ │                
    i15 : time degrees phi^(-1)
│ │ │ - -- used 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ + -- used 0.280506s (cpu); 0.261093s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │  
│ │ │  o15 : List
    
│ │ │              
    
│ │ │                
    i16 : time degrees phi
│ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ + -- used 0.0291894s (cpu); 0.0170991s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │  
│ │ │  o16 : List
    
│ │ │              
    
│ │ │                
    i17 : time describe phi
│ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ + -- used 0.00322826s (cpu); 0.00323261s (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 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ + -- used 0.00976952s (cpu); 0.00977571s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = rational map defined by forms of degree 3
│ │ │        source variety: 6-dimensional variety of degree 5 in PP^9 cut out by 5 hypersurfaces of degree 2
│ │ │        target variety: PP^6
│ │ │        dominance: true
│ │ │        birationality: true (the inverse map is already calculated)
│ │ │        projective degrees: {5, 15, 21, 17, 9, 3, 1}
│ │ │ @@ -432,41 +432,41 @@
│ │ │        degree base locus: 24
│ │ │        coefficient ring: ZZ/300007
    
│ │ │              
    
│ │ │                
    i19 : time (f,g) = graph phi^-1; f;
│ │ │ - -- used 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ + -- used 0.00962338s (cpu); 0.00962906s (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 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ + -- used 1.14177s (cpu); 0.850921s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │  
│ │ │  o21 : List
    
│ │ │              
    
│ │ │                
    i22 : time degree f
│ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ + -- used 1.4283e-05s (cpu); 1.3411e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = 1
    
│ │ │              
    
│ │ │                
    i23 : time describe f
│ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ + -- used 0.00139563s (cpu); 0.00140042s (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 0.00558593s (cpu); 0.00558112s (thread); 0s (gc)
│ │ │ │ + -- used 0.00459699s (cpu); 0.00459555s (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 0.0591643s (cpu); 0.0591742s (thread); 0s (gc)
│ │ │ │ + -- used 0.0466587s (cpu); 0.0466628s (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 0.0354136s (cpu); 0.0354203s (thread); 0s (gc)
│ │ │ │ + -- used 0.0298369s (cpu); 0.0298355s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 1
│ │ │ │  i5 : time projectiveDegrees phi
│ │ │ │ - -- used 0.683938s (cpu); 0.537126s (thread); 0s (gc)
│ │ │ │ + -- used 0.464463s (cpu); 0.3923s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : time projectiveDegrees(phi,NumDegrees=>0)
│ │ │ │ - -- used 0.223023s (cpu); 0.137433s (thread); 0s (gc)
│ │ │ │ + -- used 0.0590164s (cpu); 0.0590235s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = {5}
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time phi = toMap(phi,Dominant=>J)
│ │ │ │ - -- used 0.00354109s (cpu); 0.00354235s (thread); 0s (gc)
│ │ │ │ + -- used 0.00218726s (cpu); 0.00219061s (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 0.743183s (cpu); 0.662105s (thread); 0s (gc)
│ │ │ │ + -- used 0.482322s (cpu); 0.399664s (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 0.016456s (cpu); 0.0164579s (thread); 0s (gc)
│ │ │ │ + -- used 0.00898651s (cpu); 0.00898696s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  i10 : time degreeMap psi
│ │ │ │ - -- used 0.376507s (cpu); 0.274317s (thread); 0s (gc)
│ │ │ │ + -- used 0.149425s (cpu); 0.149431s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = 1
│ │ │ │  i11 : time projectiveDegrees psi
│ │ │ │ - -- used 5.65221s (cpu); 4.79159s (thread); 0s (gc)
│ │ │ │ + -- used 4.92152s (cpu); 4.44912s (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 0.00224049s (cpu); 0.00224156s (thread); 0s (gc)
│ │ │ │ + -- used 0.00225792s (cpu); 0.00226151s (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 0.0523877s (cpu); 0.0523973s (thread); 0s (gc)
│ │ │ │ + -- used 0.0545885s (cpu); 0.0545928s (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 0.586241s (cpu); 0.446401s (thread); 0s (gc)
│ │ │ │ + -- used 0.498961s (cpu); 0.423857s (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 0.624809s (cpu); 0.460082s (thread); 0s (gc)
│ │ │ │ + -- used 0.280506s (cpu); 0.261093s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = {5, 15, 21, 17, 9, 3, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : time degrees phi
│ │ │ │ - -- used 0.0257756s (cpu); 0.0252268s (thread); 0s (gc)
│ │ │ │ + -- used 0.0291894s (cpu); 0.0170991s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = {1, 3, 9, 17, 21, 15, 5}
│ │ │ │  
│ │ │ │  o16 : List
│ │ │ │  i17 : time describe phi
│ │ │ │ - -- used 0.00459023s (cpu); 0.00459072s (thread); 0s (gc)
│ │ │ │ + -- used 0.00322826s (cpu); 0.00323261s (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 0.0138045s (cpu); 0.0138079s (thread); 0s (gc)
│ │ │ │ + -- used 0.00976952s (cpu); 0.00977571s (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 0.0132166s (cpu); 0.0131906s (thread); 0s (gc)
│ │ │ │ + -- used 0.00962338s (cpu); 0.00962906s (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 1.60372s (cpu); 1.18114s (thread); 0s (gc)
│ │ │ │ + -- used 1.14177s (cpu); 0.850921s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = {904, 508, 268, 130, 56, 20, 5}
│ │ │ │  
│ │ │ │  o21 : List
│ │ │ │  i22 : time degree f
│ │ │ │ - -- used 2.0168e-05s (cpu); 1.9606e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.4283e-05s (cpu); 1.3411e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o22 = 1
│ │ │ │  i23 : time describe f
│ │ │ │ - -- used 0.00171686s (cpu); 0.00171795s (thread); 0s (gc)
│ │ │ │ + -- used 0.00139563s (cpu); 0.00140042s (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/Cyclotomic/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  Q3ljbG90b21pYw==
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3ljbG90b21pYyBmaWVsZHMiLCBEZXNj
│ │ │  cmlwdGlvbiA9PiAoRU17IkN5Y2xvdG9taWMifSwiIGlzIGEgcGFja2FnZSBmb3IgY3ljbG90b21p
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c291cmNlKERHQWxnZWJyYU1hcCk=
│ │ │  #:len=1484
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHNvdXJjZSBvZiBhIERHIGFsZ2Vi
│ │ │  cmEgbWFwIiwgImxpbmVudW0iID0+IDkyNTUsIElucHV0cyA9PiB7U1BBTntUVHsicGhpIn0sIiwg
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.out
│ │ │ @@ -155,15 +155,15 @@
│ │ │                                    2       2       2         2 2       2 2        2 2        2 2       2 2            2 2           2 2            2           2           2
│ │ │         Differential => {a, b, c, a T   , b T   , c T   , a*b c T   , b c T   , -a b T   , -a c T   , b c T   T   , -a c T   T   , b c T   T   , -a T   T   , c T   T   , b T   T   }
│ │ │                                      1,1     1,2     1,3         1,1       2,1        2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3 2,4     1,2 2,4
│ │ │  
│ │ │  o16 : DGAlgebra
│ │ │  
│ │ │  i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0257812s (cpu); 0.0148848s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___Basic_spoperations_spon_sp__D__G_sp__Algebras.out
│ │ │ @@ -30,15 +30,15 @@
│ │ │        Underlying algebra => R[S   ..S   ]
│ │ │                                 1,1   1,4
│ │ │        Differential => {a, b, c, d}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.028927s (cpu); 0.0183124s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i6 : describe HB
│ │ │  
│ │ │ @@ -68,15 +68,15 @@
│ │ │                                      2
│ │ │        Differential => {a, b, c, d, a S   }
│ │ │                                        1,1
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │  
│ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0322139s (cpu); 0.0202469s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra.out
│ │ │ @@ -12,15 +12,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {a, b, c}
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │  
│ │ │  i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.307926s (cpu); 0.0693884s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Algebra__Map.out
│ │ │ @@ -28,15 +28,15 @@
│ │ │  
│ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │  
│ │ │  i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.164295s (cpu); 0.031992s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module.out
│ │ │ @@ -34,15 +34,15 @@
│ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │                                     |     0    |  |     0    |
│ │ │                                     |     0    |  |     0    |
│ │ │  
│ │ │  o4 : DGModule
│ │ │  
│ │ │  i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.134531s (cpu); 0.0367374s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___H__H_sp__D__G__Module__Map.out
│ │ │ @@ -41,15 +41,15 @@
│ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │                   {2, 2} | 0 1 0 |
│ │ │                   {2, 2} | 0 0 1 |
│ │ │  
│ │ │  o4 : DGModuleMap
│ │ │  
│ │ │  i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.129394s (cpu); 0.0368988s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.out
│ │ │ @@ -49,15 +49,15 @@
│ │ │                                1                                                             {6} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d c b a |     3
│ │ │                                                                                      
│ │ │                                                                                     2
│ │ │  
│ │ │  o6 : Complex
│ │ │  
│ │ │  i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.146019s (cpu); 0.0320958s (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 0.691391s (cpu); 0.592007s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.648563s (cpu); 0.636023s (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 0.0210175s (cpu); 0.0187759s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0306578s (cpu); 0.0191141s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_get__Deg__N__Module.out
│ │ │ @@ -25,15 +25,15 @@
│ │ │        Underlying algebra => R[T   ..T   ]
│ │ │                                 1,1   1,3
│ │ │        Differential => {x, y, z}
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0264145s (cpu); 0.0154943s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │  
│ │ │  o6 : QuotientRing
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Algebra.out
│ │ │ @@ -18,15 +18,15 @@
│ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0674284s (cpu); 0.0232898s (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 0.0919266s (cpu); 0.08901s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.271239s (cpu); 0.128151s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
│ │ │  
│ │ │  i9 : numgens HA
│ │ │  
│ │ │ @@ -122,15 +122,15 @@
│ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │  
│ │ │  o15 = {1, 7, 7, 1}
│ │ │  
│ │ │  o15 : List
│ │ │  
│ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.224439s (cpu); 0.076883s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
│ │ │  
│ │ │  i17 : R = ZZ/101[a,b,c,d]
│ │ │  
│ │ │ @@ -159,14 +159,14 @@
│ │ │         Underlying algebra => S[T   ..T   ]
│ │ │                                  1,1   1,4
│ │ │         Differential => {a, b, c, d}
│ │ │  
│ │ │  o20 : DGAlgebra
│ │ │  
│ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0377198s (cpu); 0.0194923s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = HB
│ │ │  
│ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │  
│ │ │  i22 :
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_homology__Class.out
│ │ │ @@ -43,15 +43,15 @@
│ │ │  o6 = y T
│ │ │          1,2
│ │ │  
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
│ │ │  
│ │ │  i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0273023s (cpu); 0.0156337s (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 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ + -- used 0.489022s (cpu); 0.1553s (thread); 0s (gc)
│ │ │  Finding easy relations           : 
│ │ │  o6 = HKR
│ │ │  
│ │ │  o6 : QuotientRing
│ │ │  
│ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product.out
│ │ │ @@ -68,15 +68,15 @@
│ │ │                 2
│ │ │  o9 = (true, x y T   T   T    - x x y T   T   T   )
│ │ │               2 2 1,1 1,2 1,3    1 2 2 1,2 1,3 1,4
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.556432s (cpu); 0.544715s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.out
│ │ │ @@ -27,15 +27,15 @@
│ │ │                                 1,1   1,4
│ │ │        Differential => {t , t , t , t }
│ │ │                          1   2   3   4
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │  
│ │ │  i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.303128s (cpu); 0.186367s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
│ │ │  
│ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/example-output/_tor__Algebra_lp__Ring_cm__Ring_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │  
│ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.572157s (cpu); 0.463659s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
│ │ │  
│ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebra_sp__Maps.html
│ │ │ @@ -294,15 +294,15 @@
│ │ │          
    
│ │ │            
    One can also obtain the map on homology induced by a DGAlgebra map.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i17 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0257812s (cpu); 0.0148848s (thread); 0s (gc)
│ │ │  
│ │ │                            ZZ
│ │ │                           ---[a..c]
│ │ │              ZZ           101
│ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │             101  1   2           3   1     1
│ │ │                          (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -218,15 +218,15 @@
│ │ │ │                                      1,1     1,2     1,3         1,1       2,1
│ │ │ │  2,3        2,2       1,3 2,1        1,3 2,2       1,2 2,1      1,1 2,4     1,3
│ │ │ │  2,4     1,2 2,4
│ │ │ │  
│ │ │ │  o16 : DGAlgebra
│ │ │ │  One can also obtain the map on homology induced by a DGAlgebra map.
│ │ │ │  i17 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0164909s (cpu); 0.0155111s
│ │ │ │ +Finding easy relations           :  -- used 0.0257812s (cpu); 0.0148848s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                            ZZ
│ │ │ │                           ---[a..c]
│ │ │ │              ZZ           101
│ │ │ │  o17 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │             101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___Basic_spoperations_spon_sp__D__G_sp__Algebras.html
│ │ │ @@ -118,15 +118,15 @@
│ │ │          
    
│ │ │            
    One can compute the homology algebra of a DGAlgebra using the homology (or HH) command.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -179,15 +179,15 @@
│ │ │  
│ │ │  o9 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,15 +42,15 @@
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  One can compute the homology algebra of a DGAlgebra using the homology (or HH)
│ │ │ │  command.
│ │ │ │  i5 : HB = HH B
│ │ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s
│ │ │ │ +Finding easy relations           :  -- used 0.028927s (cpu); 0.0183124s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HB
│ │ │ │  
│ │ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i6 : describe HB
│ │ │ │  
│ │ │ │ @@ -87,15 +87,15 @@
│ │ │ │                                 1,1   1,4   2,1
│ │ │ │                                      2
│ │ │ │        Differential => {a, b, c, d, a S   }
│ │ │ │                                        1,1
│ │ │ │  
│ │ │ │  o9 : DGAlgebra
│ │ │ │  i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s
│ │ │ │ +Finding easy relations           :  -- used 0.0322139s (cpu); 0.0202469s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │         ZZ
│ │ │ │  o10 = ---[X ..X ]
│ │ │ │        101  1   3
│ │ │ │  
│ │ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │  
│ │ │  o2 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  o2 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {a, b, c}
│ │ │ │  
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : HA = HH A
│ │ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s
│ │ │ │ +Finding easy relations           :  -- used 0.307926s (cpu); 0.0693884s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = HA
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Algebra__Map.html
│ │ │ @@ -120,15 +120,15 @@
│ │ │  
│ │ │  o4 : DGAlgebraMap
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -255,15 +255,15 @@
│ │ │  
│ │ │  o15 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i5 : HB = HH B
│ │ │ -Finding easy relations           :  -- used 0.0198089s (cpu); 0.0185677s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.028927s (cpu); 0.0183124s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HB
│ │ │  
│ │ │  o5 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : homologyAlgebra(C,GenDegreeLimit=>4,RelDegreeLimit=>4)
│ │ │ -Finding easy relations           :  -- used 0.019996s (cpu); 0.0184306s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0322139s (cpu); 0.0202469s (thread); 0s (gc)
│ │ │  
│ │ │         ZZ
│ │ │  o10 = ---[X ..X ]
│ │ │        101  1   3
│ │ │  
│ │ │  o10 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i3 : HA = HH A
│ │ │ -Finding easy relations           :  -- used 0.0541819s (cpu); 0.0514376s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.307926s (cpu); 0.0693884s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = HA
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
    
│ │ │                
    i5 : HHg = HH g
│ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.164295s (cpu); 0.031992s (thread); 0s (gc)
│ │ │  
│ │ │                           ZZ
│ │ │                          ---[a..c]
│ │ │             ZZ           101
│ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │            101  1   2           3   1     1
│ │ │                         (c, b, a )
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,15 +43,15 @@
│ │ │ │  i4 : g = dgAlgebraMap(K1, K2, matrix{{Y_(1,2), Y_(1,3)}})
│ │ │ │  
│ │ │ │  o4 = map (R[Y   ..Y   ], R[T   ..T   ], {Y   , Y   , a, b, c})
│ │ │ │               1,1   1,3      1,1   1,2     1,2   1,3
│ │ │ │  
│ │ │ │  o4 : DGAlgebraMap
│ │ │ │  i5 : HHg = HH g
│ │ │ │ -Finding easy relations           :  -- used 0.0165828s (cpu); 0.015481s
│ │ │ │ +Finding easy relations           :  -- used 0.164295s (cpu); 0.031992s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │                           ZZ
│ │ │ │                          ---[a..c]
│ │ │ │             ZZ           101
│ │ │ │  o5 = map (---[X ..X ], ----------[X ], {X , 0, 0, 0})
│ │ │ │            101  1   2           3   1     1
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModule
    
│ │ │              
    
│ │ │                
    i5 : HM = homology M
│ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.134531s (cpu); 0.0367374s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ │  
│ │ │                                                  4
│ │ │ ├── html2text {}
│ │ │ │ @@ -57,15 +57,15 @@
│ │ │ │        Generator degrees => {{0, 0}, {2, 2}, {2, 2}}
│ │ │ │        Differentials on gens => {0, | xT_(1,1) |, | yT_(1,2) |}
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │                                     |     0    |  |     0    |
│ │ │ │  
│ │ │ │  o4 : DGModule
│ │ │ │  i5 : HM = homology M
│ │ │ │ -Finding easy relations           :  -- used 0.14103s (cpu); 0.0653344s
│ │ │ │ +Finding easy relations           :  -- used 0.134531s (cpu); 0.0367374s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = cokernel {0, 0} | X_2 X_1 0   0   0   0   |
│ │ │ │                {3, 4} | 0   0   X_1 X_2 0   0   |
│ │ │ │                {3, 4} | 0   0   0   X_1 0   X_2 |
│ │ │ │                {3, 4} | 0   0   0   0   X_2 X_1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___H__H_sp__D__G__Module__Map.html
│ │ │ @@ -132,15 +132,15 @@
│ │ │  
│ │ │  o4 : DGModuleMap
    
│ │ │              
    
│ │ │                
    i5 : h = homology idM
│ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.129394s (cpu); 0.0368988s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │       {3, 4} | 0 1 0 0 |
│ │ │       {3, 4} | 0 0 1 0 |
│ │ │       {3, 4} | 0 0 0 1 |
│ │ │  
│ │ │  o5 : Matrix
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -59,15 +59,15 @@
│ │ │ │                      1,1   1,2
│ │ │ │        Natural => {0, 0} | 1 0 0 |
│ │ │ │                   {2, 2} | 0 1 0 |
│ │ │ │                   {2, 2} | 0 0 1 |
│ │ │ │  
│ │ │ │  o4 : DGModuleMap
│ │ │ │  i5 : h = homology idM
│ │ │ │ -Finding easy relations           :  -- used 0.172027s (cpu); 0.0676646s
│ │ │ │ +Finding easy relations           :  -- used 0.129394s (cpu); 0.0368988s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = {0, 0} | 1 0 0 0 |
│ │ │ │       {3, 4} | 0 1 0 0 |
│ │ │ │       {3, 4} | 0 0 1 0 |
│ │ │ │       {3, 4} | 0 0 0 1 |
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/___The_sp__Koszul_spcomplex_spas_spa_sp__D__G_sp__Algebra.html
│ │ │ @@ -143,15 +143,15 @@
│ │ │          
    
│ │ │            
    Since the Koszul complex is a DG algebra, its homology is itself an algebra.  One can obtain this algebra using the command homology, homologyAlgebra, or HH (all commands work).  This algebra structure can detect whether or not the ring is a complete intersection or Gorenstein.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -76,15 +76,15 @@
│ │ │ │  
│ │ │ │  o6 : Complex
│ │ │ │  Since the Koszul complex is a DG algebra, its homology is itself an algebra.
│ │ │ │  One can obtain this algebra using the command homology, homologyAlgebra, or HH
│ │ │ │  (all commands work). This algebra structure can detect whether or not the ring
│ │ │ │  is a complete intersection or Gorenstein.
│ │ │ │  i7 : HKR = HH KR
│ │ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s
│ │ │ │ +Finding easy relations           :  -- used 0.146019s (cpu); 0.0320958s
│ │ │ │  (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 0.691391s (cpu); 0.592007s
│ │ │ │ +Finding easy relations           :  -- used 0.648563s (cpu); 0.636023s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = HKR'
│ │ │ │  
│ │ │ │  o10 : QuotientRing
│ │ │ │  i11 : numgens HKR'
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_cycles.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : List
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -28,15 +28,15 @@
│ │ │ │  o2 : DGAlgebra
│ │ │ │  i3 : apply(maxDegree A + 1, i -> numgens prune homology(i, A))
│ │ │ │  
│ │ │ │  o3 = {1, 4, 6, 4, 1}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : HA = homologyAlgebra A
│ │ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s
│ │ │ │ +Finding easy relations           :  -- used 0.0306578s (cpu); 0.0191141s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HA
│ │ │ │  
│ │ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  i5 : numgens HA
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_get__Deg__N__Module.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : HKR = HH KR
│ │ │ -Finding easy relations           :  -- used 0.0242787s (cpu); 0.0231763s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.146019s (cpu); 0.0320958s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = HKR
│ │ │  
│ │ │  o7 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i10 : HKR' = HH koszulComplexDGA R'
│ │ │ -Finding easy relations           :  -- used 0.691391s (cpu); 0.592007s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.648563s (cpu); 0.636023s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = HKR'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra A
│ │ │ -Finding easy relations           :  -- used 0.0210175s (cpu); 0.0187759s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0306578s (cpu); 0.0191141s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i5 : HA = HH KR;
│ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s (thread); 0s (gc)
    
│ │ │ +Finding easy relations           :  -- used 0.0264145s (cpu); 0.0154943s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : H0 = zerothHomology KR
│ │ │  
│ │ │  o6 = H0
│ │ │ ├── html2text {}
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  o4 = {Ring => R                          }
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,3
│ │ │ │        Differential => {x, y, z}
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : HA = HH KR;
│ │ │ │ -Finding easy relations           :  -- used 0.0155194s (cpu); 0.01441s
│ │ │ │ +Finding easy relations           :  -- used 0.0264145s (cpu); 0.0154943s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  i6 : H0 = zerothHomology KR
│ │ │ │  
│ │ │ │  o6 = H0
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  i7 : M1 = getDegNModule(1, H0, HA)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Algebra.html
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0223745s (cpu); 0.0212429s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0674284s (cpu); 0.0232898s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HA
│ │ │  
│ │ │  o4 : PolynomialRing, 4 skew commutative variable(s)
    
│ │ │              
    
│ │ │ @@ -153,15 +153,15 @@
│ │ │  
│ │ │  o7 : List
│ │ │              
    
│ │ │                
    i8 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0919266s (cpu); 0.08901s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.271239s (cpu); 0.128151s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = HA
│ │ │  
│ │ │  o8 : QuotientRing
    
│ │ │              
    
│ │ │                
    i16 : HA = homologyAlgebra(A)
│ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.224439s (cpu); 0.076883s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = HA
│ │ │  
│ │ │  o16 : QuotientRing
    
│ │ │              
    
│ │ │ @@ -315,15 +315,15 @@
│ │ │  
│ │ │  o20 : DGAlgebra
    
│ │ │              
    
│ │ │                
    i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0377198s (cpu); 0.0194923s (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 0.0223745s (cpu); 0.0212429s
│ │ │ │ +Finding easy relations           :  -- used 0.0674284s (cpu); 0.0232898s
│ │ │ │  (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 0.0919266s (cpu); 0.08901s
│ │ │ │ +Finding easy relations           :  -- used 0.271239s (cpu); 0.128151s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HA
│ │ │ │  
│ │ │ │  o8 : QuotientRing
│ │ │ │  i9 : numgens HA
│ │ │ │  
│ │ │ │ @@ -130,15 +130,15 @@
│ │ │ │  o14 : DGAlgebra
│ │ │ │  i15 : apply(maxDegree A + 1, i -> numgens prune homology(i,A))
│ │ │ │  
│ │ │ │  o15 = {1, 7, 7, 1}
│ │ │ │  
│ │ │ │  o15 : List
│ │ │ │  i16 : HA = homologyAlgebra(A)
│ │ │ │ -Finding easy relations           :  -- used 0.0612159s (cpu); 0.0591286s
│ │ │ │ +Finding easy relations           :  -- used 0.224439s (cpu); 0.076883s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = HA
│ │ │ │  
│ │ │ │  o16 : QuotientRing
│ │ │ │  One can check that HA has Poincare duality since R is Gorenstein.
│ │ │ │  If your DGAlgebra has generators in even degrees, then one must specify the
│ │ │ │ @@ -166,15 +166,15 @@
│ │ │ │  o20 = {Ring => S                          }
│ │ │ │         Underlying algebra => S[T   ..T   ]
│ │ │ │                                  1,1   1,4
│ │ │ │         Differential => {a, b, c, d}
│ │ │ │  
│ │ │ │  o20 : DGAlgebra
│ │ │ │  i21 : HB = homologyAlgebra(B,GenDegreeLimit=>7,RelDegreeLimit=>14)
│ │ │ │ -Finding easy relations           :  -- used 0.020179s (cpu); 0.0185169s
│ │ │ │ +Finding easy relations           :  -- used 0.0377198s (cpu); 0.0194923s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o21 = HB
│ │ │ │  
│ │ │ │  o21 : PolynomialRing, 4 skew commutative variable(s)
│ │ │ │  ********** WWaayyss ttoo uussee hhoommoollooggyyAAllggeebbrraa:: **********
│ │ │ │      * homologyAlgebra(DGAlgebra)
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_homology__Class.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  o6 : R[T   ..T   ]
│ │ │          1,1   1,3
    
│ │ │              
    
│ │ │                
    i7 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.0161623s (cpu); 0.0151328s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.0273023s (cpu); 0.0156337s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = H
│ │ │  
│ │ │  o7 : PolynomialRing, 3 skew commutative variable(s)
    
│ │ │              
    
│ │ │                
    i6 : HKR = HH(KR)
│ │ │ - -- used 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ + -- used 0.489022s (cpu); 0.1553s (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 0.114372s (cpu); 0.108291s (thread); 0s (gc)
│ │ │ │ + -- used 0.489022s (cpu); 0.1553s (thread); 0s (gc)
│ │ │ │  Finding easy relations           :
│ │ │ │  o6 = HKR
│ │ │ │  
│ │ │ │  o6 : QuotientRing
│ │ │ │  The following is the graded canonical module of R:
│ │ │ │  i7 : degList = first entries vars Q / degree / first
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product.html
│ │ │ @@ -197,15 +197,15 @@
│ │ │          
    
│ │ │            
    Given cycles z1,z2,z3 such that z1*z2 and z2*z3 are boundaries, the Massey triple product of the homology classes represented by z1,z2 and z3 is the homology class of lift12*z3 + z1*lift23.  To see this, we compute and check:
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,15 +49,15 @@
│ │ │ │        Underlying algebra => R[T   ..T   ]
│ │ │ │                                 1,1   1,4
│ │ │ │        Differential => {t , t , t , t }
│ │ │ │                          1   2   3   4
│ │ │ │  
│ │ │ │  o4 : DGAlgebra
│ │ │ │  i5 : H = HH(KR)
│ │ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s
│ │ │ │ +Finding easy relations           :  -- used 0.303128s (cpu); 0.186367s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = H
│ │ │ │  
│ │ │ │  o5 : QuotientRing
│ │ │ │  i6 : masseys = masseyTripleProduct(KR,1,1,1);
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_tor__Algebra_lp__Ring_cm__Ring_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,15 +27,15 @@
│ │ │ │  i2 : M = coker matrix {{a^3*b^3*c^3*d^3}};
│ │ │ │  i3 : S = R/ideal{a^3*b^3*c^3*d^3}
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : QuotientRing
│ │ │ │  i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s
│ │ │ │ +Finding easy relations           :  -- used 0.572157s (cpu); 0.463659s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = HB
│ │ │ │  
│ │ │ │  o4 : QuotientRing
│ │ │ │  i5 : numgens HB
│ │ ├── ./usr/share/doc/Macaulay2/DecomposableSparseSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  c29sdmVEZWNvbXBvc2FibGVTeXN0ZW0oTGlzdCk=
│ │ │  #:len=338
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│ │ ├── ./usr/share/doc/Macaulay2/Depth/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  U2VlZA==
│ │ │  #:len=219
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│ │ ├── ./usr/share/doc/Macaulay2/DeterminantalRepresentations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  Z2VuZXJhbGl6ZWRNaXhlZERpc2NyaW1pbmFudChMaXN0KQ==
│ │ │  #:len=367
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmFsaXplZE1peGVkRGlzY3JpbWluYW50LExp
│ │ ├── ./usr/share/doc/Macaulay2/DiffAlg/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  bGluZWFyQ29tYg==
│ │ │  #:len=2390
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2VuZXJpYyBsaW5lYXIgY29tYmluYXRp
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│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dGFsbHlTdW1tYW5kcw==
│ │ │  #:len=2086
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFsbHkgYSBsaXN0IG9mIG1vZHVsZXMg
│ │ │  dXAgdG8gaXNvbW9ycGhpc20iLCAibGluZW51bSIgPT4gNjgwLCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/example-output/___Direct__Summands.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i3 : F = frobeniusPushforward(1, OO_X);
│ │ │  
│ │ │  i4 : elapsedTime tallySummands summands F
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │ - -- 3.35396s elapsed
│ │ │ + -- 2.38859s elapsed
│ │ │  
│ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ @@ -32,15 +32,15 @@
│ │ │                      {1} | 0  0   -2z z   3y  x  |
│ │ │              1
│ │ │             R  => 1
│ │ │  
│ │ │  o4 : Tally
│ │ │  
│ │ │  i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ - -- 11.5598s elapsed
│ │ │ + -- 8.1221s elapsed
│ │ │  
│ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │                      {1} | x        az (-a-3)y |
│ │ │                      {1} | (2a-2)z  3y x       |
│ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │                      {1} | 3az      -y       x       |
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/example-output/_is__Indecomposable.out
│ │ │ @@ -54,14 +54,14 @@
│ │ │  
│ │ │                              19
│ │ │  o8 : S-module, quotient of S
│ │ │  
│ │ │  i9 : assert(2 == rank FHM)
│ │ │  
│ │ │  i10 : assert elapsedTime isIndecomposable FHM
│ │ │ - -- 1.00223s elapsed
│ │ │ + -- .860147s elapsed
│ │ │  
│ │ │  i11 : assert({FHM} == summands FHM)
│ │ │  
│ │ │  i12 : assert FHM.cache.isIndecomposable
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/html/_is__Indecomposable.html
│ │ │ @@ -170,15 +170,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -124,15 +124,15 @@
│ │ │ │  0     0      0    0      0      0      0      -x_4   -x_3    x_3  x_2    x_1
│ │ │ │  -x_0   |
│ │ │ │  
│ │ │ │                              19
│ │ │ │  o8 : S-module, quotient of S
│ │ │ │  i9 : assert(2 == rank FHM)
│ │ │ │  i10 : assert elapsedTime isIndecomposable FHM
│ │ │ │ - -- 1.00223s elapsed
│ │ │ │ + -- .860147s elapsed
│ │ │ │  i11 : assert({FHM} == summands FHM)
│ │ │ │  i12 : assert FHM.cache.isIndecomposable
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_i_r_e_c_t_S_u_m_m_a_n_d_s -- compute the direct summands of a module or coherent
│ │ │ │        sheaf
│ │ │ │      * _f_i_n_d_I_d_e_m_p_o_t_e_n_t_s -- construct idempotent endomorphisms
│ │ │ │  ********** WWaayyss ttoo uussee iissIInnddeeccoommppoossaabbllee:: **********
│ │ ├── ./usr/share/doc/Macaulay2/DirectSummands/html/index.html
│ │ │ @@ -111,15 +111,15 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,15 +26,15 @@
│ │ │ │  i3 : T = {{a,b},{d,e}}
│ │ │ │  
│ │ │ │  o3 = {{a, b}, {d, e}}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │ │  
│ │ │ │ -o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ │ +o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │ │                  "ring" => S
│ │ │ │                  "vertices" => {a, b, c, d, e}
│ │ │ │  
│ │ │ │  o4 : HyperGraph
│ │ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ │ │  
│ │ │ │  o5 = HyperGraph{"edges" => {{a, b, c}, {a, e}, {c, d, e}}}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/html/_random__Hyper__Graph.html
│ │ │ @@ -93,28 +93,28 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── 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 }}}
│ │ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ │ +                              1   2   4     3   4     1   2   3   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 }}}
│ │ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ │ +                              2   3   5     3   4     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/EigenSolver/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  emVyb0RpbVNvbHZlKElkZWFsKQ==
│ │ │  #:len=254
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjgzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh6ZXJvRGltU29sdmUsSWRlYWwpLCJ6ZXJvRGltU29s
│ │ ├── ./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;
│ │ │ - -- .217098s elapsed
│ │ │ + -- .199354s elapsed
│ │ │  
│ │ │  i4 : #sols -- 156 solutions
│ │ │  
│ │ │  o4 = 156
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/EigenSolver/html/index.html
│ │ │ @@ -85,15 +85,15 @@
│ │ │  
│ │ │  o2 : Ideal of QQ[a..f]
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,22 +29,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ │ + -- used 0.00290513s (cpu); 0.00290241s (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 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ │ + -- used 0.00160251s (cpu); 0.00160381s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_eliminate.html
│ │ │ @@ -102,26 +102,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,22 +30,22 @@
│ │ │ │  i3 : g = x^2+c*x+d
│ │ │ │  
│ │ │ │        2
│ │ │ │  o3 = x  + x*c + d
│ │ │ │  
│ │ │ │  o3 : R
│ │ │ │  i4 : time eliminate(x,ideal(f,g))
│ │ │ │ - -- used 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ │ + -- used 0.00268195s (cpu); 0.00267863s (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 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ │ + -- used 0.00147412s (cpu); 0.00147784s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                          2    2             2           2
│ │ │ │  o5 = ideal(- a*b*c + b*c  + a d - a*c*d + b  - 2b*d + d )
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : sylvesterMatrix(f,g,x)
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_resultant_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : z123 = masseyTripleProduct(KR,z1,z2,z3)
│ │ │ -Finding easy relations           :  -- used 0.596064s (cpu); 0.508805s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.556432s (cpu); 0.544715s (thread); 0s (gc)
│ │ │  
│ │ │               2
│ │ │  o10 = x x y z T   T   T   T
│ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │  
│ │ │  o10 : R[T   ..T   ]
│ │ │           1,1   1,5
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -90,15 +90,15 @@
│ │ │ │  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 0.596064s (cpu); 0.508805s
│ │ │ │ +Finding easy relations           :  -- used 0.556432s (cpu); 0.544715s
│ │ │ │  (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2
│ │ │ │  o10 = x x y z T   T   T   T
│ │ │ │         1 2 2   1,2 1,3 1,4 1,5
│ │ │ │  
│ │ │ │  o10 : R[T   ..T   ]
│ │ ├── ./usr/share/doc/Macaulay2/DGAlgebras/html/_massey__Triple__Product_lp__D__G__Algebra_cm__Z__Z_cm__Z__Z_cm__Z__Z_rp.html
│ │ │ @@ -124,15 +124,15 @@
│ │ │  
│ │ │  o4 : DGAlgebra
│ │ │              
    
│ │ │                
    i5 : H = HH(KR)
│ │ │ -Finding easy relations           :  -- used 0.174343s (cpu); 0.170772s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.303128s (cpu); 0.186367s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = H
│ │ │  
│ │ │  o5 : QuotientRing
    
│ │ │              
    
│ │ │                
    i4 : HB = torAlgebra(R,S,GenDegreeLimit=>4,RelDegreeLimit=>8)
│ │ │ -Finding easy relations           :  -- used 0.532566s (cpu); 0.419747s (thread); 0s (gc)
│ │ │ +Finding easy relations           :  -- used 0.572157s (cpu); 0.463659s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = HB
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
    
│ │ │                
    i9 : assert(2 == rank FHM)
    
│ │ │              
    
│ │ │                
    i10 : assert elapsedTime isIndecomposable FHM
│ │ │ - -- 1.00223s elapsed
    
│ │ │ + -- .860147s elapsed
│ │ │              
    
│ │ │                
    i11 : assert({FHM} == summands FHM)
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime tallySummands summands F
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │   -- try using changeBaseField with GF 49
│ │ │ - -- 3.35396s elapsed
│ │ │ + -- 2.38859s elapsed
│ │ │  
│ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o4 : Tally
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ - -- 11.5598s elapsed
│ │ │ + -- 8.1221s elapsed
│ │ │  
│ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │                      {1} | x        az (-a-3)y |
│ │ │                      {1} | (2a-2)z  3y x       |
│ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │                      {1} | 3az      -y       x       |
│ │ │ ├── html2text {}
│ │ │ │ @@ -54,15 +54,15 @@
│ │ │ │  i1 : R = ZZ/7[x,y,z]/(x^3+y^3+z^3);
│ │ │ │  i2 : X = Proj R;
│ │ │ │  i3 : F = frobeniusPushforward(1, OO_X);
│ │ │ │  i4 : elapsedTime tallySummands summands F
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │   -- try using changeBaseField with GF 49
│ │ │ │ - -- 3.35396s elapsed
│ │ │ │ + -- 2.38859s elapsed
│ │ │ │  
│ │ │ │  o4 = Tally{cokernel {1} | x   -2z -3y -y  0   0   | => 1}
│ │ │ │                      {1} | 3z  -y  0   x   2z  -2y |
│ │ │ │                      {1} | -3y x   z   -3z 0   0   |
│ │ │ │                      {1} | 0   0   -3y y   x   -3z |
│ │ │ │                      {1} | -2z 2y  x   0   -z  3y  |
│ │ │ │                      {1} | 0   0   -z  -3z -2y x   |
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │                      {1} | 3z -3y x   0   -2z -y |
│ │ │ │                      {1} | 0  0   -2z z   3y  x  |
│ │ │ │              1
│ │ │ │             R  => 1
│ │ │ │  
│ │ │ │  o4 : Tally
│ │ │ │  i5 : elapsedTime tallySummands summands changeBaseField(GF(7, 2), F)
│ │ │ │ - -- 11.5598s elapsed
│ │ │ │ + -- 8.1221s elapsed
│ │ │ │  
│ │ │ │  o5 = Tally{cokernel {1} | (-2a+1)y x  z       | => 1      }
│ │ │ │                      {1} | x        az (-a-3)y |
│ │ │ │                      {1} | (2a-2)z  3y x       |
│ │ │ │             cokernel {1} | (-3a-2)y x        2z      | => 1
│ │ │ │                      {1} | x        (-2a+2)z (2a-1)y |
│ │ │ │                      {1} | 3az      -y       x       |
│ │ ├── ./usr/share/doc/Macaulay2/Dmodules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  RG1vZHVsZXM=
│ │ │  #:len=758
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRC1tb2R1bGVzIHBhY2thZ2UgY29sbGVj
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│ │ ├── ./usr/share/doc/Macaulay2/EagonResolution/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  VHJhbnNwb3Nl
│ │ │  #:len=1380
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVHJhbnNwb3NlID0+IGZhbHNlLCBkZWZh
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│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y29tcGxlbWVudEdyYXBo
│ │ │  #:len=2011
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgY29tcGxlbWVudCBv
│ │ │  ZiBhIGdyYXBoIG9yIGh5cGVyZ3JhcGgiLCAibGluZW51bSIgPT4gMjA3OSwgSW5wdXRzID0+IHtT
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_delete__Edges.out
│ │ │ @@ -14,15 +14,15 @@
│ │ │  
│ │ │  o3 = {{a, b}, {d, e}}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ +o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
│ │ │  
│ │ │  i5 : h = hyperGraph {a*b*c,c*d*e,a*e}
│ │ ├── ./usr/share/doc/Macaulay2/EdgeIdeals/example-output/_random__Hyper__Graph.out
│ │ │ @@ -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 }}}
│ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ +                              1   2   4     3   4     1   2   3   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 }}}
│ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ +                              2   3   5     3   4     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/_delete__Edges.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  o3 : List
    
│ │ │              
    
│ │ │                
    i4 : gprime = deleteEdges (g,T)
│ │ │  
│ │ │ -o4 = HyperGraph{"edges" => {{b, c}, {a, e}, {c, d}}}
│ │ │ +o4 = HyperGraph{"edges" => {{c, d}, {b, c}, {a, e}}}
│ │ │                  "ring" => S
│ │ │                  "vertices" => {a, b, c, d, e}
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : randomHyperGraph(R,{3,2,4})
│ │ │  
│ │ │  o3 = HyperGraph{"edges" => {{x , x , x }, {x , x }, {x , x , x , x }}}
│ │ │ -                              2   4   5     1   5     1   2   3   4
│ │ │ +                              1   2   4     3   4     1   2   3   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 }}}
│ │ │ -                              1   2   4     1   5     2   3   4   5
│ │ │ +                              2   3   5     3   4     1   2   4   5
│ │ │                  "ring" => R
│ │ │                  "vertices" => {x , x , x , x , x }
│ │ │                                  1   2   3   4   5
│ │ │  
│ │ │  o4 : HyperGraph
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime sols = zeroDimSolve I;
│ │ │ - -- .217098s elapsed
    
│ │ │ + -- .199354s 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;
│ │ │ │ - -- .217098s elapsed
│ │ │ │ + -- .199354s 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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZWxpbWluYXRlKElkZWFsLExpc3Qp
│ │ │  #:len=200
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgxLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhlbGltaW5h
│ │ ├── ./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 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ + -- used 0.00290513s (cpu); 0.00290241s (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 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ + -- used 0.00160251s (cpu); 0.00160381s (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 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ + -- used 0.00268195s (cpu); 0.00267863s (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 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ + -- used 0.00147412s (cpu); 0.00147784s (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 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ + -- used 1.36607s (cpu); 1.13629s (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 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ + -- used 0.0147506s (cpu); 0.0147511s (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 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ + -- used 1.34177s (cpu); 1.13921s (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 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ + -- used 0.0128689s (cpu); 0.012869s (thread); 0s (gc)
│ │ │  
│ │ │                7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o6 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_discriminant_lp__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -108,26 +108,26 @@
│ │ │  
│ │ │  o3 : R
│ │ │              
    
│ │ │                
    i4 : time eliminate(x,ideal(f,g))
│ │ │ - -- used 0.00258521s (cpu); 0.0025827s (thread); 0s (gc)
│ │ │ + -- used 0.00290513s (cpu); 0.00290241s (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 0.00151386s (cpu); 0.00151487s (thread); 0s (gc)
│ │ │ + -- used 0.00160251s (cpu); 0.00160381s (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 0.00271762s (cpu); 0.00271441s (thread); 0s (gc)
│ │ │ + -- used 0.00268195s (cpu); 0.00267863s (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 0.00160798s (cpu); 0.00160903s (thread); 0s (gc)
│ │ │ + -- used 0.00147412s (cpu); 0.00147784s (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 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ + -- used 1.36607s (cpu); 1.13629s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o4 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o4 : Ideal of R
    
│ │ │              
    
│ │ │                
    i5 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ + -- used 0.0147506s (cpu); 0.0147511s (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 1.71167s (cpu); 1.39647s (thread); 0s (gc)
│ │ │ │ + -- used 1.36607s (cpu); 1.13629s (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 0.0168256s (cpu); 0.0168298s (thread); 0s (gc)
│ │ │ │ + -- used 0.0147506s (cpu); 0.0147511s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                7       8     3 5    8     6         3 4      7       3 3 2
│ │ │ │  o5 = ideal(- a b*c + a d - a b  + b  + 6a b*c + 18a b c - 7b c - 48a b c  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7
│ │ │ │       21b c  + 46a b c  - 35b c  - 15a b*c  + 35b c  - 21b c  + 7b c  - b*c  -
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Elimination/html/_sylvester__Matrix_lp__Ring__Element_cm__Ring__Element_cm__Ring__Element_rp.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o4 : R
    
│ │ │              
    
│ │ │                
    i5 : time eliminate(ideal(f,g),x)
│ │ │ - -- used 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ + -- used 1.34177s (cpu); 1.13921s (thread); 0s (gc)
│ │ │  
│ │ │              7       8     3 5    8     6         3 4      7       3 3 2  
│ │ │  o5 = ideal(a b*c - a d + a b  - b  - 6a b*c - 18a b c + 7b c + 48a b c  -
│ │ │       ------------------------------------------------------------------------
│ │ │          6 2      3 2 3      5 3      3   4      4 4      3 5     2 6      7  
│ │ │       21b c  - 46a b c  + 35b c  + 15a b*c  - 35b c  + 21b c  - 7b c  + b*c  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : time ideal resultant(f,g,x)
│ │ │ - -- used 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ + -- used 0.0128689s (cpu); 0.012869s (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 1.84957s (cpu); 1.47049s (thread); 0s (gc)
│ │ │ │ + -- used 1.34177s (cpu); 1.13921s (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 0.0175768s (cpu); 0.017584s (thread); 0s (gc)
│ │ │ │ + -- used 0.0128689s (cpu); 0.012869s (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/EliminationMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  bWF4Q29sKE1hdHJpeCk=
│ │ │  #:len=257
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWF4Q29sLE1hdHJpeCksIm1heENvbChNYXRyaXgp
│ │ ├── ./usr/share/doc/Macaulay2/EliminationTemplates/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Y29weVRlbXBsYXRl
│ │ │  #:len=2172
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29waWVzIEVsaW1pbmF0aW9uVGVtcGxh
│ │ │  dGUgb2JqZWN0IiwgImxpbmVudW0iID0+IDk4NywgSW5wdXRzID0+IHtTUEFOe1RUeyJFIn0sIiwg
│ │ ├── ./usr/share/doc/Macaulay2/EllipticCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  RWxsUG9pbnRXID09IEVsbFBvaW50Vw==
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzQ3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPT0sRWxsUG9pbnRXLEVsbFBvaW50Vyks
│ │ ├── ./usr/share/doc/Macaulay2/EllipticIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RWxsaXB0aWNJbnRlZ3JhbHM=
│ │ │  #:len=584
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiAxOihESVZ7UEFSQXtURVh7IlRoaXMg
│ │ │  cGFja2FnZSBwcm92aWRlcyBzb21lIGZ1bmN0aW9ucyBmb3IgY29tcHV0aW5nIGVsbGlwdGljIGlu
│ │ ├── ./usr/share/doc/Macaulay2/EngineTests/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  RW5naW5lVGVzdHM=
│ │ │  #:len=311
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSB0ZXN0IHN1aXRlIGZvciB0aGUgTWFj
│ │ │  YXVsYXkyIGVuZ2luZSIsICJsaW5lbnVtIiA9PiA0NzksICJmaWxlbmFtZSIgPT4gIkVuZ2luZVRl
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  RW51bWVyYXRpb25DdXJ2ZXM=
│ │ │  #:len=747
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRW51bWVyYXRpb24gb2YgcmF0aW9uYWwg
│ │ │  Y3VydmVzIHZpYSB0b3J1cyBhY3Rpb25zIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7IiJ9
│ │ ├── ./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 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ + -- used 0.0244241s (cpu); 0.0244238s (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 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ + -- used 0.316544s (cpu); 0.250914s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
│ │ │  
│ │ │  i8 : time rationalCurve(3)
│ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ + -- used 0.121674s (cpu); 0.121676s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
│ │ │  
│ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ + -- used 4.35505s (cpu); 3.80637s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
│ │ │  
│ │ │  i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ + -- used 0.202278s (cpu); 0.142025s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
│ │ │  
│ │ │  i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ + -- used 4.39847s (cpu); 3.83496s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
│ │ │  
│ │ │  i12 : time rationalCurve(4)
│ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ + -- used 1.35096s (cpu); 1.16559s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
│ │ │  
│ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ + -- used 6.14432s (cpu); 4.91462s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
│ │ │  
│ │ │  i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ + -- used 1.3986s (cpu); 1.20963s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
│ │ │  
│ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ + -- used 6.06687s (cpu); 4.84391s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
│ │ │  
│ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ + -- used 6.08951s (cpu); 4.94347s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 1139448384
│ │ │  
│ │ │  o16 : QQ
│ │ │  
│ │ │  i17 :
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_lines__Hypersurface.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Computes the number of lines on a general hypersurface of degree 2n - 3 in \mathbb P^n.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the number of lines on a general hypersurface of degree
│ │ │ │              2n - 3 in \mathbb P^n
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Computes the number of lines on a general hypersurface of degree 2n - 3 in
│ │ │ │  \mathbb P^n.
│ │ │ │  i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ │ + -- used 0.0244241s (cpu); 0.0244238s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       289139638632755625, 520764738758073845321}
│ │ │ │  
│ │ │ │  o1 : List
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/EnumerationCurves/html/_rational__Curve.html
│ │ │ @@ -157,15 +157,15 @@
│ │ │            
    The numbers of conics on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i1 : time for n from 2 to 10 list linesHypersurface(n)
│ │ │ - -- used 0.031399s (cpu); 0.0313974s (thread); 0s (gc)
│ │ │ + -- used 0.0244241s (cpu); 0.0244238s (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 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ + -- used 0.316544s (cpu); 0.250914s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │  
│ │ │  o7 : List
    
│ │ │              
    
│ │ │ @@ -173,27 +173,27 @@
│ │ │            
    For rational curves of degree 3:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ @@ -203,15 +203,15 @@
│ │ │            
    The number of rational curves of degree 3 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
    
│ │ │                
    i8 : time rationalCurve(3)
│ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ + -- used 0.121674s (cpu); 0.121676s (thread); 0s (gc)
│ │ │  
│ │ │       8564575000
│ │ │  o8 = ----------
│ │ │           27
│ │ │  
│ │ │  o8 : QQ
    
│ │ │              
    
│ │ │                
    i9 : time for D in T list rationalCurve(3,D)
│ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ + -- used 4.35505s (cpu); 3.80637s (thread); 0s (gc)
│ │ │  
│ │ │        8564575000  422690816           4834592  11239424
│ │ │  o9 = {----------, ---------, 6424365, -------, --------}
│ │ │            27          27                 3        27
│ │ │  
│ │ │  o9 : List
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time rationalCurve(3) - rationalCurve(1)/27
│ │ │ - -- used 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ + -- used 0.202278s (cpu); 0.142025s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = 317206375
│ │ │  
│ │ │  o10 : QQ
    
│ │ │              
    
│ │ │ @@ -219,15 +219,15 @@
│ │ │            
    The numbers of rational curves of degree 3 on general complete intersection Calabi-Yau threefolds can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i11 : time for D in T list rationalCurve(3,D) - rationalCurve(1,D)/27
│ │ │ - -- used 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ + -- used 4.39847s (cpu); 3.83496s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │  
│ │ │  o11 : List
    
│ │ │              
    
│ │ │ @@ -235,27 +235,27 @@
│ │ │            
    For rational curves of degree 4:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : time rationalCurve(4)
│ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ + -- used 1.35096s (cpu); 1.16559s (thread); 0s (gc)
│ │ │  
│ │ │        15517926796875
│ │ │  o12 = --------------
│ │ │              64
│ │ │  
│ │ │  o12 : QQ
    
│ │ │              
    
│ │ │                
    i13 : time rationalCurve(4,{4,2})
│ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ + -- used 6.14432s (cpu); 4.91462s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = 3883914084
│ │ │  
│ │ │  o13 : QQ
    
│ │ │              
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │            
    The number of rational curves of degree 4 on a general quintic threefold can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i14 : time rationalCurve(4) - rationalCurve(2)/8
│ │ │ - -- used 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ + -- used 1.3986s (cpu); 1.20963s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 242467530000
│ │ │  
│ │ │  o14 : QQ
    
│ │ │              
    
│ │ │ @@ -279,25 +279,25 @@
│ │ │            
    The numbers of rational curves of degree 4 on general complete intersections of types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ + -- used 6.06687s (cpu); 4.84391s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3883902528
│ │ │  
│ │ │  o15 : QQ
    
│ │ │              
    
│ │ │                
    i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ + -- used 6.08951s (cpu); 4.94347s (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 0.353246s (cpu); 0.284105s (thread); 0s (gc)
│ │ │ │ + -- used 0.316544s (cpu); 0.250914s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = {609250, 92288, 52812, 22428, 9728}
│ │ │ │  
│ │ │ │  o7 : List
│ │ │ │  For rational curves of degree 3:
│ │ │ │  i8 : time rationalCurve(3)
│ │ │ │ - -- used 0.23024s (cpu); 0.163316s (thread); 0s (gc)
│ │ │ │ + -- used 0.121674s (cpu); 0.121676s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       8564575000
│ │ │ │  o8 = ----------
│ │ │ │           27
│ │ │ │  
│ │ │ │  o8 : QQ
│ │ │ │  i9 : time for D in T list rationalCurve(3,D)
│ │ │ │ - -- used 6.4737s (cpu); 5.50773s (thread); 0s (gc)
│ │ │ │ + -- used 4.35505s (cpu); 3.80637s (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 0.232795s (cpu); 0.175565s (thread); 0s (gc)
│ │ │ │ + -- used 0.202278s (cpu); 0.142025s (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 6.08562s (cpu); 5.19424s (thread); 0s (gc)
│ │ │ │ + -- used 4.39847s (cpu); 3.83496s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {317206375, 15655168, 6424326, 1611504, 416256}
│ │ │ │  
│ │ │ │  o11 : List
│ │ │ │  For rational curves of degree 4:
│ │ │ │  i12 : time rationalCurve(4)
│ │ │ │ - -- used 1.85501s (cpu); 1.49923s (thread); 0s (gc)
│ │ │ │ + -- used 1.35096s (cpu); 1.16559s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        15517926796875
│ │ │ │  o12 = --------------
│ │ │ │              64
│ │ │ │  
│ │ │ │  o12 : QQ
│ │ │ │  i13 : time rationalCurve(4,{4,2})
│ │ │ │ - -- used 8.7345s (cpu); 6.35557s (thread); 0s (gc)
│ │ │ │ + -- used 6.14432s (cpu); 4.91462s (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 2.27006s (cpu); 1.97354s (thread); 0s (gc)
│ │ │ │ + -- used 1.3986s (cpu); 1.20963s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o14 = 242467530000
│ │ │ │  
│ │ │ │  o14 : QQ
│ │ │ │  The numbers of rational curves of degree 4 on general complete intersections of
│ │ │ │  types (4,2) and (3,3) in \mathbb P^5 can be computed as follows:
│ │ │ │  i15 : time rationalCurve(4,{4,2}) - rationalCurve(2,{4,2})/8
│ │ │ │ - -- used 8.47703s (cpu); 6.3453s (thread); 0s (gc)
│ │ │ │ + -- used 6.06687s (cpu); 4.84391s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3883902528
│ │ │ │  
│ │ │ │  o15 : QQ
│ │ │ │  i16 : time rationalCurve(4,{3,3}) - rationalCurve(2,{3,3})/8
│ │ │ │ - -- used 9.22095s (cpu); 6.77006s (thread); 0s (gc)
│ │ │ │ + -- used 6.08951s (cpu); 4.94347s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 1139448384
│ │ │ │  
│ │ │ │  o16 : QQ
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_u_l_t_i_p_l_e_C_o_v_e_r -- Multiple coverings of rational curves on Calabi-Yau
│ │ │ │        threefolds
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  YnVpbGRFTW9ub21pYWxNYXAoUmluZyxSaW5nLExpc3Qp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIzNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoYnVpbGRFTW9ub21pYWxNYXAsUmluZyxSaW5nLExp
│ │ ├── ./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 .0022445 seconds
│ │ │ -     -- used .0006884 seconds
│ │ │ +     -- used .00183378 seconds
│ │ │ +     -- used .000574381 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00370187 seconds
│ │ │ -     -- used .00494948 seconds
│ │ │ +     -- used .00342142 seconds
│ │ │ +     -- used .00448892 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00769783 seconds
│ │ │ -     -- used .118616 seconds
│ │ │ +     -- used .00707277 seconds
│ │ │ +     -- used .0918473 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0180006 seconds
│ │ │ -     -- used .319378 seconds
│ │ │ +     -- used .0176603 seconds
│ │ │ +     -- used .248368 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .039338 seconds
│ │ │ -     -- used 1.12239 seconds
│ │ │ +     -- used .0343 seconds
│ │ │ +     -- used .945058 seconds
│ │ │  (49, 217)
│ │ │  
│ │ │                                     2
│ │ │  o4 = {- y    + y   , - y   y    + y   , - y   y    + y   y   , - y   y    +
│ │ │           1,0    0,1     1,1 0,0    1,0     2,1 0,0    2,0 1,0     2,1 1,0  
│ │ │       ------------------------------------------------------------------------
│ │ │       y   y   , - y   y    + y   y   , - y   y    + y   y   , - y   y    +
│ │ ├── ./usr/share/doc/Macaulay2/EquivariantGB/html/_egb__Toric.html
│ │ │ @@ -106,34 +106,34 @@
│ │ │  o3 : RingMap R <-- S
    
│ │ │              
    
│ │ │                
    i4 : G = egbToric(m, OutFile=>stdio)
│ │ │  3
│ │ │ -     -- used .0022445 seconds
│ │ │ -     -- used .0006884 seconds
│ │ │ +     -- used .00183378 seconds
│ │ │ +     -- used .000574381 seconds
│ │ │  (9, 9)
│ │ │  new stuff found
│ │ │  4
│ │ │ -     -- used .00370187 seconds
│ │ │ -     -- used .00494948 seconds
│ │ │ +     -- used .00342142 seconds
│ │ │ +     -- used .00448892 seconds
│ │ │  (16, 26)
│ │ │  new stuff found
│ │ │  5
│ │ │ -     -- used .00769783 seconds
│ │ │ -     -- used .118616 seconds
│ │ │ +     -- used .00707277 seconds
│ │ │ +     -- used .0918473 seconds
│ │ │  (25, 60)
│ │ │  6
│ │ │ -     -- used .0180006 seconds
│ │ │ -     -- used .319378 seconds
│ │ │ +     -- used .0176603 seconds
│ │ │ +     -- used .248368 seconds
│ │ │  (36, 120)
│ │ │  7
│ │ │ -     -- used .039338 seconds
│ │ │ -     -- used 1.12239 seconds
│ │ │ +     -- used .0343 seconds
│ │ │ +     -- used .945058 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 .0022445 seconds
│ │ │ │ -     -- used .0006884 seconds
│ │ │ │ +     -- used .00183378 seconds
│ │ │ │ +     -- used .000574381 seconds
│ │ │ │  (9, 9)
│ │ │ │  new stuff found
│ │ │ │  4
│ │ │ │ -     -- used .00370187 seconds
│ │ │ │ -     -- used .00494948 seconds
│ │ │ │ +     -- used .00342142 seconds
│ │ │ │ +     -- used .00448892 seconds
│ │ │ │  (16, 26)
│ │ │ │  new stuff found
│ │ │ │  5
│ │ │ │ -     -- used .00769783 seconds
│ │ │ │ -     -- used .118616 seconds
│ │ │ │ +     -- used .00707277 seconds
│ │ │ │ +     -- used .0918473 seconds
│ │ │ │  (25, 60)
│ │ │ │  6
│ │ │ │ -     -- used .0180006 seconds
│ │ │ │ -     -- used .319378 seconds
│ │ │ │ +     -- used .0176603 seconds
│ │ │ │ +     -- used .248368 seconds
│ │ │ │  (36, 120)
│ │ │ │  7
│ │ │ │ -     -- used .039338 seconds
│ │ │ │ -     -- used 1.12239 seconds
│ │ │ │ +     -- used .0343 seconds
│ │ │ │ +     -- used .945058 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/EuclideanDistanceDegree/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=40
│ │ │  bnVtZXJpY1dlaWdodEVERGVncmVlKC4uLixTdWJtb2RlbD0+Li4uKQ==
│ │ │  #:len=341
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│ │ ├── ./usr/share/doc/Macaulay2/ExampleSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  aGVhcnQ=
│ │ │  #:len=1516
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gOC1kaW1lbnNpb25hbCBlY29ub21p
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│ │ ├── ./usr/share/doc/Macaulay2/ExteriorExtensions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  YmFzZXM=
│ │ │  #:len=602
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│ │ ├── ./usr/share/doc/Macaulay2/ExteriorIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
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│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  c29sdmVNYWNhdWxheUV4cGFuc2lvbihMaXN0KQ==
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDMwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/ExteriorModules/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  YWxtb3N0U3RhYmxlTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=292
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhbG1vc3RTdGFibGVNb2R1bGUsTW9kdWxlKSwiYWxt
│ │ ├── ./usr/share/doc/Macaulay2/FGLM/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  ZmdsbShJZGVhbCxSaW5nKQ==
│ │ │  #:len=211
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzk2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmZ2xtLElkZWFsLFJpbmcpLCJmZ2xtKElkZWFsLFJp
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlTWlub3JzKFpaLE1hdHJpeCk=
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3NCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVjdXJzaXZlTWlub3JzLFpaLE1hdHJpeCksInJl
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Fast__Minors__Strategy__Tutorial.out
│ │ │ @@ -414,50 +414,50 @@
│ │ │                 3 2 4     3 6
│ │ │  o27 = ideal(12x x x  - 4x x )
│ │ │                 3 7 9     3 9
│ │ │  
│ │ │  o27 : Ideal of S
│ │ │  
│ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ + -- used 0.0519705s (cpu); 0.0544062s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
│ │ │  
│ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ + -- used 0.250956s (cpu); 0.188982s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
│ │ │  
│ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ + -- used 0.355188s (cpu); 0.293825s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
│ │ │  
│ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ + -- used 0.166683s (cpu); 0.169616s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
│ │ │  
│ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ + -- used 0.232028s (cpu); 0.169064s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
│ │ │  
│ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ + -- used 0.258848s (cpu); 0.19744s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
│ │ │  
│ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ + -- used 0.123619s (cpu); 0.124027s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
│ │ │  
│ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ + -- used 11.0934s (cpu); 8.87264s (thread); 0s (gc)
│ │ │  
│ │ │  o35 = 1
│ │ │  
│ │ │  i36 : peek StrategyDefault
│ │ │  
│ │ │  o36 = OptionTable{GRevLexLargest => 0      }
│ │ │                    GRevLexSmallest => 16
│ │ │ @@ -488,15 +488,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ + -- used 0.380774s (cpu); 0.322506s (thread); 0s (gc)
│ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │  
│ │ │  o37 : Ideal of S
│ │ │  
│ │ │  i38 : peek StrategyDefaultNonRandom
│ │ │  
│ │ │  o38 = OptionTable{GRevLexLargest => 0      }
│ │ │ @@ -536,15 +536,15 @@
│ │ │  i41 : ptsStratGeometric = new OptionTable from (options chooseGoodMinors)#PointOptions;
│ │ │  
│ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │  
│ │ │  o42 = true
│ │ │  
│ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ + -- used 0.689331s (cpu); 0.566722s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
│ │ │  
│ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │  
│ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │                    DimensionFunction => dim
│ │ │ @@ -559,49 +559,49 @@
│ │ │  o44 : OptionTable
│ │ │  
│ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │  
│ │ │  o45 = false
│ │ │  
│ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ + -- used 0.574447s (cpu); 0.458716s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
│ │ │  
│ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ + -- used 2.09388s (cpu); 1.84434s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
│ │ │  
│ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ + -- used 1.18495s (cpu); 1.00427s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
│ │ │  
│ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
│ │ │ + -- used 2.91444s (cpu); 2.71878s (thread); 0s (gc)
│ │ │  
│ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
│ │ │ + -- used 2.39803s (cpu); 1.91006s (thread); 0s (gc)
│ │ │  
│ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ + -- used 0.290842s (cpu); 0.293267s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
│ │ │  
│ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
│ │ │ + -- used 2.83106s (cpu); 2.194s (thread); 0s (gc)
│ │ │  
│ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
│ │ │ + -- used 3.39483s (cpu); 2.82774s (thread); 0s (gc)
│ │ │  
│ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ + -- used 50.639s (cpu); 43.1714s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
│ │ │  
│ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ + -- used 2.79141s (cpu); 2.21665s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
│ │ │  
│ │ │  i56 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/___Regular__In__Codimension__Tutorial.out
│ │ │ @@ -7,20 +7,20 @@
│ │ │  o2 : Ideal of S
│ │ │  
│ │ │  i3 : dim (S/J)
│ │ │  
│ │ │  o3 = 4
│ │ │  
│ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ + -- used 3.20956s (cpu); 1.83345s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
│ │ │  
│ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ + -- used 12.9122s (cpu); 7.69404s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time regularInCodimension(1, S/J, Verbose=>true)
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ @@ -127,15 +127,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 3.38226s (cpu); 2.03164s (thread); 0s (gc)
│ │ │  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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -175,15 +175,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ + -- used 0.259069s (cpu); 0.147426s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -197,15 +197,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ + -- used 0.268901s (cpu); 0.148663s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -219,15 +219,15 @@
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ + -- used 0.425667s (cpu); 0.248564s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 6, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -262,15 +262,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ + -- used 1.24082s (cpu); 0.794053s (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
│ │ │  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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 5, and computed = 5
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -313,15 +313,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ + -- used 0.882226s (cpu); 0.556667s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
│ │ ├── ./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)
│ │ │ - -- 4.1651s elapsed
│ │ │ + -- 2.88772s 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)
│ │ │ - -- 1.67457s elapsed
│ │ │ + -- 1.2043s 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 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ + -- used 0.00110915s (cpu); 0.00269432s (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 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ + -- used 0.0040313s (cpu); 0.00270732s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ + -- used 9.0212e-05s (cpu); 0.00263502s (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 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ + -- used 0.303186s (cpu); 0.182563s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ + -- used 0.0817542s (cpu); 0.0246287s (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 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ + -- used 0.492104s (cpu); 0.424466s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
│ │ │  
│ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ + -- used 1.20314s (cpu); 1.08155s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : I1 == I2
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/example-output/_regular__In__Codimension.out
│ │ │ @@ -19,44 +19,44 @@
│ │ │  i6 : S = T/I;
│ │ │  
│ │ │  i7 : dim S
│ │ │  
│ │ │  o7 = 3
│ │ │  
│ │ │  i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ + -- used 0.914639s (cpu); 0.56796s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : time regularInCodimension(2, S)
│ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
│ │ │ + -- used 7.86533s (cpu); 5.06723s (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 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ + -- used 0.0162322s (cpu); 0.0179455s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
│ │ │  
│ │ │  i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ + -- used 0.489791s (cpu); 0.269348s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
│ │ │  
│ │ │  i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ + -- used 0.312116s (cpu); 0.201081s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
│ │ │  
│ │ │  i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ + -- used 0.343079s (cpu); 0.17906s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = true
│ │ │  
│ │ │  i16 : time regularInCodimension(2, S, Verbose=>true)
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ @@ -381,15 +381,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc)
│ │ │ + -- used 7.93361s (cpu); 5.14841s (thread); 0s (gc)
│ │ │  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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -462,15 +462,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc)
│ │ │ + -- used 1.54612s (cpu); 1.02916s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 9, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -486,68 +486,68 @@
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 30, and computed = 25
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 1
│ │ │  regularInCodimension:  Loop completed, submatrices considered = 30, and computed = 25.  singular locus dimension appears to be = 1
│ │ │  
│ │ │  i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ + -- used 1.48633s (cpu); 0.875826s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : StrategyCurrent#Random = 0;
│ │ │  
│ │ │  i20 : StrategyCurrent#LexSmallest = 100;
│ │ │  
│ │ │  i21 : StrategyCurrent#LexSmallestTerm = 0;
│ │ │  
│ │ │  i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ + -- used 0.509945s (cpu); 0.298963s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
│ │ │  
│ │ │  i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ + -- used 0.56291s (cpu); 0.291157s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ + -- used 0.368988s (cpu); 0.250928s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
│ │ │  
│ │ │  i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ + -- used 0.41993s (cpu); 0.248302s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = true
│ │ │  
│ │ │  i26 : StrategyCurrent#LexSmallest = 0;
│ │ │  
│ │ │  i27 : StrategyCurrent#LexSmallestTerm = 100;
│ │ │  
│ │ │  i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
│ │ │ + -- used 2.86732s (cpu); 1.70858s (thread); 0s (gc)
│ │ │  
│ │ │  i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
│ │ │ + -- used 2.79408s (cpu); 1.63083s (thread); 0s (gc)
│ │ │  
│ │ │  i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ + -- used 0.306272s (cpu); 0.201183s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
│ │ │  
│ │ │  i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ + -- used 0.468831s (cpu); 0.314544s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
│ │ │  
│ │ │  i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ + -- used 1.80094s (cpu); 1.23912s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
│ │ │  
│ │ │  i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ + -- used 1.49844s (cpu); 1.01402s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = true
│ │ │  
│ │ │  i34 :
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Fast__Minors__Strategy__Tutorial.html
│ │ │ @@ -577,71 +577,71 @@
│ │ │          
    
│ │ │            
    Here the $1$ passed to the function says how many minors to compute.  For instance, let's compute 8 minors for each of these strategies and see if that was enough to verify that the ring is regular in codimension 1.  In other words, if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/J$ has dimension 3).
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
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│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ + -- used 0.0519705s (cpu); 0.0544062s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = 2
    
│ │ │              
    
│ │ │                
    i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ + -- used 0.250956s (cpu); 0.188982s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = 3
    
│ │ │              
    
│ │ │                
    i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ + -- used 0.355188s (cpu); 0.293825s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 1
    
│ │ │              
    
│ │ │                
    i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ + -- used 0.166683s (cpu); 0.169616s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 2
    
│ │ │              
    
│ │ │                
    i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ + -- used 0.232028s (cpu); 0.169064s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = 3
    
│ │ │              
    
│ │ │                
    i33 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallestTerm))
│ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ + -- used 0.258848s (cpu); 0.19744s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 3
    
│ │ │              
    
│ │ │                
    i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ + -- used 0.123619s (cpu); 0.124027s (thread); 0s (gc)
│ │ │  
│ │ │  o34 = 3
    
│ │ │              
    
│ │ │                
    i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ + -- used 11.0934s (cpu); 8.87264s (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,
    
│ │ │ @@ -688,15 +688,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ + -- used 0.380774s (cpu); 0.322506s (thread); 0s (gc)
│ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │  
│ │ │  o37 : Ideal of S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -779,15 +779,15 @@
│ │ │  
│ │ │  o42 = true
    
│ │ │              
    
│ │ │                
    i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratGeometric))
│ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ + -- used 0.689331s (cpu); 0.566722s (thread); 0s (gc)
│ │ │  
│ │ │  o43 = 2
    
│ │ │              
    
│ │ │                
    i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that value
│ │ │ @@ -811,15 +811,15 @@
│ │ │  
│ │ │  o45 = false
    
│ │ │              
    
│ │ │                
    i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points, PointOptions=>ptsStratRational))
│ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ + -- used 0.574447s (cpu); 0.458716s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = 2
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Other options may also be passed to the RandomPoints package via the PointOptions option.
    
│ │ │ @@ -827,71 +827,71 @@
│ │ │          
    
│ │ │            
    regularInCodimension:  It is reasonable to think that you should find a few minors (with one strategy or another), and see if perhaps the minors you have computed so far are enough to verify our ring is regular in codimension 1.  This is exactly what regularInCodimension does.  One can control at a fine level how frequently new minors are computed, and how frequently the dimension of what we have computed so far is checked, by the option codimCheckFunction.  For more on that, see RegularInCodimensionTutorial and regularInCodimension.  Let us finish running regularInCodimension on our example with several different strategies.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i47 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefault)
│ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ + -- used 2.09388s (cpu); 1.84434s (thread); 0s (gc)
│ │ │  
│ │ │  o47 = true
    
│ │ │              
    
│ │ │                
    i48 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultNonRandom)
│ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ + -- used 1.18495s (cpu); 1.00427s (thread); 0s (gc)
│ │ │  
│ │ │  o48 = true
    
│ │ │              
    
│ │ │                
    i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
    
│ │ │ + -- used 2.91444s (cpu); 2.71878s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i50 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallest)
│ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
    
│ │ │ + -- used 2.39803s (cpu); 1.91006s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i51 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>LexSmallestTerm)
│ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ + -- used 0.290842s (cpu); 0.293267s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = true
    
│ │ │              
    
│ │ │                
    i52 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallest)
│ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
    
│ │ │ + -- used 2.83106s (cpu); 2.194s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i53 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>GRevLexSmallestTerm)
│ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
    
│ │ │ + -- used 3.39483s (cpu); 2.82774s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ + -- used 50.639s (cpu); 43.1714s (thread); 0s (gc)
│ │ │  
│ │ │  o54 = true
    
│ │ │              
    
│ │ │                
    i55 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>StrategyDefaultWithPoints)
│ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ + -- used 2.79141s (cpu); 2.21665s (thread); 0s (gc)
│ │ │  
│ │ │  o55 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If regularInCodimension outputs nothing, then it couldn't verify that the ring was regular in that codimension.  We set MaxMinors => 100 to keep it from running too long with an ineffective strategy.  Again, even though GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular example, in others they perform better than other strategies.  Note similar considerations also apply to projDim.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -438,44 +438,44 @@
│ │ │ │  o27 : Ideal of S
│ │ │ │  Here the $1$ passed to the function says how many minors to compute. For
│ │ │ │  instance, let's compute 8 minors for each of these strategies and see if that
│ │ │ │  was enough to verify that the ring is regular in codimension 1. In other words,
│ │ │ │  if the dimension of $J$ plus the ideal of partial minors is $\leq 1$ (since $S/
│ │ │ │  J$ has dimension 3).
│ │ │ │  i28 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Random))
│ │ │ │ - -- used 0.0399774s (cpu); 0.0434466s (thread); 0s (gc)
│ │ │ │ + -- used 0.0519705s (cpu); 0.0544062s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = 2
│ │ │ │  i29 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallest))
│ │ │ │ - -- used 0.221823s (cpu); 0.174555s (thread); 0s (gc)
│ │ │ │ + -- used 0.250956s (cpu); 0.188982s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = 3
│ │ │ │  i30 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexSmallestTerm))
│ │ │ │ - -- used 0.358134s (cpu); 0.30248s (thread); 0s (gc)
│ │ │ │ + -- used 0.355188s (cpu); 0.293825s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 1
│ │ │ │  i31 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>LexLargest))
│ │ │ │ - -- used 0.169855s (cpu); 0.172074s (thread); 0s (gc)
│ │ │ │ + -- used 0.166683s (cpu); 0.169616s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o31 = 2
│ │ │ │  i32 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexSmallest))
│ │ │ │ - -- used 0.2399s (cpu); 0.172785s (thread); 0s (gc)
│ │ │ │ + -- used 0.232028s (cpu); 0.169064s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o32 = 3
│ │ │ │  i33 : time dim (J + chooseGoodMinors(8, 6, M, J,
│ │ │ │  Strategy=>GRevLexSmallestTerm))
│ │ │ │ - -- used 0.286617s (cpu); 0.214112s (thread); 0s (gc)
│ │ │ │ + -- used 0.258848s (cpu); 0.19744s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o33 = 3
│ │ │ │  i34 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>GRevLexLargest))
│ │ │ │ - -- used 0.135747s (cpu); 0.13629s (thread); 0s (gc)
│ │ │ │ + -- used 0.123619s (cpu); 0.124027s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o34 = 3
│ │ │ │  i35 : time dim (J + chooseGoodMinors(8, 6, M, J, Strategy=>Points))
│ │ │ │ - -- used 12.1654s (cpu); 9.70719s (thread); 0s (gc)
│ │ │ │ + -- used 11.0934s (cpu); 8.87264s (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
│ │ │ │ @@ -518,15 +518,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.415617s (cpu); 0.354159s (thread); 0s (gc)
│ │ │ │ + -- used 0.380774s (cpu); 0.322506s (thread); 0s (gc)
│ │ │ │  chooseGoodMinors: found =20, attempted = 22
│ │ │ │  
│ │ │ │  o37 : Ideal of S
│ │ │ │  we can see different minors being chosen via different strategies.
│ │ │ │  Note, if one asks chooseGoodMinors for more than one minor, then any time a
│ │ │ │  Points strategy is selected, the point is found on $J$ plus the ideal of all
│ │ │ │  minors computed thus far.
│ │ │ │ @@ -587,15 +587,15 @@
│ │ │ │  i41 : ptsStratGeometric = new OptionTable from (options
│ │ │ │  chooseGoodMinors)#PointOptions;
│ │ │ │  i42 : ptsStratGeometric#ExtendField --look at the default value
│ │ │ │  
│ │ │ │  o42 = true
│ │ │ │  i43 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratGeometric))
│ │ │ │ - -- used 0.731279s (cpu); 0.584895s (thread); 0s (gc)
│ │ │ │ + -- used 0.689331s (cpu); 0.566722s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o43 = 2
│ │ │ │  i44 : ptsStratRational = ptsStratGeometric++{ExtendField=>false} --change that
│ │ │ │  value
│ │ │ │  
│ │ │ │  o44 = OptionTable{DecompositionStrategy => Decompose}
│ │ │ │                    DimensionFunction => dim
│ │ │ │ @@ -609,60 +609,60 @@
│ │ │ │  
│ │ │ │  o44 : OptionTable
│ │ │ │  i45 : ptsStratRational.ExtendField --look at our changed value
│ │ │ │  
│ │ │ │  o45 = false
│ │ │ │  i46 : time dim (J + chooseGoodMinors(1, 6, M, J, Strategy=>Points,
│ │ │ │  PointOptions=>ptsStratRational))
│ │ │ │ - -- used 0.628132s (cpu); 0.473813s (thread); 0s (gc)
│ │ │ │ + -- used 0.574447s (cpu); 0.458716s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o46 = 2
│ │ │ │  Other options may also be passed to the _R_a_n_d_o_m_P_o_i_n_t_s package via the
│ │ │ │  _P_o_i_n_t_O_p_t_i_o_n_s option.
│ │ │ │  rreegguullaarrIInnCCooddiimmeennssiioonn:: It is reasonable to think that you should find a few
│ │ │ │  minors (with one strategy or another), and see if perhaps the minors you have
│ │ │ │  computed so far are enough to verify our ring is regular in codimension 1. This
│ │ │ │  is exactly what regularInCodimension does. One can control at a fine level how
│ │ │ │  frequently new minors are computed, and how frequently the dimension of what we
│ │ │ │  have computed so far is checked, by the option codimCheckFunction. For more on
│ │ │ │  that, see _R_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n_T_u_t_o_r_i_a_l and _r_e_g_u_l_a_r_I_n_C_o_d_i_m_e_n_s_i_o_n. Let us finish
│ │ │ │  running regularInCodimension on our example with several different strategies.
│ │ │ │  i47 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefault)
│ │ │ │ - -- used 2.16725s (cpu); 1.89894s (thread); 0s (gc)
│ │ │ │ + -- used 2.09388s (cpu); 1.84434s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o47 = true
│ │ │ │  i48 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultNonRandom)
│ │ │ │ - -- used 1.2841s (cpu); 1.07067s (thread); 0s (gc)
│ │ │ │ + -- used 1.18495s (cpu); 1.00427s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o48 = true
│ │ │ │  i49 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Random)
│ │ │ │ - -- used 2.82661s (cpu); 2.5335s (thread); 0s (gc)
│ │ │ │ + -- used 2.91444s (cpu); 2.71878s (thread); 0s (gc)
│ │ │ │  i50 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallest)
│ │ │ │ - -- used 2.82185s (cpu); 2.12806s (thread); 0s (gc)
│ │ │ │ + -- used 2.39803s (cpu); 1.91006s (thread); 0s (gc)
│ │ │ │  i51 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>LexSmallestTerm)
│ │ │ │ - -- used 0.282679s (cpu); 0.282995s (thread); 0s (gc)
│ │ │ │ + -- used 0.290842s (cpu); 0.293267s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o51 = true
│ │ │ │  i52 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallest)
│ │ │ │ - -- used 2.96165s (cpu); 2.31539s (thread); 0s (gc)
│ │ │ │ + -- used 2.83106s (cpu); 2.194s (thread); 0s (gc)
│ │ │ │  i53 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>GRevLexSmallestTerm)
│ │ │ │ - -- used 3.46682s (cpu); 2.82813s (thread); 0s (gc)
│ │ │ │ + -- used 3.39483s (cpu); 2.82774s (thread); 0s (gc)
│ │ │ │  i54 : time regularInCodimension(1, S/J, MaxMinors => 100, Strategy=>Points)
│ │ │ │ - -- used 63.8248s (cpu); 51.3047s (thread); 0s (gc)
│ │ │ │ + -- used 50.639s (cpu); 43.1714s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o54 = true
│ │ │ │  i55 : time regularInCodimension(1, S/J, MaxMinors => 100,
│ │ │ │  Strategy=>StrategyDefaultWithPoints)
│ │ │ │ - -- used 2.98865s (cpu); 2.35025s (thread); 0s (gc)
│ │ │ │ + -- used 2.79141s (cpu); 2.21665s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o55 = true
│ │ │ │  If regularInCodimension outputs nothing, then it couldn't verify that the ring
│ │ │ │  was regular in that codimension. We set MaxMinors => 100 to keep it from
│ │ │ │  running too long with an ineffective strategy. Again, even though
│ │ │ │  GRevLexSmallest and GRevLexSmallestTerm are not effective in this particular
│ │ │ │  example, in others they perform better than other strategies. Note similar
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Regular__In__Codimension__Tutorial.html
│ │ │ @@ -86,23 +86,23 @@
│ │ │          
    
│ │ │            
    It is the cone over $P^2 \times E$ where $E$ is an elliptic curve.  We have embedded it with a Segre embedding inside $P^8$.  In particular, this example is even regular in codimension 3.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : time regularInCodimension(1, S/J)
│ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ + -- used 3.20956s (cpu); 1.83345s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = true
    
│ │ │              
    
│ │ │                
    i5 : time regularInCodimension(2, S/J)
│ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ + -- used 12.9122s (cpu); 7.69404s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the ideal made up of a small number of minors of the Jacobian matrix. In this example, instead of computing all relevant 1465128 minors to compute the singular locus, and then trying to compute the dimension of the ideal they generate, we instead compute a few of them.  regularInCodimension returns true if it verified that the ring is regular in codim 1 or 2 (respectively) and null if not.  Because of the randomness that exists in terms of selecting minors, the execution time can actually vary quite a bit.   Let's take a look at what is occurring by using the Verbose option.  We go through the output and explain what each line is telling us.
    
│ │ │ @@ -217,15 +217,15 @@
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ + -- used 3.38226s (cpu); 2.03164s (thread); 0s (gc)
│ │ │  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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 9
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -273,15 +273,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ + -- used 0.259069s (cpu); 0.147426s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -306,15 +306,15 @@
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ + -- used 0.268901s (cpu); 0.148663s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 10, and computed = 10
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -339,15 +339,15 @@
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ + -- used 0.425667s (cpu); 0.248564s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 3, and computed = 3
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 6, and computed = 6
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -393,15 +393,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing Random
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ + -- used 1.24082s (cpu); 0.794053s (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
│ │ │  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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 5, and computed = 5
│ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ @@ -452,15 +452,15 @@
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ + -- used 0.882226s (cpu); 0.556667s (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
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 7, and computed = 7
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 11, and computed = 11
│ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices considered: 15, and computed = 14
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,19 +24,19 @@
│ │ │ │  i3 : dim (S/J)
│ │ │ │  
│ │ │ │  o3 = 4
│ │ │ │  It is the cone over $P^2 \times E$ where $E$ is an elliptic curve. We have
│ │ │ │  embedded it with a Segre embedding inside $P^8$. In particular, this example is
│ │ │ │  even regular in codimension 3.
│ │ │ │  i4 : time regularInCodimension(1, S/J)
│ │ │ │ - -- used 3.94924s (cpu); 2.5242s (thread); 0s (gc)
│ │ │ │ + -- used 3.20956s (cpu); 1.83345s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : time regularInCodimension(2, S/J)
│ │ │ │ - -- used 14.4632s (cpu); 9.06112s (thread); 0s (gc)
│ │ │ │ + -- used 12.9122s (cpu); 7.69404s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  We try to verify that $S/J$ is regular in codimension 1 or 2 by computing the
│ │ │ │  ideal made up of a small number of minors of the Jacobian matrix. In this
│ │ │ │  example, instead of computing all relevant 1465128 minors to compute the
│ │ │ │  singular locus, and then trying to compute the dimension of the ideal they
│ │ │ │  generate, we instead compute a few of them. regularInCodimension returns true
│ │ │ │ @@ -152,15 +152,15 @@
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │ - -- used 3.66634s (cpu); 2.3306s (thread); 0s (gc)
│ │ │ │ + -- used 3.38226s (cpu); 2.03164s (thread); 0s (gc)
│ │ │ │  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
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 6
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -213,15 +213,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │ - -- used 0.31421s (cpu); 0.194894s (thread); 0s (gc)
│ │ │ │ + -- used 0.259069s (cpu); 0.147426s (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
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -244,15 +244,15 @@
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.265969s (cpu); 0.170823s (thread); 0s (gc)
│ │ │ │ + -- used 0.268901s (cpu); 0.148663s (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
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -282,15 +282,15 @@
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │ - -- used 0.436288s (cpu); 0.287639s (thread); 0s (gc)
│ │ │ │ + -- used 0.425667s (cpu); 0.248564s (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
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 3, and computed = 3
│ │ │ │  regularInCodimension:  isCodimAtLeast failed, computing codim.
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 3
│ │ │ │ @@ -343,15 +343,15 @@
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing Random
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │ - -- used 1.22231s (cpu); 0.809378s (thread); 0s (gc)
│ │ │ │ + -- used 1.24082s (cpu); 0.794053s (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
│ │ │ │  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
│ │ │ │ @@ -410,15 +410,15 @@
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing LexSmallest
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallestTerm
│ │ │ │   -- internalChooseMinor: Choosing GRevLexSmallest
│ │ │ │   -- internalChooseMinor: Choosing RandomNonZero
│ │ │ │ - -- used 1.07283s (cpu); 0.688614s (thread); 0s (gc)
│ │ │ │ + -- used 0.882226s (cpu); 0.556667s (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
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ │ │  considered: 7, and computed = 7
│ │ │ │  regularInCodimension:  partial singular locus dimension computed, = 4
│ │ │ │  regularInCodimension:  Loop step, about to compute dimension.  Submatrices
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/___Strategy__Default.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │              
    		
│ │ │                
    i1 : T=ZZ/7[a..i]/ideal(f*h-e*i,c*h-b*i,f*g-d*i,e*g-d*h,c*g-a*i,b*g-a*h,c*e-b*f,c*d-a*f,b*d-a*e,g^3-h^2*i-g*i^2,d*g^2-e*h*i-d*i^2,a*g^2-b*h*i-a*i^2,d^2*g-e^2*i-d*f*i,a*d*g-b*e*i-a*f*i,a^2*g-b^2*i-a*c*i,d^3-e^2*f-d*f^2,a*d^2-b*e*f-a*f^2,a^2*d-b^2*f-a*c*f,c^3+f^3-i^3,b*c^2+e*f^2-h*i^2,a*c^2+d*f^2-g*i^2,b^2*c+e^2*f-h^2*i,a*b*c+d*e*f-g*h*i,a^2*c+d^2*f-g^2*i,b^3+e^3-h^3,a*b^2+d*e^2-g*h^2,a^2*b+d^2*e-g^2*h,a^3+e^2*f+d*f^2-h^2*i-g*i^2);
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime regularInCodimension(1, T, Strategy=>StrategyDefault)
│ │ │ - -- 4.1651s elapsed
│ │ │ + -- 2.88772s elapsed
│ │ │  
│ │ │  o2 = true
    
│ │ │              
    
│ │ │  In this particular example, on one machine, we list average time to completion of each of the above strategies after 100 runs.        
      
│ │ │            
    • StrategyDefault: 1.65 seconds
      • 
│ │ │ @@ -127,15 +127,15 @@
│ │ │            
    • StrategyPoints: choose all submatrices via Points.
      • 
│ │ │            
    • StrategyDefaultWithPoints: like StrategyDefault but replaces the Random and RandomNonZero submatrices as with matrices chosen as in Points.
      • 
│ │ │          
    
│ │ │  Additionally, a MutableHashTable named StrategyCurrent is also exported.  It begins as the default strategy, but the user can modify it.

    Using a single heuristic  Alternatively, if the user only wants to use say LexSmallestTerm they can set, Strategy to point to that symbol, instead of a creating a custom strategy HashTable.  For example:         
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime regularInCodimension(1, T, Strategy=>LexSmallestTerm)
│ │ │ - -- 1.67457s elapsed
│ │ │ + -- 1.2043s 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)
│ │ │ │ - -- 4.1651s elapsed
│ │ │ │ + -- 2.88772s 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)
│ │ │ │ - -- 1.67457s elapsed
│ │ │ │ + -- 1.2043s elapsed
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _S_t_r_a_t_e_g_y_D_e_f_a_u_l_t is an _o_p_t_i_o_n_ _t_a_b_l_e.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/FastMinors.m2:2037:0.
│ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_is__Codim__At__Least.html
│ │ │ @@ -119,15 +119,15 @@
│ │ │  
│ │ │  o6 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isCodimAtLeast(3, J)
│ │ │ - -- used 0.000947066s (cpu); 0.00263955s (thread); 0s (gc)
│ │ │ + -- used 0.00110915s (cpu); 0.00269432s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    The function works by computing gb(I, PairLimit=>f(i)) for successive values of i. Here f(i) is a function that takes t, some approximation of the base degree value of the polynomial ring (for example, in a standard graded polynomial ring, this is probably expected to be \{1\}). And i is a counting variable. You can provide your own function by calling isCodimAtLeast(n, I, SPairsFunction=>( (i) -> f(i) ), the default function is SPairsFunction=>i->ceiling(1.5^i) Perhaps more commonly however, the user may want to instead tell the function to compute for larger values of i. This is done via the option PairLimit. This is the max value of i to be plugged into SPairsFunction before the function gives up. In other words, PairLimit=>5 will tell the function to check codimension 5 times.

    │ │ │ @@ -141,24 +141,24 @@ │ │ │ o8 : Ideal of ---[x , x , x , x , x , x , x , x , x , x , x , x ] │ │ │ 127 11 8 1 9 12 6 5 10 2 4 3 7 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time isCodimAtLeast(5, I, PairLimit => 5, Verbose=>true)
│ │ │ - -- used 0.00400074s (cpu); 0.00251447s (thread); 0s (gc)
│ │ │ + -- used 0.0040313s (cpu); 0.00270732s (thread); 0s (gc)
│ │ │  isCodimAtLeast: Computing codim of monomials based on ideal generators.
│ │ │  
│ │ │  o9 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false)
│ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc)
│ │ │ + -- used 9.0212e-05s (cpu); 0.00263502s (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 0.000947066s (cpu); 0.00263955s (thread); 0s (gc) │ │ │ │ + -- used 0.00110915s (cpu); 0.00269432s (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 0.00400074s (cpu); 0.00251447s (thread); 0s (gc) │ │ │ │ + -- used 0.0040313s (cpu); 0.00270732s (thread); 0s (gc) │ │ │ │ isCodimAtLeast: Computing codim of monomials based on ideal generators. │ │ │ │ │ │ │ │ o9 = true │ │ │ │ i10 : time isCodimAtLeast(5, I, PairLimit => 200, Verbose=>false) │ │ │ │ - -- used 6.5373e-05s (cpu); 0.0022963s (thread); 0s (gc) │ │ │ │ + -- used 9.0212e-05s (cpu); 0.00263502s (thread); 0s (gc) │ │ │ │ │ │ │ │ o10 = true │ │ │ │ Notice in the first case the function returned null, because the depth of │ │ │ │ search was not high enough. It only computed codim 5 times. The second returned │ │ │ │ true, but it did so as soon as the answer was found (and before we hit the │ │ │ │ PairLimit limit). │ │ │ │ ********** WWaayyss ttoo uussee iissCCooddiimmAAttLLeeaasstt:: ********** │ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_proj__Dim.html │ │ │ @@ -104,23 +104,23 @@ │ │ │ │ │ │ o3 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time projDim(module I, Strategy=>StrategyRandom)
│ │ │ - -- used 0.296971s (cpu); 0.159933s (thread); 0s (gc)
│ │ │ + -- used 0.303186s (cpu); 0.182563s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1)
│ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc)
│ │ │ + -- used 0.0817542s (cpu); 0.0246287s (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 0.296971s (cpu); 0.159933s (thread); 0s (gc) │ │ │ │ + -- used 0.303186s (cpu); 0.182563s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 1 │ │ │ │ i5 : time projDim(module I, Strategy=>StrategyRandom, MinDimension => 1) │ │ │ │ - -- used 0.00988389s (cpu); 0.0120097s (thread); 0s (gc) │ │ │ │ + -- used 0.0817542s (cpu); 0.0246287s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 1 │ │ │ │ The option MaxMinors can be used to control how many minors are computed at │ │ │ │ each step. If this is not specified, the number of minors is a function of the │ │ │ │ dimension $d$ of the polynomial ring and the possible minors $c$. Specifically │ │ │ │ it is 10 * d + 2 * log_1.3(c). Otherwise the user can set the option MaxMinors │ │ │ │ => ZZ to specify that a fixed integer is used for each step. Alternatively, the │ │ ├── ./usr/share/doc/Macaulay2/FastMinors/html/_recursive__Minors.html │ │ │ @@ -97,23 +97,23 @@ │ │ │ 6 7 │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time I2 = recursiveMinors(4, M, Threads=>0);
│ │ │ - -- used 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ + -- used 0.492104s (cpu); 0.424466s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ + -- used 1.20314s (cpu); 1.08155s (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 0.555663s (cpu); 0.481264s (thread); 0s (gc)
│ │ │ │ + -- used 0.492104s (cpu); 0.424466s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : time I1 = minors(4, M, Strategy=>Cofactor);
│ │ │ │ - -- used 1.57748s (cpu); 1.37879s (thread); 0s (gc)
│ │ │ │ + -- used 1.20314s (cpu); 1.08155s (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
│ │ │ @@ -139,23 +139,23 @@
│ │ │  
│ │ │  o7 = 3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time regularInCodimension(1, S)
│ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc)
│ │ │ + -- used 0.914639s (cpu); 0.56796s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time regularInCodimension(2, S)
│ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc)
    │ │ │ + -- used 7.86533s (cpu); 5.06723s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    There are numerous examples where regularInCodimension is several orders of magnitude faster that calls of dim singularLocus.

    │ │ │
    │ │ │
    │ │ │ @@ -173,39 +173,39 @@ │ │ │ │ │ │ o11 = 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time (dim singularLocus (R))
│ │ │ - -- used 0.0279591s (cpu); 0.0261627s (thread); 0s (gc)
│ │ │ + -- used 0.0162322s (cpu); 0.0179455s (thread); 0s (gc)
│ │ │  
│ │ │  o12 = -1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time regularInCodimension(2, R)
│ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc)
│ │ │ + -- used 0.489791s (cpu); 0.269348s (thread); 0s (gc)
│ │ │  
│ │ │  o13 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time regularInCodimension(2, R)
│ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc)
│ │ │ + -- used 0.312116s (cpu); 0.201081s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time regularInCodimension(2, R)
│ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc)
│ │ │ + -- used 0.343079s (cpu); 0.17906s (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.

    │ │ │ @@ -538,15 +538,15 @@ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc) │ │ │ + -- used 7.93361s (cpu); 5.14841s (thread); 0s (gc) │ │ │ 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 │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 9, and computed = 9 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 11, and computed = 10 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ @@ -627,15 +627,15 @@ │ │ │ -- internalChooseMinor: Choosing Random │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ -- internalChooseMinor: Choosing RandomNonZero │ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc) │ │ │ + -- used 1.54612s (cpu); 1.02916s (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 │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 9, and computed = 9 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices considered: 11, and computed = 11 │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ @@ -659,15 +659,15 @@ │ │ │
    │ │ │

    If you set the option VerifyNonRegular => true, then Macaulay2 will try to verify that the ring is not regular in codimension n. Turning this on means that when the set where the minors computed so far has codimension n, then it evaluates the matrix at the generic point of a minimal prime of that set. If that evaluated Jacobian matrix has too low of a rank, then one has verified that variety is not regular in codimemsion n. We consider the same example as above, but notice now the function returns false instead of true. This sometimes can be slower and sometimes can be faster.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i18 : time regularInCodimension(2, S, VerifyNonRegular=>true)
│ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc)
│ │ │ + -- used 1.48633s (cpu); 0.875826s (thread); 0s (gc)
│ │ │  
│ │ │  o18 = false
    │ │ │
    │ │ │
    │ │ │

    This function has many options which allow you to fine tune the strategy used to find interesting minors. You can pass it a HashTable specifying the strategy via the option Strategy. See LexSmallest for how to construct this HashTable. The default strategy is StrategyDefault, which seems to work well on the examples we have explored. However, caution must be exercised, because, even in the examples above, certain strategies work well while others do not. In the Abelian surface example, LexSmallest works very well, while LexSmallestTerm does not even typically correctly identify the ring as nonsingular (this is 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.

    │ │ │ @@ -687,39 +687,39 @@ │ │ │ │ │ │ 
    i21 : StrategyCurrent#LexSmallestTerm = 0;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc)
│ │ │ + -- used 0.509945s (cpu); 0.298963s (thread); 0s (gc)
│ │ │  
│ │ │  o22 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc)
│ │ │ + -- used 0.56291s (cpu); 0.291157s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc)
│ │ │ + -- used 0.368988s (cpu); 0.250928s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc)
│ │ │ + -- used 0.41993s (cpu); 0.248302s (thread); 0s (gc)
│ │ │  
│ │ │  o25 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : StrategyCurrent#LexSmallest = 0;
    │ │ │ @@ -729,51 +729,51 @@ │ │ │ │ │ │ 
    i27 : StrategyCurrent#LexSmallestTerm = 100;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc)
    │ │ │ + -- used 2.86732s (cpu); 1.70858s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent)
│ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc)
    │ │ │ + -- used 2.79408s (cpu); 1.63083s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc)
│ │ │ + -- used 0.306272s (cpu); 0.201183s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent)
│ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc)
│ │ │ + -- used 0.468831s (cpu); 0.314544s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc)
│ │ │ + -- used 1.80094s (cpu); 1.23912s (thread); 0s (gc)
│ │ │  
│ │ │  o32 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom)
│ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc)
│ │ │ + -- used 1.49844s (cpu); 1.01402s (thread); 0s (gc)
│ │ │  
│ │ │  o33 = 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 {} │ │ │ │ @@ -77,19 +77,19 @@ │ │ │ │ │ │ │ │ o5 : Ideal of T │ │ │ │ i6 : S = T/I; │ │ │ │ i7 : dim S │ │ │ │ │ │ │ │ o7 = 3 │ │ │ │ i8 : time regularInCodimension(1, S) │ │ │ │ - -- used 0.921914s (cpu); 0.617916s (thread); 0s (gc) │ │ │ │ + -- used 0.914639s (cpu); 0.56796s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = true │ │ │ │ i9 : time regularInCodimension(2, S) │ │ │ │ - -- used 10.1044s (cpu); 6.75793s (thread); 0s (gc) │ │ │ │ + -- used 7.86533s (cpu); 5.06723s (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. │ │ │ │ @@ -97,27 +97,27 @@ │ │ │ │ (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 0.0279591s (cpu); 0.0261627s (thread); 0s (gc) │ │ │ │ + -- used 0.0162322s (cpu); 0.0179455s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 = -1 │ │ │ │ i13 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.559038s (cpu); 0.316642s (thread); 0s (gc) │ │ │ │ + -- used 0.489791s (cpu); 0.269348s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.441343s (cpu); 0.303101s (thread); 0s (gc) │ │ │ │ + -- used 0.312116s (cpu); 0.201081s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : time regularInCodimension(2, R) │ │ │ │ - -- used 0.424787s (cpu); 0.217971s (thread); 0s (gc) │ │ │ │ + -- used 0.343079s (cpu); 0.17906s (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) │ │ │ │ @@ -445,15 +445,15 @@ │ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ │ -- internalChooseMinor: Choosing LexSmallest │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ - -- used 9.54962s (cpu); 6.2698s (thread); 0s (gc) │ │ │ │ + -- used 7.93361s (cpu); 5.14841s (thread); 0s (gc) │ │ │ │ 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 │ │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices │ │ │ │ considered: 9, and computed = 9 │ │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ │ @@ -550,15 +550,15 @@ │ │ │ │ -- internalChooseMinor: Choosing Random │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallest │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing LexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing GRevLexSmallestTerm │ │ │ │ -- internalChooseMinor: Choosing RandomNonZero │ │ │ │ - -- used 1.69942s (cpu); 1.13047s (thread); 0s (gc) │ │ │ │ + -- used 1.54612s (cpu); 1.02916s (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 │ │ │ │ regularInCodimension: Loop step, about to compute dimension. Submatrices │ │ │ │ considered: 9, and computed = 9 │ │ │ │ regularInCodimension: isCodimAtLeast failed, computing codim. │ │ │ │ regularInCodimension: partial singular locus dimension computed, = 1 │ │ │ │ @@ -589,15 +589,15 @@ │ │ │ │ that when the set where the minors computed so far has codimension n, then it │ │ │ │ evaluates the matrix at the generic point of a minimal prime of that set. If │ │ │ │ that evaluated Jacobian matrix has too low of a rank, then one has verified │ │ │ │ that variety is not regular in codimemsion n. We consider the same example as │ │ │ │ above, but notice now the function returns false instead of true. This │ │ │ │ sometimes can be slower and sometimes can be faster. │ │ │ │ i18 : time regularInCodimension(2, S, VerifyNonRegular=>true) │ │ │ │ - -- used 1.63296s (cpu); 0.972645s (thread); 0s (gc) │ │ │ │ + -- used 1.48633s (cpu); 0.875826s (thread); 0s (gc) │ │ │ │ │ │ │ │ o18 = false │ │ │ │ This function has many options which allow you to fine tune the strategy used │ │ │ │ to find interesting minors. You can pass it a HashTable specifying the strategy │ │ │ │ via the option Strategy. See _L_e_x_S_m_a_l_l_e_s_t for how to construct this HashTable. │ │ │ │ The default strategy is StrategyDefault, which seems to work well on the │ │ │ │ examples we have explored. However, caution must be exercised, because, even in │ │ │ │ @@ -607,49 +607,49 @@ │ │ │ │ because there are a small number of entries with nonzero constant terms, which │ │ │ │ are selected repeatedly). However, in our first example, the LexSmallestTerm is │ │ │ │ much faster, and Random does not perform well at all. │ │ │ │ i19 : StrategyCurrent#Random = 0; │ │ │ │ i20 : StrategyCurrent#LexSmallest = 100; │ │ │ │ i21 : StrategyCurrent#LexSmallestTerm = 0; │ │ │ │ i22 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.597363s (cpu); 0.322568s (thread); 0s (gc) │ │ │ │ + -- used 0.509945s (cpu); 0.298963s (thread); 0s (gc) │ │ │ │ │ │ │ │ o22 = true │ │ │ │ i23 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.706068s (cpu); 0.366625s (thread); 0s (gc) │ │ │ │ + -- used 0.56291s (cpu); 0.291157s (thread); 0s (gc) │ │ │ │ │ │ │ │ o23 = true │ │ │ │ i24 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.49826s (cpu); 0.35781s (thread); 0s (gc) │ │ │ │ + -- used 0.368988s (cpu); 0.250928s (thread); 0s (gc) │ │ │ │ │ │ │ │ o24 = true │ │ │ │ i25 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.532771s (cpu); 0.31747s (thread); 0s (gc) │ │ │ │ + -- used 0.41993s (cpu); 0.248302s (thread); 0s (gc) │ │ │ │ │ │ │ │ o25 = true │ │ │ │ i26 : StrategyCurrent#LexSmallest = 0; │ │ │ │ i27 : StrategyCurrent#LexSmallestTerm = 100; │ │ │ │ i28 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 3.58881s (cpu); 2.13991s (thread); 0s (gc) │ │ │ │ + -- used 2.86732s (cpu); 1.70858s (thread); 0s (gc) │ │ │ │ i29 : time regularInCodimension(2, R, Strategy=>StrategyCurrent) │ │ │ │ - -- used 3.21572s (cpu); 1.86136s (thread); 0s (gc) │ │ │ │ + -- used 2.79408s (cpu); 1.63083s (thread); 0s (gc) │ │ │ │ i30 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.328318s (cpu); 0.225783s (thread); 0s (gc) │ │ │ │ + -- used 0.306272s (cpu); 0.201183s (thread); 0s (gc) │ │ │ │ │ │ │ │ o30 = true │ │ │ │ i31 : time regularInCodimension(1, S, Strategy=>StrategyCurrent) │ │ │ │ - -- used 0.566594s (cpu); 0.371181s (thread); 0s (gc) │ │ │ │ + -- used 0.468831s (cpu); 0.314544s (thread); 0s (gc) │ │ │ │ │ │ │ │ o31 = true │ │ │ │ i32 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 2.2427s (cpu); 1.62597s (thread); 0s (gc) │ │ │ │ + -- used 1.80094s (cpu); 1.23912s (thread); 0s (gc) │ │ │ │ │ │ │ │ o32 = true │ │ │ │ i33 : time regularInCodimension(1, S, Strategy=>StrategyRandom) │ │ │ │ - -- used 1.78805s (cpu); 1.22068s (thread); 0s (gc) │ │ │ │ + -- used 1.49844s (cpu); 1.01402s (thread); 0s (gc) │ │ │ │ │ │ │ │ o33 = 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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bmV4dERlZ3JlZShNYXRyaXgsWlosUmluZyk= │ │ │ #:len=288 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhuZXh0RGVncmVlLE1hdHJpeCxaWixSaW5nKSwibmV4 │ │ ├── ./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 0.00253889s (cpu); 0.00253616s (thread); 0s (gc) │ │ │ + -- used 0.00241787s (cpu); 0.00241593s (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 0.00662056s (cpu); 0.00662794s (thread); 0s (gc) │ │ │ + -- used 0.00566586s (cpu); 0.00566901s (thread); 0s (gc) │ │ │ │ │ │ o16 : Ideal of R │ │ │ │ │ │ i17 : I==J │ │ │ │ │ │ o17 = true │ │ ├── ./usr/share/doc/Macaulay2/FiniteFittingIdeals/html/___Fitting_spideals_spof_spfinite_spmodules.html │ │ │ @@ -207,26 +207,26 @@ │ │ │ 8 8 │ │ │ o14 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time I=co1Fitting(K3)
│ │ │ - -- used 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ + -- used 0.00241787s (cpu); 0.00241593s (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 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ + -- used 0.00566586s (cpu); 0.00566901s (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 0.00253889s (cpu); 0.00253616s (thread); 0s (gc)
│ │ │ │ + -- used 0.00241787s (cpu); 0.00241593s (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 0.00662056s (cpu); 0.00662794s (thread); 0s (gc)
│ │ │ │ + -- used 0.00566586s (cpu); 0.00566901s (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/FirstPackage/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  Rmlyc3RQYWNrYWdl
│ │ │  #:len=509
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZXhhbXBsZSBNYWNhdWxheTIgcGFj
│ │ │  a2FnZSIsICJsaW5lbnVtIiA9PiA1MywgImZpbGVuYW1lIiA9PiAiRmlyc3RQYWNrYWdlLm0yIiwg
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  dmFsdWUodWludDE2KQ==
│ │ │  #:len=273
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTcxOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodmFsdWUsdWludDE2KSwidmFsdWUodWludDE2KSIs
│ │ ├── ./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 => 0x7f40f21d9050}
│ │ │ +o2 = int32{Address => 0x7f850e975650}
│ │ │  
│ │ │  i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9050
│ │ │ +o3 = 0x7f850e975650
│ │ │  
│ │ │  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 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ +o3 = {0x7f850e9b0fd0, 0x7f850e9b0fc0, 0x7f850e9b0fb0}
│ │ │  
│ │ │  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 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ +o2 = {0x7f850e9b0de0, 0x7f850e9b0dd0, 0x7f850e9b0dc0}
│ │ │  
│ │ │  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 = 0x7f40f26a3da0
│ │ │ +o1 = 0x7f850ee220b0
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f40f26a3da0
│ │ │ +o2 = 0x7f850ee220b0
│ │ │  
│ │ │  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 = 0x7f40f21d97f0
│ │ │ +o2 = 0x7f850e975cd0
│ │ │  
│ │ │  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 = 0x7f40f21d91f0
│ │ │ +o1 = 0x7f850e98f9f0
│ │ │  
│ │ │  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" => 6.91261e-310}
│ │ │ +o2 = HashTable{"bar" => 6.92726e-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 => 0x7f40f21d9960}
│ │ │ +o2 = int32{Address => 0x7f850e9754f0}
│ │ │  
│ │ │  i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9960
│ │ │ +o3 = 0x7f850e9754f0
│ │ │  
│ │ │  o3 : Pointer
│ │ │  
│ │ │  i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f40f21d9965
│ │ │ +o4 = 0x7f850e9754f5
│ │ │  
│ │ │  o4 : Pointer
│ │ │  
│ │ │  i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f40f21d995d
│ │ │ +o5 = 0x7f850e9754ed
│ │ │  
│ │ │  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{0x7f40f9b43ac0, mpfr}
│ │ │ +o2 = SharedLibrary{0x7f85231deac0, 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 = 0x7f40f21d9220
│ │ │ +o2 = 0x7f850e9756d0
│ │ │  
│ │ │  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 = 0x562fbd063080
│ │ │ +o1 = 0x56009c05e080
│ │ │  
│ │ │  o1 : Pointer
│ │ │  
│ │ │  i2 : address int 5
│ │ │  
│ │ │ -o2 = 0x7f40f21d95f0
│ │ │ +o2 = 0x7f850e975b00
│ │ │  
│ │ │  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 = 0x7f0558070f60
│ │ │ +o17 = 0x7f22f8070f60
│ │ │  
│ │ │  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 = 0x7f40f50ca360
│ │ │ +o1 = 0x7f850f41fca0
│ │ │  
│ │ │  o1 : ForeignObject of type void*
│ │ │  
│ │ │  i2 : ptr = getMemory(8, Atomic => true)
│ │ │  
│ │ │ -o2 = 0x7f40f21d9ae0
│ │ │ +o2 = 0x7f850e94f180
│ │ │  
│ │ │  o2 : ForeignObject of type void*
│ │ │  
│ │ │  i3 : ptr = getMemory int
│ │ │  
│ │ │ -o3 = 0x7f40f21d9a00
│ │ │ +o3 = 0x7f850e94f010
│ │ │  
│ │ │  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 0x7f40dc0848b0
│ │ │ -freeing memory at 0x7f40dc084870
│ │ │ -freeing memory at 0x7f40dc084830
│ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ -freeing memory at 0x7f40dc084810
│ │ │ -freeing memory at 0x7f40dc084850
│ │ │ -freeing memory at 0x7f40dc084890
│ │ │ +freeing memory at 0x7f84f8084890
│ │ │ +freeing memory at 0x7f84f80848b0
│ │ │ +freeing memory at 0x7f84f8084870
│ │ │ +freeing memory at 0x7f84f80841d0
│ │ │ +freeing memory at 0x7f84f8084810
│ │ │ +freeing memory at 0x7f84f8084830
│ │ │ +freeing memory at 0x7f84f8084850
│ │ │ +freeing memory at 0x7f84f80841f0
│ │ │ +freeing memory at 0x7f84f80848d0
│ │ │  
│ │ │  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 = 0x7f40f21fd990
│ │ │ +o5 = 0x7f850e98fc30
│ │ │  
│ │ │  o5 : ForeignObject of type void*
│ │ │  
│ │ │  i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f40f21fd990
│ │ │ +o6 = 0x7f850e98fc30
│ │ │  
│ │ │  o6 : Pointer
│ │ │  
│ │ │  i7 : x = charstar "Hello, world!"
│ │ │  
│ │ │  o7 = Hello, world!
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Object.html
│ │ │ @@ -69,27 +69,27 @@
│ │ │  o1 : ForeignObject of type int32
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9050}
    │ │ │ +o2 = int32{Address => 0x7f850e975650} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    To get this, use address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9050
│ │ │ +o3 = 0x7f850e975650
│ │ │  
│ │ │  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 => 0x7f40f21d9050} │ │ │ │ +o2 = int32{Address => 0x7f850e975650} │ │ │ │ To get this, use _a_d_d_r_e_s_s. │ │ │ │ i3 : address x │ │ │ │ │ │ │ │ -o3 = 0x7f40f21d9050 │ │ │ │ +o3 = 0x7f850e975650 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Use _c_l_a_s_s to determine the type of the object. │ │ │ │ i4 : class x │ │ │ │ │ │ │ │ o4 = int32 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type.html │ │ │ @@ -79,15 +79,15 @@ │ │ │ o2 : ForeignObject of type char** │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o3 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70}
│ │ │ +o3 = {0x7f850e9b0fd0, 0x7f850e9b0fc0, 0x7f850e9b0fb0}
│ │ │  
│ │ │  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 = {0x7f40f2217e90, 0x7f40f2217e80, 0x7f40f2217e70} │ │ │ │ +o3 = {0x7f850e9b0fd0, 0x7f850e9b0fc0, 0x7f850e9b0fb0} │ │ │ │ │ │ │ │ o3 : ForeignObject of type void** │ │ │ │ Foreign pointer arrays may be subscripted using __. │ │ │ │ i4 : x = charstarstar {"foo", "bar", "baz"} │ │ │ │ │ │ │ │ o4 = {foo, bar, baz} │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Array__Type_sp__Visible__List.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ o1 : ForeignObject of type char** │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : voidstarstar {address int 0, address int 1, address int 2}
│ │ │  
│ │ │ -o2 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ +o2 = {0x7f850e9b0de0, 0x7f850e9b0dd0, 0x7f850e9b0dc0}
│ │ │  
│ │ │  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 = {0x7f40f2217d00, 0x7f40f2217cf0, 0x7f40f2217ce0}
│ │ │ │ +o2 = {0x7f850e9b0de0, 0x7f850e9b0dd0, 0x7f850e9b0dc0}
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type void**
│ │ │ │  i3 : int2star = foreignPointerArrayType(2 * int)
│ │ │ │  
│ │ │ │  o3 = int32[2]*
│ │ │ │  
│ │ │ │  o3 : ForeignPointerArrayType
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Pointer__Type_sp__Pointer.html
│ │ │ @@ -78,24 +78,24 @@
│ │ │            
    To cast a Macaulay2 pointer to a foreign object with a pointer type, give the type followed by the pointer.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = address int 0
│ │ │  
│ │ │ -o1 = 0x7f40f26a3da0
│ │ │ +o1 = 0x7f850ee220b0
│ │ │  
│ │ │  o1 : Pointer
    │ │ │
    │ │ │ 
    i2 : voidstar ptr
│ │ │  
│ │ │ -o2 = 0x7f40f26a3da0
│ │ │ +o2 = 0x7f850ee220b0
│ │ │  
│ │ │  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 = 0x7f40f26a3da0 │ │ │ │ +o1 = 0x7f850ee220b0 │ │ │ │ │ │ │ │ o1 : Pointer │ │ │ │ i2 : voidstar ptr │ │ │ │ │ │ │ │ -o2 = 0x7f40f26a3da0 │ │ │ │ +o2 = 0x7f850ee220b0 │ │ │ │ │ │ │ │ o2 : ForeignObject of type void* │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _F_o_r_e_i_g_n_P_o_i_n_t_e_r_T_y_p_e_ _P_o_i_n_t_e_r -- cast a Macaulay2 pointer to a foreign │ │ │ │ pointer │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp__Pointer.html │ │ │ @@ -87,15 +87,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d97f0
│ │ │ +o2 = 0x7f850e975cd0
│ │ │  
│ │ │  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 = 0x7f40f21d97f0
│ │ │ │ +o2 = 0x7f850e975cd0
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : int ptr
│ │ │ │  
│ │ │ │  o3 = 5
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Type_sp_st_spvoidstar.html
│ │ │ @@ -78,15 +78,15 @@
│ │ │            
    This is syntactic sugar for T value ptr (see ForeignType Pointer) for dereferencing pointers.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = voidstar address int 5
│ │ │  
│ │ │ -o1 = 0x7f40f21d91f0
│ │ │ +o1 = 0x7f850e98f9f0
│ │ │  
│ │ │  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 = 0x7f40f21d91f0
│ │ │ │ +o1 = 0x7f850e98f9f0
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type void*
│ │ │ │  i2 : int * ptr
│ │ │ │  
│ │ │ │  o2 = 5
│ │ │ │  
│ │ │ │  o2 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Foreign__Union__Type_sp__Thing.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  o1 : ForeignUnionType
    │ │ │
    │ │ │ 
    i2 : myunion 27
│ │ │  
│ │ │ -o2 = HashTable{"bar" => 6.91261e-310}
│ │ │ +o2 = HashTable{"bar" => 6.92726e-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" => 6.91261e-310} │ │ │ │ +o2 = HashTable{"bar" => 6.92726e-310} │ │ │ │ "foo" => 27 │ │ │ │ │ │ │ │ o2 : ForeignObject of type myunion │ │ │ │ i3 : myunion pi │ │ │ │ │ │ │ │ o3 = HashTable{"bar" => 3.14159 } │ │ │ │ "foo" => 1413754136 │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Pointer.html │ │ │ @@ -69,50 +69,50 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │
    │ │ │ 
    i2 : peek x
│ │ │  
│ │ │ -o2 = int32{Address => 0x7f40f21d9960}
    │ │ │ +o2 = int32{Address => 0x7f850e9754f0} │ │ │
    │ │ │
    │ │ │

    These pointers can be accessed using address.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i3 : ptr = address x
│ │ │  
│ │ │ -o3 = 0x7f40f21d9960
│ │ │ +o3 = 0x7f850e9754f0
│ │ │  
│ │ │  o3 : Pointer
    │ │ │
    │ │ │
    │ │ │

    Simple arithmetic can be performed on pointers.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : ptr + 5
│ │ │  
│ │ │ -o4 = 0x7f40f21d9965
│ │ │ +o4 = 0x7f850e9754f5
│ │ │  
│ │ │  o4 : Pointer
    │ │ │
    │ │ │ 
    i5 : ptr - 3
│ │ │  
│ │ │ -o5 = 0x7f40f21d995d
│ │ │ +o5 = 0x7f850e9754ed
│ │ │  
│ │ │  o5 : Pointer
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,30 +10,30 @@ │ │ │ │ i1 : x = int 20 │ │ │ │ │ │ │ │ o1 = 20 │ │ │ │ │ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ i2 : peek x │ │ │ │ │ │ │ │ -o2 = int32{Address => 0x7f40f21d9960} │ │ │ │ +o2 = int32{Address => 0x7f850e9754f0} │ │ │ │ These pointers can be accessed using _a_d_d_r_e_s_s. │ │ │ │ i3 : ptr = address x │ │ │ │ │ │ │ │ -o3 = 0x7f40f21d9960 │ │ │ │ +o3 = 0x7f850e9754f0 │ │ │ │ │ │ │ │ o3 : Pointer │ │ │ │ Simple arithmetic can be performed on pointers. │ │ │ │ i4 : ptr + 5 │ │ │ │ │ │ │ │ -o4 = 0x7f40f21d9965 │ │ │ │ +o4 = 0x7f850e9754f5 │ │ │ │ │ │ │ │ o4 : Pointer │ │ │ │ i5 : ptr - 3 │ │ │ │ │ │ │ │ -o5 = 0x7f40f21d995d │ │ │ │ +o5 = 0x7f850e9754ed │ │ │ │ │ │ │ │ o5 : Pointer │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _n_u_l_l_P_o_i_n_t_e_r -- the null pointer │ │ │ │ * _a_d_d_r_e_s_s -- pointer to type or object │ │ │ │ * _F_o_r_e_i_g_n_T_y_p_e_ _P_o_i_n_t_e_r -- dereference a pointer │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa ppooiinntteerr:: ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/___Shared__Library.html │ │ │ @@ -69,15 +69,15 @@ │ │ │ o1 : SharedLibrary │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : peek mpfr
│ │ │  
│ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr}
    │ │ │ +o2 = SharedLibrary{0x7f85231deac0, mpfr} │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    Menu

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ i1 : mpfr = openSharedLibrary "mpfr" │ │ │ │ │ │ │ │ o1 = mpfr │ │ │ │ │ │ │ │ o1 : SharedLibrary │ │ │ │ i2 : peek mpfr │ │ │ │ │ │ │ │ -o2 = SharedLibrary{0x7f40f9b43ac0, mpfr} │ │ │ │ +o2 = SharedLibrary{0x7f85231deac0, mpfr} │ │ │ │ ******** MMeennuu ******** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * _o_p_e_n_S_h_a_r_e_d_L_i_b_r_a_r_y -- open a shared library │ │ │ │ ********** MMeetthhooddss tthhaatt uussee aa sshhaarreedd lliibbrraarryy:: ********** │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) -- see │ │ │ │ _f_o_r_e_i_g_n_F_u_n_c_t_i_o_n -- construct a foreign function │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/__st_spvoidstar_sp_eq_sp__Thing.html │ │ │ @@ -83,15 +83,15 @@ │ │ │ o1 : ForeignObject of type int32 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : ptr = address x
│ │ │  
│ │ │ -o2 = 0x7f40f21d9220
│ │ │ +o2 = 0x7f850e9756d0
│ │ │  
│ │ │  o2 : Pointer
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : *ptr = int 6
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,15 +16,15 @@
│ │ │ │  i1 : x = int 5
│ │ │ │  
│ │ │ │  o1 = 5
│ │ │ │  
│ │ │ │  o1 : ForeignObject of type int32
│ │ │ │  i2 : ptr = address x
│ │ │ │  
│ │ │ │ -o2 = 0x7f40f21d9220
│ │ │ │ +o2 = 0x7f850e9756d0
│ │ │ │  
│ │ │ │  o2 : Pointer
│ │ │ │  i3 : *ptr = int 6
│ │ │ │  
│ │ │ │  o3 = 6
│ │ │ │  
│ │ │ │  o3 : ForeignObject of type int32
│ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_address.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │            
      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 = 0x562fbd063080
│ │ │ +o1 = 0x56009c05e080
│ │ │  
│ │ │  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 = 0x7f40f21d95f0
│ │ │ +o2 = 0x7f850e975b00
│ │ │  
│ │ │  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 = 0x562fbd063080 │ │ │ │ +o1 = 0x56009c05e080 │ │ │ │ │ │ │ │ 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 = 0x7f40f21d95f0 │ │ │ │ +o2 = 0x7f850e975b00 │ │ │ │ │ │ │ │ o2 : Pointer │ │ │ │ ********** WWaayyss ttoo uussee aaddddrreessss:: ********** │ │ │ │ * address(ForeignObject) │ │ │ │ * address(ForeignType) │ │ │ │ * address(Nothing) (missing documentation) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_foreign__Function.html │ │ │ @@ -237,15 +237,15 @@ │ │ │ o16 : ForeignFunction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : x = malloc 8
│ │ │  
│ │ │ -o17 = 0x7f0558070f60
│ │ │ +o17 = 0x7f22f8070f60
│ │ │  
│ │ │  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 = 0x7f0558070f60 │ │ │ │ +o17 = 0x7f22f8070f60 │ │ │ │ │ │ │ │ o17 : ForeignObject of type void* │ │ │ │ i18 : registerFinalizer(x, free) │ │ │ │ ********** WWaayyss ttoo uussee ffoorreeiiggnnFFuunnccttiioonn:: ********** │ │ │ │ * foreignFunction(Pointer,String,ForeignType,VisibleList) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,ForeignType) │ │ │ │ * foreignFunction(SharedLibrary,String,ForeignType,VisibleList) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_get__Memory.html │ │ │ @@ -82,43 +82,43 @@ │ │ │

    Allocate n bytes of memory using the GC garbage collector.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : ptr = getMemory 8
│ │ │  
│ │ │ -o1 = 0x7f40f50ca360
│ │ │ +o1 = 0x7f850f41fca0
│ │ │  
│ │ │  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 = 0x7f40f21d9ae0
│ │ │ +o2 = 0x7f850e94f180
│ │ │  
│ │ │  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 = 0x7f40f21d9a00
│ │ │ +o3 = 0x7f850e94f010
│ │ │  
│ │ │  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 = 0x7f40f50ca360 │ │ │ │ +o1 = 0x7f850f41fca0 │ │ │ │ │ │ │ │ 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 = 0x7f40f21d9ae0 │ │ │ │ +o2 = 0x7f850e94f180 │ │ │ │ │ │ │ │ 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 = 0x7f40f21d9a00 │ │ │ │ +o3 = 0x7f850e94f010 │ │ │ │ │ │ │ │ o3 : ForeignObject of type void* │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _r_e_g_i_s_t_e_r_F_i_n_a_l_i_z_e_r_(_F_o_r_e_i_g_n_O_b_j_e_c_t_,_F_u_n_c_t_i_o_n_) -- register a finalizer for a │ │ │ │ foreign object │ │ │ │ ********** WWaayyss ttoo uussee ggeettMMeemmoorryy:: ********** │ │ │ │ * getMemory(ForeignType) │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_register__Finalizer_lp__Foreign__Object_cm__Function_rp.html │ │ │ @@ -105,23 +105,23 @@ │ │ │ │ │ │ 
    i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer))
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : collectGarbage()
│ │ │ -freeing memory at 0x7f40dc0848b0
│ │ │ -freeing memory at 0x7f40dc084870
│ │ │ -freeing memory at 0x7f40dc084830
│ │ │ -freeing memory at 0x7f40dc0841d0
│ │ │ -freeing memory at 0x7f40dc0841f0
│ │ │ -freeing memory at 0x7f40dc0848d0
│ │ │ -freeing memory at 0x7f40dc084810
│ │ │ -freeing memory at 0x7f40dc084850
│ │ │ -freeing memory at 0x7f40dc084890
    │ │ │ +freeing memory at 0x7f84f8084890 │ │ │ +freeing memory at 0x7f84f80848b0 │ │ │ +freeing memory at 0x7f84f8084870 │ │ │ +freeing memory at 0x7f84f80841d0 │ │ │ +freeing memory at 0x7f84f8084810 │ │ │ +freeing memory at 0x7f84f8084830 │ │ │ +freeing memory at 0x7f84f8084850 │ │ │ +freeing memory at 0x7f84f80841f0 │ │ │ +freeing memory at 0x7f84f80848d0 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -31,23 +31,23 @@ │ │ │ │ i3 : finalizer = x -> (print("freeing memory at " | net x); free x) │ │ │ │ │ │ │ │ o3 = finalizer │ │ │ │ │ │ │ │ o3 : FunctionClosure │ │ │ │ i4 : for i to 9 do (x := malloc 8; registerFinalizer(x, finalizer)) │ │ │ │ i5 : collectGarbage() │ │ │ │ -freeing memory at 0x7f40dc0848b0 │ │ │ │ -freeing memory at 0x7f40dc084870 │ │ │ │ -freeing memory at 0x7f40dc084830 │ │ │ │ -freeing memory at 0x7f40dc0841d0 │ │ │ │ -freeing memory at 0x7f40dc0841f0 │ │ │ │ -freeing memory at 0x7f40dc0848d0 │ │ │ │ -freeing memory at 0x7f40dc084810 │ │ │ │ -freeing memory at 0x7f40dc084850 │ │ │ │ -freeing memory at 0x7f40dc084890 │ │ │ │ +freeing memory at 0x7f84f8084890 │ │ │ │ +freeing memory at 0x7f84f80848b0 │ │ │ │ +freeing memory at 0x7f84f8084870 │ │ │ │ +freeing memory at 0x7f84f80841d0 │ │ │ │ +freeing memory at 0x7f84f8084810 │ │ │ │ +freeing memory at 0x7f84f8084830 │ │ │ │ +freeing memory at 0x7f84f8084850 │ │ │ │ +freeing memory at 0x7f84f80841f0 │ │ │ │ +freeing memory at 0x7f84f80848d0 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_t_M_e_m_o_r_y -- allocate memory using the garbage collector │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _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 │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/ForeignFunctions/html/_value_lp__Foreign__Object_rp.html │ │ │ @@ -121,24 +121,24 @@ │ │ │

      Foreign pointer objects are converted to Pointer objects.

      │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : x = voidstar address int 5
│ │ │  
│ │ │ -o5 = 0x7f40f21fd990
│ │ │ +o5 = 0x7f850e98fc30
│ │ │  
│ │ │  o5 : ForeignObject of type void*
    │ │ │
    │ │ │ 
    i6 : value x
│ │ │  
│ │ │ -o6 = 0x7f40f21fd990
│ │ │ +o6 = 0x7f850e98fc30
│ │ │  
│ │ │  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 = 0x7f40f21fd990 │ │ │ │ +o5 = 0x7f850e98fc30 │ │ │ │ │ │ │ │ o5 : ForeignObject of type void* │ │ │ │ i6 : value x │ │ │ │ │ │ │ │ -o6 = 0x7f40f21fd990 │ │ │ │ +o6 = 0x7f850e98fc30 │ │ │ │ │ │ │ │ o6 : Pointer │ │ │ │ Foreign string objects are converted to strings. │ │ │ │ i7 : x = charstar "Hello, world!" │ │ │ │ │ │ │ │ o7 = Hello, world! │ │ ├── ./usr/share/doc/Macaulay2/FormalGroupLaws/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ c2VyaWVzKFJpbmdFbGVtZW50LFpaKQ== │ │ │ #:len=1234 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29uc3RydWN0aW5nIGEgZm9ybWFsIHNl │ │ │ cmllcyIsICJsaW5lbnVtIiA9PiAzNzIsIElucHV0cyA9PiB7U1BBTntUVHsicyJ9LCIsICIsU1BB │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ dG9yaWNHcm9lYm5lcg== │ │ │ #:len=2522 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsY3VsYXRlcyBhIEdyb2VibmVyIGJh │ │ │ c2lzIG9mIHRoZSB0b3JpYyBpZGVhbCBJX0EsIGdpdmVuIEE7IGludm9rZXMgXCJncm9lYm5lclwi │ │ ├── ./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/M2-15423-0/0 │ │ │ +o2 = /tmp/M2-19394-0/0 │ │ │ │ │ │ i3 : F = openOut(s) │ │ │ │ │ │ -o3 = /tmp/M2-15423-0/0 │ │ │ +o3 = /tmp/M2-19394-0/0 │ │ │ │ │ │ o3 : File │ │ │ │ │ │ i4 : putMatrix(F,A) │ │ │ │ │ │ i5 : close(F) │ │ │ │ │ │ -o5 = /tmp/M2-15423-0/0 │ │ │ +o5 = /tmp/M2-19394-0/0 │ │ │ │ │ │ o5 : File │ │ │ │ │ │ i6 : getMatrix(s) │ │ │ │ │ │ o6 = | 1 1 1 1 | │ │ │ | 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/FourTiTwo/html/_put__Matrix.html │ │ │ @@ -84,36 +84,36 @@ │ │ │ o1 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : s = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-15423-0/0
    │ │ │ +o2 = /tmp/M2-19394-0/0 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : F = openOut(s)
│ │ │  
│ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ +o3 = /tmp/M2-19394-0/0
│ │ │  
│ │ │  o3 : File
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : putMatrix(F,A)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : close(F)
│ │ │  
│ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ +o5 = /tmp/M2-19394-0/0
│ │ │  
│ │ │  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/M2-15423-0/0
│ │ │ │ +o2 = /tmp/M2-19394-0/0
│ │ │ │  i3 : F = openOut(s)
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-15423-0/0
│ │ │ │ +o3 = /tmp/M2-19394-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : putMatrix(F,A)
│ │ │ │  i5 : close(F)
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-15423-0/0
│ │ │ │ +o5 = /tmp/M2-19394-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : getMatrix(s)
│ │ │ │  
│ │ │ │  o6 = | 1 1 1 1 |
│ │ │ │       | 1 2 3 4 |
│ │ ├── ./usr/share/doc/Macaulay2/FourierMotzkin/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Zm91cmllck1vdHpraW4=
│ │ │  #:len=3387
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW50ZXJjaGFuZ2UgaW5lcXVhbGl0eS9n
│ │ │  ZW5lcmF0b3IgcmVwcmVzZW50YXRpb24gb2YgYSBwb2x5aGVkcmFsIGNvbmUiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=32
│ │ │  aXNGUFQoLi4uLFFHb3JlbnN0ZWluSW5kZXg9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0ZQVCxRR29yZW5zdGVpbkluZGV4XSwiaXNGUFQo
│ │ ├── ./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 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ + -- used 1.59834s (cpu); 1.10112s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
│ │ │  
│ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ + -- used 0.901604s (cpu); 0.678875s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
│ │ │  
│ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ + -- used 0.758585s (cpu); 0.552892s (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 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ + -- used 0.00497671s (cpu); 0.00497499s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
│ │ │  
│ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ + -- used 0.304256s (cpu); 0.239791s (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 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ + -- used 0.248974s (cpu); 0.188436s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
│ │ │  
│ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ + -- used 1.27854s (cpu); 1.00993s (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 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ + -- used 0.706071s (cpu); 0.567819s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
│ │ │  
│ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ + -- used 1.07331s (cpu); 0.805762s (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 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ + -- used 1.43663s (cpu); 1.17058s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
│ │ │  
│ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ + -- used 0.485752s (cpu); 0.426025s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = 97
│ │ │  
│ │ │  i32 : R = ZZ/7[x,y];
│ │ │  
│ │ │  i33 : f = (x - 1)^3 - (y - 2)^2;
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_fpt.html
│ │ │ @@ -368,37 +368,37 @@
│ │ │          
    
│ │ │            
    The computations performed when FinalAttempt is set to true are often slow, and often fail to improve the estimate, and for this reason, this option should be used sparingly. It is often more effective to increase the values of Attempts or DepthOfSearch, instead.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -228,29 +228,29 @@
│ │ │ │  
│ │ │ │  o26 : List
│ │ │ │  The computations performed when FinalAttempt is set to true are often slow, and
│ │ │ │  often fail to improve the estimate, and for this reason, this option should be
│ │ │ │  used sparingly. It is often more effective to increase the values of Attempts
│ │ │ │  or DepthOfSearch, instead.
│ │ │ │  i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ │ + -- used 1.59834s (cpu); 1.10112s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = {.142067, .144}
│ │ │ │  
│ │ │ │  o27 : List
│ │ │ │  i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ │ + -- used 0.901604s (cpu); 0.678875s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o28 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o28 : QQ
│ │ │ │  i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ │ + -- used 0.758585s (cpu); 0.552892s (thread); 0s (gc)
│ │ │ │  
│ │ │ │        1
│ │ │ │  o29 = -
│ │ │ │        7
│ │ │ │  
│ │ │ │  o29 : QQ
│ │ │ │  As seen in several examples above, when the exact answer is not found, a list
│ │ ├── ./usr/share/doc/Macaulay2/FrobeniusThresholds/html/_frobenius__Nu.html
│ │ │ @@ -197,23 +197,23 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i27 : time numeric fpt(f, DepthOfSearch => 3, FinalAttempt => true)
│ │ │ - -- used 1.56927s (cpu); 1.12322s (thread); 0s (gc)
│ │ │ + -- used 1.59834s (cpu); 1.10112s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = {.142067, .144}
│ │ │  
│ │ │  o27 : List
    
│ │ │              
    
│ │ │                
    i28 : time fpt(f, DepthOfSearch => 3, Attempts => 7)
│ │ │ - -- used 1.0262s (cpu); 0.753238s (thread); 0s (gc)
│ │ │ + -- used 0.901604s (cpu); 0.678875s (thread); 0s (gc)
│ │ │  
│ │ │        1
│ │ │  o28 = -
│ │ │        7
│ │ │  
│ │ │  o28 : QQ
    
│ │ │              
    
│ │ │                
    i29 : time fpt(f, DepthOfSearch => 4)
│ │ │ - -- used 0.740369s (cpu); 0.533854s (thread); 0s (gc)
│ │ │ + -- used 0.758585s (cpu); 0.552892s (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 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ + -- used 0.00497671s (cpu); 0.00497499s (thread); 0s (gc)
│ │ │  
│ │ │  o15 = 3756
    
│ │ │              
    
│ │ │                
    i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ + -- used 0.304256s (cpu); 0.239791s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 3756
    
│ │ │              
    
│ │ │          
    
│ │ │            
    The valid values for the option ContainmentTest are FrobeniusPower, FrobeniusRoot, and StandardPower. The default value of this option depends on what is passed to frobeniusNu. Indeed, by default, ContainmentTest is set to FrobeniusRoot if frobeniusNu is passed a ring element $f$, and is set to StandardPower if frobeniusNu is passed an ideal $I$. We describe the consequences of setting ContainmentTest to each of these values below.
    
│ │ │ @@ -230,23 +230,23 @@
│ │ │              
│ │ │                
    i18 : f = x^3 + y^3 + z^3 + x*y*z;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i19 : time frobeniusNu(4, f) -- ContainmentTest is set to FrobeniusRoot, by default
│ │ │ - -- used 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ + -- used 0.248974s (cpu); 0.188436s (thread); 0s (gc)
│ │ │  
│ │ │  o19 = 499
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ + -- used 1.27854s (cpu); 1.00993s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = 499
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Finally, when ContainmentTest is set to FrobeniusPower, then instead of producing the invariant $\nu_I^J(p^e)$ as defined above, frobeniusNu instead outputs the maximal integer $n$ such that the $n$^{th} (generalized) Frobenius power of $I$ is not contained in the $p^e$-th Frobenius power of $J$. Here, the $n$^{th} Frobenius power of $I$, when $n$ is a nonnegative integer, is as defined in the paper Frobenius Powers by Hernández, Teixeira, and Witt, which can be computed with the function frobeniusPower, from the TestIdeals package. In particular, frobeniusNu(e,I,J) and frobeniusNu(e,I,J,ContainmentTest=>FrobeniusPower) need not agree. However, they will agree when $I$ is a principal ideal.
    
│ │ │ @@ -292,46 +292,46 @@
│ │ │              
│ │ │                
    i26 : f = x^2*y^4 + y^2*z^7 + z^2*x^8;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time frobeniusNu(5, f) -- uses binary search (default)
│ │ │ - -- used 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ + -- used 0.706071s (cpu); 0.567819s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = 1124
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ + -- used 1.07331s (cpu); 0.805762s (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 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ + -- used 1.43663s (cpu); 1.17058s (thread); 0s (gc)
│ │ │  
│ │ │  o30 = 97
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets luckier
│ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ + -- used 0.485752s (cpu); 0.426025s (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 0.00433815s (cpu); 0.00433416s (thread); 0s (gc)
│ │ │ │ + -- used 0.00497671s (cpu); 0.00497499s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 3756
│ │ │ │  i16 : time frobeniusNu(3, f, UseSpecialAlgorithms => false)
│ │ │ │ - -- used 0.293694s (cpu); 0.229445s (thread); 0s (gc)
│ │ │ │ + -- used 0.304256s (cpu); 0.239791s (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 0.252728s (cpu); 0.176544s (thread); 0s (gc)
│ │ │ │ + -- used 0.248974s (cpu); 0.188436s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o19 = 499
│ │ │ │  i20 : time frobeniusNu(4, f, ContainmentTest => StandardPower)
│ │ │ │ - -- used 1.81045s (cpu); 1.35141s (thread); 0s (gc)
│ │ │ │ + -- used 1.27854s (cpu); 1.00993s (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 0.778907s (cpu); 0.635009s (thread); 0s (gc)
│ │ │ │ + -- used 0.706071s (cpu); 0.567819s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = 1124
│ │ │ │  i28 : time frobeniusNu(5, f, Search => Linear)
│ │ │ │ - -- used 1.08232s (cpu); 0.827197s (thread); 0s (gc)
│ │ │ │ + -- used 1.07331s (cpu); 0.805762s (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 2.16055s (cpu); 1.60855s (thread); 0s (gc)
│ │ │ │ + -- used 1.43663s (cpu); 1.17058s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = 97
│ │ │ │  i31 : time frobeniusNu(2, M, M^2, Search => Linear) -- but linear search gets
│ │ │ │  luckier
│ │ │ │ - -- used 0.636757s (cpu); 0.499453s (thread); 0s (gc)
│ │ │ │ + -- used 0.485752s (cpu); 0.426025s (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/FunctionFieldDesingularization/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  YXJjcw==
│ │ │  #:len=3089
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbnRzIG5vZGUgbGFiZWxzIGZvciB0
│ │ │  aGUgZGVzaW5ndWxhcml6YXRpb24gdHJlZSIsICJsaW5lbnVtIiA9PiA2NTIsIElucHV0cyA9PiB7
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=10
│ │ │  UlJFRk1ldGhvZA==
│ │ │  #:len=209
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjEzMCwgInVuZG9jdW1lbnRlZCIgPT4g
│ │ │  dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiUlJFRk1l
│ │ ├── ./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 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ + -- used 5.65026s (cpu); 0.943858s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
│ │ │  
│ │ │  i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ + -- used 4.66055s (cpu); 1.31207s (thread); 0s (gc)
│ │ │  
│ │ │  o28 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o28 : KClass
│ │ │  
│ │ │  i29 :
│ │ ├── ./usr/share/doc/Macaulay2/GKMVarieties/html/_orbit__Closure.html
│ │ │ @@ -391,25 +391,25 @@
│ │ │                 3       4
│ │ │  o26 : Matrix QQ  <-- QQ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i27 : time C = orbitClosure(X,Mat)
│ │ │ - -- used 0.68501s (cpu); 0.426849s (thread); 0s (gc)
│ │ │ + -- used 5.65026s (cpu); 0.943858s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = an "equivariant K-class" on a GKM variety 
│ │ │  
│ │ │  o27 : KClass
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i28 : time C = orbitClosure(X,Mat, RREFMethod => true)
│ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc)
│ │ │ + -- used 4.66055s (cpu); 1.31207s (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 0.68501s (cpu); 0.426849s (thread); 0s (gc) │ │ │ │ + -- used 5.65026s (cpu); 0.943858s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = an "equivariant K-class" on a GKM variety │ │ │ │ │ │ │ │ o27 : KClass │ │ │ │ i28 : time C = orbitClosure(X,Mat, RREFMethod => true) │ │ │ │ - -- used 2.0599s (cpu); 1.05036s (thread); 0s (gc) │ │ │ │ + -- used 4.66055s (cpu); 1.31207s (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/GameTheory/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ cHJvYmFiaWxpdHlSaW5nKExpc3Qp │ │ │ #:len=262 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU0Niwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocHJvYmFiaWxpdHlSaW5nLExpc3QpLCJwcm9iYWJp │ │ ├── ./usr/share/doc/Macaulay2/GenericInitialIdeal/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ Z2luKElkZWFsKQ== │ │ │ #:len=242 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTczLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnaW4sSWRlYWwpLCJnaW4oSWRlYWwpIiwiR2VuZXJp │ │ ├── ./usr/share/doc/Macaulay2/GeometricDecomposability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ SXNJZGVhbFVubWl4ZWQ= │ │ │ #:len=1529 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAic3BlY2lmeSB3aGV0aGVyIGFuIGlkZWFs │ │ │ IGlzIHVubWl4ZWQiLCAibGluZW51bSIgPT4gMTkwNCwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsi │ │ ├── ./usr/share/doc/Macaulay2/GradedLieAlgebras/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ WlogXyBFeHRBbGdlYnJh │ │ │ #:len=932 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSB6ZXJvIGVsZW1lbnQiLCBE │ │ │ ZXNjcmlwdGlvbiA9PiAoRElWe0hFQURFUjJ7IlN5bm9wc2lzIn0sVUx7TEl7REx7ImNsYXNzIiA9 │ │ ├── ./usr/share/doc/Macaulay2/GraphicalModels/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ VmFyaWFibGVOYW1l │ │ │ #:len=616 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl │ │ │ IGluZGV0ZXJtaW5hdGUgbmFtZSBpbiBtYXJrb3ZSaW5nIiwgImxpbmVudW0iID0+IDE4ODgsIFNl │ │ ├── ./usr/share/doc/Macaulay2/GraphicalModelsMLE/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c29sdmVyTUxFKExpc3QsR3JhcGgp │ │ │ #:len=274 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk1OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc29sdmVyTUxFLExpc3QsR3JhcGgpLCJzb2x2ZXJN │ │ ├── ./usr/share/doc/Macaulay2/Graphics/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ cGljdHVyZVpvbmUoU3BoZXJlKQ== │ │ │ #:len=747 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCB0aGUgem9uZSB0aGF0IGNvbnRh │ │ │ aW5zIHRoZSBzcGhlcmUiLCAibGluZW51bSIgPT4gMTExNywgSW5wdXRzID0+IHtTUEFOe1RUeyJz │ │ ├── ./usr/share/doc/Macaulay2/Graphs/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ cmV2ZXJzZUJyZWFkdGhGaXJzdFNlYXJjaA== │ │ │ #:len=1565 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicnVucyBhIHJldmVyc2UgYnJlYWR0aCBm │ │ │ aXJzdCBzZWFyY2ggb24gdGhlIGRpZ3JhcGggc3RhcnRpbmcgYXQgYSBzcGVjaWZpZWQgbm9kZSIs │ │ ├── ./usr/share/doc/Macaulay2/Graphs/example-output/_new__Digraph.out │ │ │ @@ -32,12 +32,12 @@ │ │ │ 5 => {6} │ │ │ 6 => {} │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ i3 : keys H │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ +o3 = {newDigraph, digraph, map} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Graphs/html/_new__Digraph.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ o2 : SortedDigraph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : keys H
│ │ │  
│ │ │ -o3 = {digraph, map, newDigraph}
│ │ │ +o3 = {newDigraph, digraph, map}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 4 => {} │ │ │ │ 5 => {6} │ │ │ │ 6 => {} │ │ │ │ │ │ │ │ o2 : SortedDigraph │ │ │ │ i3 : keys H │ │ │ │ │ │ │ │ -o3 = {digraph, map, newDigraph} │ │ │ │ +o3 = {newDigraph, digraph, map} │ │ │ │ │ │ │ │ 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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ ZmluZFdlaWdodENvbnN0cmFpbnRz │ │ │ #:len=2630 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyBhIG1hdHJpeCBvZiB3ZWln │ │ │ aHQgY29uc3RyYWludHMiLCAibGluZW51bSIgPT4gNjkxLCBJbnB1dHMgPT4ge1NQQU57VFR7Ik0i │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ c2V0V2Fsa1RyYWNlKFpaKQ== │ │ │ #:len=251 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTYxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzZXRXYWxrVHJhY2UsWlopLCJzZXRXYWxrVHJhY2Uo │ │ ├── ./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 │ │ │ - -- 3.51847s elapsed │ │ │ + -- 1.76845s elapsed │ │ │ │ │ │ o5 = GroebnerBasis[status: done; S-pairs encountered up to degree 16] │ │ │ │ │ │ o5 : GroebnerBasis │ │ │ │ │ │ i6 : elapsedTime groebnerWalk(gb I1, R2) │ │ │ - -- 3.08455s elapsed │ │ │ + -- 1.45439s elapsed │ │ │ │ │ │ o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 0] │ │ │ │ │ │ o6 : GroebnerBasis │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/GroebnerWalk/html/index.html │ │ │ @@ -97,30 +97,30 @@ │ │ │ │ │ │ o4 : Ideal of R2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime gb I2
│ │ │ - -- 3.51847s elapsed
│ │ │ + -- 1.76845s 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)
│ │ │ - -- 3.08455s elapsed
│ │ │ + -- 1.45439s 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 │ │ │ │ - -- 3.51847s elapsed │ │ │ │ + -- 1.76845s 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) │ │ │ │ - -- 3.08455s elapsed │ │ │ │ + -- 1.45439s 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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=23 │ │ │ aWRlYWxPZlByb2plY3RpdmVQb2ludHM= │ │ │ #:len=1240 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGlkZWFsIG9mIHNl │ │ │ dCBvZiBwb2ludHMiLCAibGluZW51bSIgPT4gNDI0LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi │ │ ├── ./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{0, 2}, Point{2, 4}, Point{1, 0}} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/example-output/_ideal__Of__Projective__Points.out │ │ │ @@ -30,17 +30,17 @@ │ │ │ 2 2 │ │ │ + x*y - 6x*z ) │ │ │ │ │ │ o4 : Ideal of S │ │ │ │ │ │ i5 : X2 = hadamardPower(X,2) │ │ │ │ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1}, │ │ │ +o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, │ │ │ ------------------------------------------------------------------------ │ │ │ - Point{1, 1, 0}, Point{1, 2, 0}} │ │ │ + Point{0, 1, 1}, Point{1, 1, 0}} │ │ │ │ │ │ o5 : List │ │ │ │ │ │ i6 : I2 == idealOfProjectivePoints(X2,S) │ │ │ │ │ │ o6 = true │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_hadamard__Product_lp__List_cm__List_rp.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ 
    i2 : M = {point{1,0}, point{2,2}};
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : hadamardProduct(L,M)
│ │ │  
│ │ │ -o3 = {Point{1, 0}, Point{0, 2}, Point{2, 4}}
│ │ │ +o3 = {Point{0, 2}, Point{2, 4}, Point{1, 0}}
│ │ │  
│ │ │  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{0, 2}, Point{2, 4}, Point{1, 0}} │ │ │ │ │ │ │ │ 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- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/Hadamard.m2:345:0. │ │ ├── ./usr/share/doc/Macaulay2/Hadamard/html/_ideal__Of__Projective__Points.html │ │ │ @@ -121,17 +121,17 @@ │ │ │ o4 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : X2 = hadamardPower(X,2)
│ │ │  
│ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ +o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ +     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ ├── html2text {}
│ │ │ │ @@ -39,17 +39,17 @@
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │            2       2
│ │ │ │       + x*y  - 6x*z )
│ │ │ │  
│ │ │ │  o4 : Ideal of S
│ │ │ │  i5 : X2 = hadamardPower(X,2)
│ │ │ │  
│ │ │ │ -o5 = {Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0}, Point{0, 1, 1},
│ │ │ │ +o5 = {Point{1, 2, 0}, Point{1, 4, 1}, Point{0, 2, -1}, Point{0, 1, 0},
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Point{1, 1, 0}, Point{1, 2, 0}}
│ │ │ │ +     Point{0, 1, 1}, Point{1, 1, 0}}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : I2 == idealOfProjectivePoints(X2,S)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  ********** WWaayyss ttoo uussee iiddeeaallOOffPPrroojjeeccttiivveePPooiinnttss:: **********
│ │ │ │      * idealOfProjectivePoints(List,Ring)
│ │ ├── ./usr/share/doc/Macaulay2/HigherCIOperators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Y2lPcGVyYXRvclJlc29sdXRpb24=
│ │ │  #:len=2606
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiXCJsaWZ0IHJlc29sdXRpb24gZnJvbSBj
│ │ │  b21wbGV0ZSBpbnRlcnNlY3Rpb24gdXNpbmcgaGlnaGVyIGNpLW9wZXJhdG9yc1wiIiwgImxpbmVu
│ │ ├── ./usr/share/doc/Macaulay2/HighestWeights/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  R3JvdXBBY3Rpbmc=
│ │ │  #:len=621
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│ │ │  ZiB0aGUgZ3JvdXAgYWN0aW5nIG9uIGEgcmluZyIsICJsaW5lbnVtIiA9PiA4MywgU2VlQWxzbyA9
│ │ ├── ./usr/share/doc/Macaulay2/HodgeIntegrals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  a2FwcGE=
│ │ │  #:len=1396
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTXVtZm9yZC1Nb3JpdGEtTWlsbGVyIGNs
│ │ │  YXNzZXMiLCAibGluZW51bSIgPT4gNzE1LCBJbnB1dHMgPT4ge1NQQU57VFR7ImEifSwiLCAiLFNQ
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  ZXVsZXJPcGVyYXRvcnM=
│ │ │  #:len=1959
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXVsZXIgT3BlcmF0b3JzIiwgImxpbmVu
│ │ │  dW0iID0+IDE0MCwgSW5wdXRzID0+IHtTUEFOe1RUeyJBIn0sIiwgIixTUEFOeyJhICIsVE8ye25l
│ │ ├── ./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.
│ │ │ - -- .000004839s elapsed
│ │ │ - -- .000004879s elapsed
│ │ │ - -- .00000513s elapsed
│ │ │ - -- .0000051s elapsed
│ │ │ - -- .000003987s elapsed
│ │ │ + -- .000005544s elapsed
│ │ │ + -- .000007518s elapsed
│ │ │ + -- .000005112s elapsed
│ │ │ + -- .000004878s elapsed
│ │ │ + -- .000004246s elapsed
│ │ │  
│ │ │       +----------------------------------------------------+
│ │ │       |   1 5   5 5                                        |
│ │ │       | - - - - - -                                        |
│ │ │       |   2 2   2 2                                        |
│ │ │  o6 = |x   x x   x                                         |
│ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/example-output/_solve__Frobenius__Ideal.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  i2 : I = ideal(t_1+t_2+t_3+t_4+t_5, t_1+t_2-t_4, t_2+t_3-t_4, t_1*t_3, t_2*t_4);
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000006092s elapsed
│ │ │ + -- .000006185s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -26,15 +26,15 @@
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │  
│ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000007755s elapsed
│ │ │ + -- .000005572s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_css__Lead__Term.html
│ │ │ @@ -139,19 +139,19 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : netList cssLeadTerm(Hbeta, w)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000004839s elapsed
│ │ │ - -- .000004879s elapsed
│ │ │ - -- .00000513s elapsed
│ │ │ - -- .0000051s elapsed
│ │ │ - -- .000003987s elapsed
│ │ │ + -- .000005544s elapsed
│ │ │ + -- .000007518s elapsed
│ │ │ + -- .000005112s elapsed
│ │ │ + -- .000004878s elapsed
│ │ │ + -- .000004246s 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.
│ │ │ │ - -- .000004839s elapsed
│ │ │ │ - -- .000004879s elapsed
│ │ │ │ - -- .00000513s elapsed
│ │ │ │ - -- .0000051s elapsed
│ │ │ │ - -- .000003987s elapsed
│ │ │ │ + -- .000005544s elapsed
│ │ │ │ + -- .000007518s elapsed
│ │ │ │ + -- .000005112s elapsed
│ │ │ │ + -- .000004878s elapsed
│ │ │ │ + -- .000004246s elapsed
│ │ │ │  
│ │ │ │       +----------------------------------------------------+
│ │ │ │       |   1 5   5 5                                        |
│ │ │ │       | - - - - - -                                        |
│ │ │ │       |   2 2   2 2                                        |
│ │ │ │  o6 = |x   x x   x                                         |
│ │ │ │       | 1   2 4   5                                        |
│ │ ├── ./usr/share/doc/Macaulay2/HolonomicSystems/html/_solve__Frobenius__Ideal.html
│ │ │ @@ -91,15 +91,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : solveFrobeniusIdeal I
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000006092s elapsed
│ │ │ + -- .000006185s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX 
│ │ │ @@ -120,15 +120,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : solveFrobeniusIdeal(I, W)
│ │ │  Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
│ │ │  Converting to Naive algorithm.
│ │ │ - -- .000007755s elapsed
│ │ │ + -- .000005572s elapsed
│ │ │  
│ │ │                                                                               
│ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │                  0        1        2       3        0       1       2       4 
│ │ │       ------------------------------------------------------------------------
│ │ │       1             1    2   1             1             1             1    2
│ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,15 +20,15 @@
│ │ │ │  t_2*t_4);
│ │ │ │  
│ │ │ │  o2 : Ideal of R
│ │ │ │  i3 : solveFrobeniusIdeal I
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000006092s elapsed
│ │ │ │ + -- .000006185s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o3 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ │ │ @@ -40,15 +40,15 @@
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : W = makeWeylAlgebra(QQ[x_1..x_5]);
│ │ │ │  i5 : solveFrobeniusIdeal(I, W)
│ │ │ │  Warning:  F4 Algorithm not available over current coefficient ring or
│ │ │ │  inhomogeneous ideal.
│ │ │ │  Converting to Naive algorithm.
│ │ │ │ - -- .000007755s elapsed
│ │ │ │ + -- .000005572s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  o5 = {1, - 2logX  + 3logX  - 2logX  + logX , - logX  + logX  - logX  + logX ,
│ │ │ │                  0        1        2       3        0       1       2       4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       1             1    2   1             1             1             1    2
│ │ │ │       -logX logX  - -logX  + -logX logX  + -logX logX  + -logX logX  + -logX
│ │ ├── ./usr/share/doc/Macaulay2/HomotopyLieAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  YWxsZ2VucyhER0FsZ2VicmEsWlop
│ │ │  #:len=269
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  cmVzdHJpY3Rpb24oQXJyYW5nZW1lbnQsTGlzdCk=
│ │ │  #:len=343
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjc3OCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmVzdHJpY3Rpb24sQXJyYW5nZW1lbnQsTGlzdCks
│ │ ├── ./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 = {y, x, x - y}
│ │ │ +o14 = {x - y, y, x}
│ │ │  
│ │ │  o14 : Hyperplane Arrangement 
│ │ │  
│ │ │  i15 : assert isCentral cA''
│ │ │  
│ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.out
│ │ │ @@ -10,35 +10,35 @@
│ │ │  
│ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : restriction(A,1)
│ │ │  
│ │ │  o4 = {x, z, x, x - z}
│ │ │  
│ │ │  o4 : Hyperplane Arrangement 
│ │ │  
│ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │  
│ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
│ │ │  
│ │ │  i9 : prune image der(A,m)
│ │ │  
│ │ │        3
│ │ │  o9 = R
│ │ │ @@ -59,16 +59,16 @@
│ │ │  
│ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │  
│ │ │  i12 : A = arrangement "bracelet";
│ │ │  
│ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
│ │ │  
│ │ │  i14 : C = restriction(A,0)
│ │ │  
│ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_trim_lp__Arrangement_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │  
│ │ │  o2 : Hyperplane Arrangement 
│ │ │  
│ │ │  i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {x + y, y, x}
│ │ │ +o3 = {y, x, x + y}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
│ │ │  
│ │ │  i4 : assert(ring A' === R)
│ │ │  
│ │ │  i5 : assert(trim A' == A')
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/example-output/_type__B_lp__Z__Z_cm__Ring_rp.out
│ │ │ @@ -33,16 +33,16 @@
│ │ │  o5 = {x , x  + x , x  + x , x }
│ │ │         1   1    2   1    2   2
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
│ │ │  
│ │ │  i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       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
│ │ │ @@ -172,15 +172,15 @@
│ │ │  o13 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : cA'' = trim cone(A, x)
│ │ │  
│ │ │ -o14 = {y, x, x - y}
│ │ │ +o14 = {x - y, y, x}
│ │ │  
│ │ │  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 = {y, x, x - y}
│ │ │ │ +o14 = {x - y, y, x}
│ │ │ │  
│ │ │ │  o14 : Hyperplane Arrangement
│ │ │ │  i15 : assert isCentral cA''
│ │ │ │  i16 : assert(# hyperplanes cA'' =!= 1 + # hyperplanes A)
│ │ │ │  When the second input is a _S_y_m_b_o_l, this method creates a new ring from the
│ │ │ │  underlying ring of $A$ by adjoining the symbol as a variable and constructs the
│ │ │ │  cone in this new ring.
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_euler__Restriction_lp__Central__Arrangement_cm__List_cm__Z__Z_rp.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │  
│ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : restriction(A,1)
│ │ │ @@ -118,15 +118,15 @@
│ │ │  o4 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │  
│ │ │ -o5 = {x - z, z, x}
│ │ │ +o5 = {z, x, x - z}
│ │ │  
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$, where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the Euler restriction is free as well, and the modules of logarithmic derivations form a short exact sequence.  See the paper of Abe, Terao and Wakefield for details.
    
│ │ │ @@ -137,15 +137,15 @@
│ │ │                
    i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │  
│ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │  
│ │ │  o8 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : prune image der(A,m)
│ │ │ @@ -186,16 +186,16 @@
│ │ │                
    i12 : A = arrangement "bracelet";
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │  
│ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │  
│ │ │  o13 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : C = restriction(A,0)
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,36 +33,36 @@
│ │ │ │  i2 : A = arrangement {x,y,z,x-y,x-z}
│ │ │ │  
│ │ │ │  o2 = {x, y, z, x - y, x - z}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : (A'',m'') = eulerRestriction(A,{1,1,1,1,1},1)
│ │ │ │  
│ │ │ │ -o3 = ({x - z, z, x}, {1, 1, 1})
│ │ │ │ +o3 = ({z, x, x - z}, {1, 1, 1})
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : restriction(A,1)
│ │ │ │  
│ │ │ │  o4 = {x, z, x, x - z}
│ │ │ │  
│ │ │ │  o4 : Hyperplane Arrangement
│ │ │ │  i5 : trim oo -- same underlying simple arrangement, different multiplicities
│ │ │ │  
│ │ │ │ -o5 = {x - z, z, x}
│ │ │ │ +o5 = {z, x, x - z}
│ │ │ │  
│ │ │ │  o5 : Hyperplane Arrangement
│ │ │ │  If $({\mathcal A},m)$ is a free multiarrangement and so is $({\mathcal A},m')$,
│ │ │ │  where $m'$ is obtained from $m$ by lowering a single multiplicity by one, the
│ │ │ │  Euler restriction is free as well, and the modules of _l_o_g_a_r_i_t_h_m_i_c_ _d_e_r_i_v_a_t_i_o_n_s
│ │ │ │  form a short exact sequence. See the paper of Abe, Terao and Wakefield for
│ │ │ │  details.
│ │ │ │  i6 : m = {2,2,2,2,1}; m' = {2,2,2,1,1};
│ │ │ │  i8 : (A'',m'') = eulerRestriction(A,m,3)
│ │ │ │  
│ │ │ │ -o8 = ({y - z, z, y}, {1, 2, 3})
│ │ │ │ +o8 = ({z, y, y - z}, {2, 3, 1})
│ │ │ │  
│ │ │ │  o8 : Sequence
│ │ │ │  i9 : prune image der(A,m)
│ │ │ │  
│ │ │ │        3
│ │ │ │  o9 = R
│ │ │ │  
│ │ │ │ @@ -80,16 +80,16 @@
│ │ │ │  
│ │ │ │  o11 : QQ[y..z]-module, free, degrees {2:3}
│ │ │ │  It may be the case that the Euler restriction is free, while the naive
│ │ │ │  restriction is not:
│ │ │ │  i12 : A = arrangement "bracelet";
│ │ │ │  i13 : (B,m) = eulerRestriction(A,{1,1,1,1,1,1,1,1,1},0)
│ │ │ │  
│ │ │ │ -o13 = ({x , x  + x , x , x , x  + x  + x , x  + x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ -         2   3    4   3   4   2    3    4   2    4
│ │ │ │ +o13 = ({x  + x , x , x , x  + x  + x , x  + x , x }, {1, 1, 1, 1, 1, 1})
│ │ │ │ +         3    4   3   4   2    3    4   2    4   2
│ │ │ │  
│ │ │ │  o13 : Sequence
│ │ │ │  i14 : C = restriction(A,0)
│ │ │ │  
│ │ │ │  o14 = {x , x , x , x  + x , x  + x , x  + x , x  + x , x  + x  + x }
│ │ │ │          2   3   4   2    4   3    4   2    4   3    4   2    3    4
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_trim_lp__Arrangement_rp.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  o2 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : A' = trim A
│ │ │  
│ │ │ -o3 = {x + y, y, x}
│ │ │ +o3 = {y, x, x + y}
│ │ │  
│ │ │  o3 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : assert(ring A' === R)
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  i2 : A = arrangement{x,x,0_R,y,y,y,x+y,x+y,x+y,x+y,x+y}
│ │ │ │  
│ │ │ │  o2 = {x, x, 0, y, y, y, x + y, x + y, x + y, x + y, x + y}
│ │ │ │  
│ │ │ │  o2 : Hyperplane Arrangement
│ │ │ │  i3 : A' = trim A
│ │ │ │  
│ │ │ │ -o3 = {x + y, y, x}
│ │ │ │ +o3 = {y, x, x + y}
│ │ │ │  
│ │ │ │  o3 : Hyperplane Arrangement
│ │ │ │  i4 : assert(ring A' === R)
│ │ │ │  i5 : assert(trim A' == A')
│ │ │ │  i6 : assert(trim A' == A')
│ │ │ │  Some natural operations produce non-simple hyperplane arrangements.
│ │ │ │  i7 : A'' = restriction(A, y)
│ │ ├── ./usr/share/doc/Macaulay2/HyperplaneArrangements/html/_type__B_lp__Z__Z_cm__Ring_rp.html
│ │ │ @@ -130,16 +130,16 @@
│ │ │  o5 : Hyperplane Arrangement 
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : trim A3
│ │ │  
│ │ │ -o6 = {x , x , x  + x }
│ │ │ -       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/IncidenceCorrespondenceCohomology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  cmVjdXJzaXZlRGl2aWRlZENvaG9tb2xvZ3k=
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│ │ ├── ./usr/share/doc/Macaulay2/IntegerProgramming/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026
│ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
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│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=36
│ │ │  YWRhcHRlZE1vbm9taWFsT3JkZXIoLi4uLEZpZWxkPT4uLi4p
│ │ │  #:len=286
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTI2LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  aW50ZWdyYWxDbG9zdXJlKC4uLixMaW1pdD0+Li4uKQ==
│ │ │  #:len=1120
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZG8gYSBwYXJ0aWFsIGludGVncmFsIGNs
│ │ │  b3N1cmUiLCAibGluZW51bSIgPT4gMTM5NywgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./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 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ + -- used 0.365813s (cpu); 0.286036s (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 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ + -- used 0.459947s (cpu); 0.296919s (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 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ + -- used 0.486008s (cpu); 0.323538s (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 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ + -- used 0.411283s (cpu); 0.320842s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
│ │ │  
│ │ │  i22 : netList (ideal R')_*
│ │ │  
│ │ │ @@ -266,15 +266,15 @@
│ │ │  i25 : R = S/f
│ │ │  
│ │ │  o25 = R
│ │ │  
│ │ │  o25 : QuotientRing
│ │ │  
│ │ │  i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ + -- used 0.783755s (cpu); 0.531534s (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 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ + -- used 0.475759s (cpu); 0.31473s (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 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ + -- used 0.045068s (cpu); 0.045068s (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 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ + -- used 0.158146s (cpu); 0.0740257s (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 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ + -- used 0.0658035s (cpu); 0.0658005s (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 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ + -- used 0.0499354s (cpu); 0.0497789s (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 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ + -- used 0.163208s (cpu); 0.0690922s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
│ │ │  
│ │ │  i57 : icFractions R
│ │ │  
│ │ │ @@ -633,15 +633,15 @@
│ │ │  i66 : R = S/I
│ │ │  
│ │ │  o66 = R
│ │ │  
│ │ │  o66 : QuotientRing
│ │ │  
│ │ │  i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ + -- used 0.061662s (cpu); 0.0616648s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
│ │ │  
│ │ │  i68 : icFractions R
│ │ │  
│ │ │ @@ -722,15 +722,15 @@
│ │ │  i77 : R = S/I
│ │ │  
│ │ │  o77 = R
│ │ │  
│ │ │  o77 : QuotientRing
│ │ │  
│ │ │  i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ + -- used 0.462522s (cpu); 0.3029s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
│ │ │  
│ │ │  i79 : icFractions R
│ │ │  
│ │ │ @@ -750,15 +750,15 @@
│ │ │  i81 : R = S/sub(I,S)
│ │ │  
│ │ │  o81 = R
│ │ │  
│ │ │  o81 : QuotientRing
│ │ │  
│ │ │  i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ + -- used 0.502344s (cpu); 0.321716s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
│ │ │  
│ │ │  i83 : icFractions R
│ │ │  
│ │ │ @@ -778,20 +778,20 @@
│ │ │  i85 : R = S/sub(I,S)
│ │ │  
│ │ │  o85 = R
│ │ │  
│ │ │  o85 : QuotientRing
│ │ │  
│ │ │  i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ + [jacobian time .000537355 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ + [step 0:   time .213813 sec  #fractions 6]
│ │ │ + [step 1:   time .24629 sec  #fractions 6]
│ │ │ + -- used 0.463889s (cpu); 0.263509s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
│ │ │  
│ │ │  i87 : icFractions R
│ │ │  
│ │ │ @@ -811,20 +811,20 @@
│ │ │  i89 : R = S/sub(I,S)
│ │ │  
│ │ │  o89 = R
│ │ │  
│ │ │  o89 : QuotientRing
│ │ │  
│ │ │  i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ + [jacobian time .000521638 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ + [step 0:   time .192362 sec  #fractions 6]
│ │ │ + [step 1:   time .254282 sec  #fractions 6]
│ │ │ + -- used 0.450305s (cpu); 0.28162s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
│ │ │  
│ │ │  i91 : icFractions R
│ │ │  
│ │ │ @@ -844,20 +844,20 @@
│ │ │  i93 : R = S/sub(I,S)
│ │ │  
│ │ │  o93 = R
│ │ │  
│ │ │  o93 : QuotientRing
│ │ │  
│ │ │  i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ + [jacobian time .00071598 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ + [step 0:   time .243077 sec  #fractions 6]
│ │ │ + [step 1:   time .675484 sec  #fractions 6]
│ │ │ + -- used 0.922337s (cpu); 0.567458s (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 .000522279 sec #minors 3]
│ │ │ + [jacobian time .000526737 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ -      idlizer1:  .00747003 seconds
│ │ │ -      idlizer2:  .00844957 seconds
│ │ │ -      minpres:   .00811702 seconds
│ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00252093 seconds
│ │ │ +      idlizer1:  .0079105 seconds
│ │ │ +      idlizer2:  .00874379 seconds
│ │ │ +      minpres:   .00798809 seconds
│ │ │ +  time .038948 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ -      idlizer1:  .0119562 seconds
│ │ │ -      idlizer2:  .0111 seconds
│ │ │ -      minpres:   .0117312 seconds
│ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00241193 seconds
│ │ │ +      idlizer1:  .013298 seconds
│ │ │ +      idlizer2:  .0114285 seconds
│ │ │ +      minpres:   .0113229 seconds
│ │ │ +  time .0492339 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ -      idlizer1:  .0169512 seconds
│ │ │ -      idlizer2:  .0143372 seconds
│ │ │ -      minpres:   .0137525 seconds
│ │ │ -  time .203496 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0954895 seconds
│ │ │ +      idlizer1:  .0112346 seconds
│ │ │ +      idlizer2:  .00979558 seconds
│ │ │ +      minpres:   .00891309 seconds
│ │ │ +  time .136133 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ -      idlizer1:  .0141692 seconds
│ │ │ -      idlizer2:  .0166105 seconds
│ │ │ -      minpres:   .0188805 seconds
│ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00244948 seconds
│ │ │ +      idlizer1:  .0138735 seconds
│ │ │ +      idlizer2:  .0131458 seconds
│ │ │ +      minpres:   .0182919 seconds
│ │ │ +  time .0611503 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ -      idlizer1:  .00935896 seconds
│ │ │ -      idlizer2:  .0181192 seconds
│ │ │ -      minpres:   .0119314 seconds
│ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00312647 seconds
│ │ │ +      idlizer1:  .0108913 seconds
│ │ │ +      idlizer2:  .0198098 seconds
│ │ │ +      minpres:   .0117986 seconds
│ │ │ +  time .0613114 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ -      idlizer1:  .00849894 seconds
│ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .0027537 seconds
│ │ │ +      idlizer1:  .00841786 seconds
│ │ │ +  time .018532 sec  #fractions 5]
│ │ │ + -- used 0.36952s (cpu); 0.301764s (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 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ + -- used 1.21947s (cpu); 0.691576s (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 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ + -- used 1.2477s (cpu); 0.529009s (thread); 0s (gc)
│ │ │  
│ │ │               2 2              2 2                2          2   2     
│ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │       ------------------------------------------------------------------------
│ │ │             2   3               2 2     2   5
│ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o3 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time R' = integralClosure R
│ │ │ - -- used 0.387896s (cpu); 0.303595s (thread); 0s (gc)
│ │ │ + -- used 0.365813s (cpu); 0.286036s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = R'
│ │ │  
│ │ │  o4 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -191,15 +191,15 @@
│ │ │  
│ │ │  o9 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.508329s (cpu); 0.342291s (thread); 0s (gc)
│ │ │ + -- used 0.459947s (cpu); 0.296919s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = R'
│ │ │  
│ │ │  o10 : QuotientRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -278,15 +278,15 @@
│ │ │  
│ │ │  o15 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.507202s (cpu); 0.349308s (thread); 0s (gc)
│ │ │ + -- used 0.486008s (cpu); 0.323538s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = R'
│ │ │  
│ │ │  o16 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -353,15 +353,15 @@ │ │ │ │ │ │ o20 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time R' = integralClosure(R, Strategy => SimplifyFractions)
│ │ │ - -- used 0.400249s (cpu); 0.317688s (thread); 0s (gc)
│ │ │ + -- used 0.411283s (cpu); 0.320842s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = R'
│ │ │  
│ │ │  o21 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -428,15 +428,15 @@ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i26 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc)
│ │ │ + -- used 0.783755s (cpu); 0.531534s (thread); 0s (gc)
│ │ │  
│ │ │  o26 = R'
│ │ │  
│ │ │  o26 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -503,15 +503,15 @@ │ │ │ │ │ │ o30 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i31 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.508579s (cpu); 0.336293s (thread); 0s (gc)
│ │ │ + -- used 0.475759s (cpu); 0.31473s (thread); 0s (gc)
│ │ │  
│ │ │  o31 = R'
│ │ │  
│ │ │  o31 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -578,15 +578,15 @@ │ │ │ │ │ │ o35 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i36 : time R' = integralClosure R
│ │ │ - -- used 0.0428345s (cpu); 0.0428212s (thread); 0s (gc)
│ │ │ + -- used 0.045068s (cpu); 0.045068s (thread); 0s (gc)
│ │ │  
│ │ │  o36 = R'
│ │ │  
│ │ │  o36 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -648,15 +648,15 @@ │ │ │ │ │ │ o40 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.167523s (cpu); 0.0828249s (thread); 0s (gc)
│ │ │ + -- used 0.158146s (cpu); 0.0740257s (thread); 0s (gc)
│ │ │  
│ │ │  o41 = R'
│ │ │  
│ │ │  o41 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -700,15 +700,15 @@ │ │ │ │ │ │ o45 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i46 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.0639871s (cpu); 0.0639869s (thread); 0s (gc)
│ │ │ + -- used 0.0658035s (cpu); 0.0658005s (thread); 0s (gc)
│ │ │  
│ │ │  o46 = R'
│ │ │  
│ │ │  o46 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -751,15 +751,15 @@ │ │ │ │ │ │ o50 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i51 : time R' = integralClosure (R, Strategy => RadicalCodim1)
│ │ │ - -- used 0.0434306s (cpu); 0.0434315s (thread); 0s (gc)
│ │ │ + -- used 0.0499354s (cpu); 0.0497789s (thread); 0s (gc)
│ │ │  
│ │ │  o51 = R'
│ │ │  
│ │ │  o51 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -803,15 +803,15 @@ │ │ │ │ │ │ o55 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i56 : time R' = integralClosure (R, Strategy => Vasconcelos)
│ │ │ - -- used 0.175803s (cpu); 0.0898169s (thread); 0s (gc)
│ │ │ + -- used 0.163208s (cpu); 0.0690922s (thread); 0s (gc)
│ │ │  
│ │ │  o56 = R'
│ │ │  
│ │ │  o56 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -938,15 +938,15 @@ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i67 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc)
│ │ │ + -- used 0.061662s (cpu); 0.0616648s (thread); 0s (gc)
│ │ │  
│ │ │  o67 = R'
│ │ │  
│ │ │  o67 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1062,15 +1062,15 @@ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i78 : time R' = integralClosure(R, Strategy => Radical)
│ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc)
│ │ │ + -- used 0.462522s (cpu); 0.3029s (thread); 0s (gc)
│ │ │  
│ │ │  o78 = R'
│ │ │  
│ │ │  o78 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1104,15 +1104,15 @@ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i82 : time R' = integralClosure(R, Strategy => AllCodimensions)
│ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc)
│ │ │ + -- used 0.502344s (cpu); 0.321716s (thread); 0s (gc)
│ │ │  
│ │ │  o82 = R'
│ │ │  
│ │ │  o82 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1146,20 +1146,20 @@ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1)
│ │ │ - [jacobian time .000532358 sec #minors 4]
│ │ │ + [jacobian time .000537355 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .228455 sec  #fractions 6]
│ │ │ - [step 1:   time .246733 sec  #fractions 6]
│ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc)
│ │ │ + [step 0:   time .213813 sec  #fractions 6]
│ │ │ + [step 1:   time .24629 sec  #fractions 6]
│ │ │ + -- used 0.463889s (cpu); 0.263509s (thread); 0s (gc)
│ │ │  
│ │ │  o86 = R'
│ │ │  
│ │ │  o86 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1193,20 +1193,20 @@ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1)
│ │ │ - [jacobian time .000530094 sec #minors 4]
│ │ │ + [jacobian time .000521638 sec #minors 4]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .210442 sec  #fractions 6]
│ │ │ - [step 1:   time .269174 sec  #fractions 6]
│ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc)
│ │ │ + [step 0:   time .192362 sec  #fractions 6]
│ │ │ + [step 1:   time .254282 sec  #fractions 6]
│ │ │ + -- used 0.450305s (cpu); 0.28162s (thread); 0s (gc)
│ │ │  
│ │ │  o90 = R'
│ │ │  
│ │ │  o90 : QuotientRing
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -1243,20 +1243,20 @@ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, StartWithOneMinor}, Verbosity => 1)
│ │ │ - [jacobian time .000774762 sec #minors 1]
│ │ │ + [jacobian time .00071598 sec #minors 1]
│ │ │  integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │ - [step 0:   time .271111 sec  #fractions 6]
│ │ │ - [step 1:   time .874897 sec  #fractions 6]
│ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc)
│ │ │ + [step 0:   time .243077 sec  #fractions 6]
│ │ │ + [step 1:   time .675484 sec  #fractions 6]
│ │ │ + -- used 0.922337s (cpu); 0.567458s (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 0.387896s (cpu); 0.303595s (thread); 0s (gc) │ │ │ │ + -- used 0.365813s (cpu); 0.286036s (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 0.508329s (cpu); 0.342291s (thread); 0s (gc) │ │ │ │ + -- used 0.459947s (cpu); 0.296919s (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 0.507202s (cpu); 0.349308s (thread); 0s (gc) │ │ │ │ + -- used 0.486008s (cpu); 0.323538s (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 0.400249s (cpu); 0.317688s (thread); 0s (gc) │ │ │ │ + -- used 0.411283s (cpu); 0.320842s (thread); 0s (gc) │ │ │ │ │ │ │ │ o21 = R' │ │ │ │ │ │ │ │ o21 : QuotientRing │ │ │ │ i22 : netList (ideal R')_* │ │ │ │ │ │ │ │ +------------------------------------------------------------------------ │ │ │ │ @@ -363,15 +363,15 @@ │ │ │ │ o24 : Ideal of S │ │ │ │ i25 : R = S/f │ │ │ │ │ │ │ │ o25 = R │ │ │ │ │ │ │ │ o25 : QuotientRing │ │ │ │ i26 : time R' = integralClosure (R, Strategy => RadicalCodim1) │ │ │ │ - -- used 0.79625s (cpu); 0.550685s (thread); 0s (gc) │ │ │ │ + -- used 0.783755s (cpu); 0.531534s (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 0.508579s (cpu); 0.336293s (thread); 0s (gc) │ │ │ │ + -- used 0.475759s (cpu); 0.31473s (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 0.0428345s (cpu); 0.0428212s (thread); 0s (gc) │ │ │ │ + -- used 0.045068s (cpu); 0.045068s (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 0.167523s (cpu); 0.0828249s (thread); 0s (gc) │ │ │ │ + -- used 0.158146s (cpu); 0.0740257s (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 0.0639871s (cpu); 0.0639869s (thread); 0s (gc) │ │ │ │ + -- used 0.0658035s (cpu); 0.0658005s (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 0.0434306s (cpu); 0.0434315s (thread); 0s (gc) │ │ │ │ + -- used 0.0499354s (cpu); 0.0497789s (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 0.175803s (cpu); 0.0898169s (thread); 0s (gc) │ │ │ │ + -- used 0.163208s (cpu); 0.0690922s (thread); 0s (gc) │ │ │ │ │ │ │ │ o56 = R' │ │ │ │ │ │ │ │ o56 : QuotientRing │ │ │ │ i57 : icFractions R │ │ │ │ │ │ │ │ 2 │ │ │ │ @@ -755,15 +755,15 @@ │ │ │ │ o65 : BettiTally │ │ │ │ i66 : R = S/I │ │ │ │ │ │ │ │ o66 = R │ │ │ │ │ │ │ │ o66 : QuotientRing │ │ │ │ i67 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.0622734s (cpu); 0.0622745s (thread); 0s (gc) │ │ │ │ + -- used 0.061662s (cpu); 0.0616648s (thread); 0s (gc) │ │ │ │ │ │ │ │ o67 = R' │ │ │ │ │ │ │ │ o67 : QuotientRing │ │ │ │ i68 : icFractions R │ │ │ │ │ │ │ │ 2 2 │ │ │ │ @@ -839,15 +839,15 @@ │ │ │ │ o76 : BettiTally │ │ │ │ i77 : R = S/I │ │ │ │ │ │ │ │ o77 = R │ │ │ │ │ │ │ │ o77 : QuotientRing │ │ │ │ i78 : time R' = integralClosure(R, Strategy => Radical) │ │ │ │ - -- used 0.500553s (cpu); 0.333412s (thread); 0s (gc) │ │ │ │ + -- used 0.462522s (cpu); 0.3029s (thread); 0s (gc) │ │ │ │ │ │ │ │ o78 = R' │ │ │ │ │ │ │ │ o78 : QuotientRing │ │ │ │ i79 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -863,15 +863,15 @@ │ │ │ │ o80 : PolynomialRing │ │ │ │ i81 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o81 = R │ │ │ │ │ │ │ │ o81 : QuotientRing │ │ │ │ i82 : time R' = integralClosure(R, Strategy => AllCodimensions) │ │ │ │ - -- used 0.543183s (cpu); 0.38548s (thread); 0s (gc) │ │ │ │ + -- used 0.502344s (cpu); 0.321716s (thread); 0s (gc) │ │ │ │ │ │ │ │ o82 = R' │ │ │ │ │ │ │ │ o82 : QuotientRing │ │ │ │ i83 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -887,20 +887,20 @@ │ │ │ │ o84 : PolynomialRing │ │ │ │ i85 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o85 = R │ │ │ │ │ │ │ │ o85 : QuotientRing │ │ │ │ i86 : time R' = integralClosure (R, Strategy => RadicalCodim1, Verbosity => 1) │ │ │ │ - [jacobian time .000532358 sec #minors 4] │ │ │ │ + [jacobian time .000537355 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .228455 sec #fractions 6] │ │ │ │ - [step 1: time .246733 sec #fractions 6] │ │ │ │ - -- used 0.478873s (cpu); 0.304753s (thread); 0s (gc) │ │ │ │ + [step 0: time .213813 sec #fractions 6] │ │ │ │ + [step 1: time .24629 sec #fractions 6] │ │ │ │ + -- used 0.463889s (cpu); 0.263509s (thread); 0s (gc) │ │ │ │ │ │ │ │ o86 = R' │ │ │ │ │ │ │ │ o86 : QuotientRing │ │ │ │ i87 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -916,20 +916,20 @@ │ │ │ │ o88 : PolynomialRing │ │ │ │ i89 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o89 = R │ │ │ │ │ │ │ │ o89 : QuotientRing │ │ │ │ i90 : time R' = integralClosure (R, Strategy => Vasconcelos, Verbosity => 1) │ │ │ │ - [jacobian time .000530094 sec #minors 4] │ │ │ │ + [jacobian time .000521638 sec #minors 4] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .210442 sec #fractions 6] │ │ │ │ - [step 1: time .269174 sec #fractions 6] │ │ │ │ - -- used 0.483329s (cpu); 0.298388s (thread); 0s (gc) │ │ │ │ + [step 0: time .192362 sec #fractions 6] │ │ │ │ + [step 1: time .254282 sec #fractions 6] │ │ │ │ + -- used 0.450305s (cpu); 0.28162s (thread); 0s (gc) │ │ │ │ │ │ │ │ o90 = R' │ │ │ │ │ │ │ │ o90 : QuotientRing │ │ │ │ i91 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 3 2 │ │ │ │ @@ -948,20 +948,20 @@ │ │ │ │ i93 : R = S/sub(I,S) │ │ │ │ │ │ │ │ o93 = R │ │ │ │ │ │ │ │ o93 : QuotientRing │ │ │ │ i94 : time R' = integralClosure (R, Strategy => {Vasconcelos, │ │ │ │ StartWithOneMinor}, Verbosity => 1) │ │ │ │ - [jacobian time .000774762 sec #minors 1] │ │ │ │ + [jacobian time .00071598 sec #minors 1] │ │ │ │ integral closure nvars 4 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ - [step 0: time .271111 sec #fractions 6] │ │ │ │ - [step 1: time .874897 sec #fractions 6] │ │ │ │ - -- used 1.15027s (cpu); 0.732279s (thread); 0s (gc) │ │ │ │ + [step 0: time .243077 sec #fractions 6] │ │ │ │ + [step 1: time .675484 sec #fractions 6] │ │ │ │ + -- used 0.922337s (cpu); 0.567458s (thread); 0s (gc) │ │ │ │ │ │ │ │ o94 = R' │ │ │ │ │ │ │ │ o94 : QuotientRing │ │ │ │ i95 : icFractions R │ │ │ │ │ │ │ │ 2 2 2 2 3 2 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp..._cm__Verbosity_eq_gt..._rp.html │ │ │ @@ -76,52 +76,52 @@ │ │ │ │ │ │ 
    i1 : R = QQ[x,y,z]/ideal(x^8-z^6-y^2*z^4-z^3);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time R' = integralClosure(R, Verbosity => 2)
│ │ │ - [jacobian time .000522279 sec #minors 3]
│ │ │ + [jacobian time .000526737 sec #minors 3]
│ │ │  integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2
│ │ │  
│ │ │   [step 0: 
│ │ │ -      radical (use minprimes) .00233339 seconds
│ │ │ -      idlizer1:  .00747003 seconds
│ │ │ -      idlizer2:  .00844957 seconds
│ │ │ -      minpres:   .00811702 seconds
│ │ │ -  time .0371527 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00252093 seconds
│ │ │ +      idlizer1:  .0079105 seconds
│ │ │ +      idlizer2:  .00874379 seconds
│ │ │ +      minpres:   .00798809 seconds
│ │ │ +  time .038948 sec  #fractions 4]
│ │ │   [step 1: 
│ │ │ -      radical (use minprimes) .00248298 seconds
│ │ │ -      idlizer1:  .0119562 seconds
│ │ │ -      idlizer2:  .0111 seconds
│ │ │ -      minpres:   .0117312 seconds
│ │ │ -  time .0484066 sec  #fractions 4]
│ │ │ +      radical (use minprimes) .00241193 seconds
│ │ │ +      idlizer1:  .013298 seconds
│ │ │ +      idlizer2:  .0114285 seconds
│ │ │ +      minpres:   .0113229 seconds
│ │ │ +  time .0492339 sec  #fractions 4]
│ │ │   [step 2: 
│ │ │ -      radical (use minprimes) .00413852 seconds
│ │ │ -      idlizer1:  .0169512 seconds
│ │ │ -      idlizer2:  .0143372 seconds
│ │ │ -      minpres:   .0137525 seconds
│ │ │ -  time .203496 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .0954895 seconds
│ │ │ +      idlizer1:  .0112346 seconds
│ │ │ +      idlizer2:  .00979558 seconds
│ │ │ +      minpres:   .00891309 seconds
│ │ │ +  time .136133 sec  #fractions 5]
│ │ │   [step 3: 
│ │ │ -      radical (use minprimes) .00279962 seconds
│ │ │ -      idlizer1:  .0141692 seconds
│ │ │ -      idlizer2:  .0166105 seconds
│ │ │ -      minpres:   .0188805 seconds
│ │ │ -  time .0680688 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00244948 seconds
│ │ │ +      idlizer1:  .0138735 seconds
│ │ │ +      idlizer2:  .0131458 seconds
│ │ │ +      minpres:   .0182919 seconds
│ │ │ +  time .0611503 sec  #fractions 5]
│ │ │   [step 4: 
│ │ │ -      radical (use minprimes) .0025556 seconds
│ │ │ -      idlizer1:  .00935896 seconds
│ │ │ -      idlizer2:  .0181192 seconds
│ │ │ -      minpres:   .0119314 seconds
│ │ │ -  time .0555899 sec  #fractions 5]
│ │ │ +      radical (use minprimes) .00312647 seconds
│ │ │ +      idlizer1:  .0108913 seconds
│ │ │ +      idlizer2:  .0198098 seconds
│ │ │ +      minpres:   .0117986 seconds
│ │ │ +  time .0613114 sec  #fractions 5]
│ │ │   [step 5: 
│ │ │ -      radical (use minprimes) .00261851 seconds
│ │ │ -      idlizer1:  .00849894 seconds
│ │ │ -  time .0184597 sec  #fractions 5]
│ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc)
│ │ │ +      radical (use minprimes) .0027537 seconds
│ │ │ +      idlizer1:  .00841786 seconds
│ │ │ +  time .018532 sec  #fractions 5]
│ │ │ + -- used 0.36952s (cpu); 0.301764s (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 .000522279 sec #minors 3] │ │ │ │ + [jacobian time .000526737 sec #minors 3] │ │ │ │ integral closure nvars 3 numgens 1 is S2 codim 1 codimJ 2 │ │ │ │ │ │ │ │ [step 0: │ │ │ │ - radical (use minprimes) .00233339 seconds │ │ │ │ - idlizer1: .00747003 seconds │ │ │ │ - idlizer2: .00844957 seconds │ │ │ │ - minpres: .00811702 seconds │ │ │ │ - time .0371527 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00252093 seconds │ │ │ │ + idlizer1: .0079105 seconds │ │ │ │ + idlizer2: .00874379 seconds │ │ │ │ + minpres: .00798809 seconds │ │ │ │ + time .038948 sec #fractions 4] │ │ │ │ [step 1: │ │ │ │ - radical (use minprimes) .00248298 seconds │ │ │ │ - idlizer1: .0119562 seconds │ │ │ │ - idlizer2: .0111 seconds │ │ │ │ - minpres: .0117312 seconds │ │ │ │ - time .0484066 sec #fractions 4] │ │ │ │ + radical (use minprimes) .00241193 seconds │ │ │ │ + idlizer1: .013298 seconds │ │ │ │ + idlizer2: .0114285 seconds │ │ │ │ + minpres: .0113229 seconds │ │ │ │ + time .0492339 sec #fractions 4] │ │ │ │ [step 2: │ │ │ │ - radical (use minprimes) .00413852 seconds │ │ │ │ - idlizer1: .0169512 seconds │ │ │ │ - idlizer2: .0143372 seconds │ │ │ │ - minpres: .0137525 seconds │ │ │ │ - time .203496 sec #fractions 5] │ │ │ │ + radical (use minprimes) .0954895 seconds │ │ │ │ + idlizer1: .0112346 seconds │ │ │ │ + idlizer2: .00979558 seconds │ │ │ │ + minpres: .00891309 seconds │ │ │ │ + time .136133 sec #fractions 5] │ │ │ │ [step 3: │ │ │ │ - radical (use minprimes) .00279962 seconds │ │ │ │ - idlizer1: .0141692 seconds │ │ │ │ - idlizer2: .0166105 seconds │ │ │ │ - minpres: .0188805 seconds │ │ │ │ - time .0680688 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00244948 seconds │ │ │ │ + idlizer1: .0138735 seconds │ │ │ │ + idlizer2: .0131458 seconds │ │ │ │ + minpres: .0182919 seconds │ │ │ │ + time .0611503 sec #fractions 5] │ │ │ │ [step 4: │ │ │ │ - radical (use minprimes) .0025556 seconds │ │ │ │ - idlizer1: .00935896 seconds │ │ │ │ - idlizer2: .0181192 seconds │ │ │ │ - minpres: .0119314 seconds │ │ │ │ - time .0555899 sec #fractions 5] │ │ │ │ + radical (use minprimes) .00312647 seconds │ │ │ │ + idlizer1: .0108913 seconds │ │ │ │ + idlizer2: .0198098 seconds │ │ │ │ + minpres: .0117986 seconds │ │ │ │ + time .0613114 sec #fractions 5] │ │ │ │ [step 5: │ │ │ │ - radical (use minprimes) .00261851 seconds │ │ │ │ - idlizer1: .00849894 seconds │ │ │ │ - time .0184597 sec #fractions 5] │ │ │ │ - -- used 0.435062s (cpu); 0.361171s (thread); 0s (gc) │ │ │ │ + radical (use minprimes) .0027537 seconds │ │ │ │ + idlizer1: .00841786 seconds │ │ │ │ + time .018532 sec #fractions 5] │ │ │ │ + -- used 0.36952s (cpu); 0.301764s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = R' │ │ │ │ │ │ │ │ o2 : QuotientRing │ │ │ │ i3 : trim ideal R' │ │ │ │ │ │ │ │ 3 2 2 2 4 4 │ │ ├── ./usr/share/doc/Macaulay2/IntegralClosure/html/_integral__Closure_lp__Ideal_cm__Ring__Element_cm__Z__Z_rp.html │ │ │ @@ -114,29 +114,29 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time integralClosure J
│ │ │ - -- used 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ + -- used 1.21947s (cpu); 0.691576s (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 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ + -- used 1.2477s (cpu); 0.529009s (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 1.00348s (cpu); 0.739924s (thread); 0s (gc)
│ │ │ │ + -- used 1.21947s (cpu); 0.691576s (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 0.733674s (cpu); 0.56695s (thread); 0s (gc)
│ │ │ │ + -- used 1.2477s (cpu); 0.529009s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               2 2              2 2                2          2   2
│ │ │ │  o5 = ideal (b c  - 16000a*c, a c  - 16000b*c, a*b c - 16000a , a b*c -
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │             2   3               2 2     2   5
│ │ │ │       16000b , a c - 16000a*b, a b  + 3c , a b + 15997a*c)
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  c2NocmVpZXJHcmFwaA==
│ │ │  #:len=1712
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NocmVpZXIgZ3JhcGggb2YgYSBmaW5p
│ │ │  dGUgZ3JvdXAiLCAibGluZW51bSIgPT4gMjYyLCBJbnB1dHMgPT4ge1NQQU57VFR7IkcifSwiLCAi
│ │ ├── ./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)
│ │ │ - -- .00739572s elapsed
│ │ │ + -- .00285722s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -51,10 +51,10 @@
│ │ │           0   1
│ │ │  
│ │ │  i6 : T.cache.?equivariantHilbert
│ │ │  
│ │ │  o6 = true
│ │ │  
│ │ │  i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000676866s elapsed
│ │ │ + -- .000676728s elapsed
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_hironaka__Decomposition.out
│ │ │ @@ -5,13 +5,13 @@
│ │ │  o1 = QQ[x..y] <- <| 0 -1 |>
│ │ │                    | 1 0  |
│ │ │  
│ │ │  o1 : FiniteGroupAction
│ │ │  
│ │ │  i2 : hironakaDecomposition C4
│ │ │  
│ │ │ -        2    2   2 2        3       3
│ │ │ -o2 = ({x  + y , x y }, {1, x y - x*y })
│ │ │ +        2    2     3       3        4    4
│ │ │ +o2 = ({x  + y , - x y + x*y }, {1, x  + y })
│ │ │  
│ │ │  o2 : Sequence
│ │ │  
│ │ │  i3 :
│ │ ├── ./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 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ + -- used 0.449998s (cpu); 0.327006s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   4    4
│ │ │ -o4 = {z , x  + y , x  + y }
│ │ │ +       2   2    2   2 2
│ │ │ +o4 = {z , x  + y , x y }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ + -- used 0.386015s (cpu); 0.316236s (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 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ + -- used 0.10863s (cpu); 0.0419354s (thread); 0s (gc)
│ │ │  
│ │ │            3       3
│ │ │  o6 = {1, x y - x*y }
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ + -- used 1.13054s (cpu); 0.868848s (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
│ │ │ - -- .422203s elapsed
│ │ │ + -- .341975s 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.
│ │ │  
│ │ │ - -- .339469s elapsed
│ │ │ + -- .241384s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_invariants_lp..._cm__Strategy_eq_gt..._rp.out
│ │ │ @@ -14,28 +14,28 @@
│ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : elapsedTime invariants S4
│ │ │ - -- .735714s elapsed
│ │ │ + -- .384335s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │        4
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .126449s elapsed
│ │ │ + -- .0655101s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants.out
│ │ │ @@ -16,16 +16,16 @@
│ │ │                    | 0 0 1 |  | 1 0 0 |
│ │ │                    | 1 0 0 |  | 0 0 1 |
│ │ │  
│ │ │  o3 : FiniteGroupAction
│ │ │  
│ │ │  i4 : primaryInvariants S3
│ │ │  
│ │ │ -                                   3    3    3
│ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ +                  2    2    2   2       2    2     2       2      2
│ │ │ +o4 = {x + y + z, x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : K=GF(101)
│ │ │  
│ │ │  o5 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/example-output/_primary__Invariants_lp..._cm__Dade_eq_gt..._rp.out
│ │ │ @@ -77,16 +77,16 @@
│ │ │                2 4              5              5          6
│ │ │       69234300y z  + 13824000x*z  + 13824000y*z  + 810000z }
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : primaryInvariants(S3)
│ │ │  
│ │ │ -                  2    2    2
│ │ │ -o5 = {x + y + z, x  + y  + z , x*y*z}
│ │ │ +                                   3    3    3
│ │ │ +o5 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : K=GF(101)
│ │ │  
│ │ │  o6 = K
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_equivariant__Hilbert.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o4 = false
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime equivariantHilbertSeries(T, Order => 5)
│ │ │ - -- .00739572s elapsed
│ │ │ + -- .00285722s elapsed
│ │ │  
│ │ │                    -1    -1       2 2              -2    -1 -1    -2  2  
│ │ │  o5 = 1 + (ζ ζ  + ζ   + ζ  )T + (ζ ζ  + ζ  + ζ  + ζ   + ζ  ζ   + ζ  )T  +
│ │ │             0 1    1     0        0 1    0    1    1     0  1     0      
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3    2        2      -1        -3    -1      -1 -2    -2 -1    -3  3
│ │ │       (ζ ζ  + ζ ζ  + ζ ζ  + ζ ζ   + 1 + ζ   + ζ  ζ  + ζ  ζ   + ζ  ζ   + ζ  )T 
│ │ │ @@ -129,15 +129,15 @@
│ │ │  
│ │ │  o6 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime equivariantHilbertSeries(T, Order => 5);
│ │ │ - -- .000676866s elapsed
    │ │ │ + -- .000676728s 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) │ │ │ │ - -- .00739572s elapsed │ │ │ │ + -- .00285722s elapsed │ │ │ │ │ │ │ │ -1 -1 2 2 -2 -1 -1 -2 2 │ │ │ │ o5 = 1 + (ζ ζ + ζ + ζ )T + (ζ ζ + ζ + ζ + ζ + ζ ζ + ζ │ │ │ │ )T + │ │ │ │ 0 1 1 0 0 1 0 1 1 0 1 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 3 3 2 2 -1 -3 -1 -1 -2 -2 -1 -3 3 │ │ │ │ @@ -57,13 +57,13 @@ │ │ │ │ │ │ │ │ o5 : ZZ[ζ ..ζ ][T] │ │ │ │ 0 1 │ │ │ │ i6 : T.cache.?equivariantHilbert │ │ │ │ │ │ │ │ o6 = true │ │ │ │ i7 : elapsedTime equivariantHilbertSeries(T, Order => 5); │ │ │ │ - -- .000676866s elapsed │ │ │ │ + -- .000676728s elapsed │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _e_q_u_i_v_a_r_i_a_n_t_H_i_l_b_e_r_t is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/InvariantRing/AbelianGroupsDoc.m2:225:0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hironaka__Decomposition.html │ │ │ @@ -92,16 +92,16 @@ │ │ │ o1 : FiniteGroupAction │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : hironakaDecomposition C4
│ │ │  
│ │ │ -        2    2   2 2        3       3
│ │ │ -o2 = ({x  + y , x y }, {1, x y - x*y })
│ │ │ +        2    2     3       3        4    4
│ │ │ +o2 = ({x  + y , - x y + x*y }, {1, x  + y })
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │

    From the output one sees that QQ[x,y]C4=QQ[f1f2]$\oplus$QQ[f1f2](x4+y4), where f1=x2+y2 and f2=xy3-x3y.

    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -39,16 +39,16 @@ │ │ │ │ │ │ │ │ o1 = QQ[x..y] <- <| 0 -1 |> │ │ │ │ | 1 0 | │ │ │ │ │ │ │ │ o1 : FiniteGroupAction │ │ │ │ i2 : hironakaDecomposition C4 │ │ │ │ │ │ │ │ - 2 2 2 2 3 3 │ │ │ │ -o2 = ({x + y , x y }, {1, x y - x*y }) │ │ │ │ + 2 2 3 3 4 4 │ │ │ │ +o2 = ({x + y , - x y + x*y }, {1, x + y }) │ │ │ │ │ │ │ │ o2 : Sequence │ │ │ │ From the output one sees that QQ[x,y]C4=QQ[f1f2]$\oplus$QQ[f1f2](x4+y4), where │ │ │ │ f1=x2+y2 and f2=xy3-x3y. │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently hironakaDecomposition works only with polynomial rings and matrices │ │ │ │ over fields of characteristic 0. │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_hsop_spalgorithms.html │ │ │ @@ -97,26 +97,26 @@ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : time P1=primaryInvariants C4xC2
│ │ │ - -- used 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ + -- used 0.449998s (cpu); 0.327006s (thread); 0s (gc)
│ │ │  
│ │ │ -       2   2    2   4    4
│ │ │ -o4 = {z , x  + y , x  + y }
│ │ │ +       2   2    2   2 2
│ │ │ +o4 = {z , x  + y , x y }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ + -- used 0.386015s (cpu); 0.316236s (thread); 0s (gc)
│ │ │  
│ │ │                     8                 7                   6 2  
│ │ │  o5 = {656100000000x  - 4738500000000x y + 10209037500000x y  -
│ │ │       ------------------------------------------------------------------------
│ │ │                     5 3                  4 4                 3 5  
│ │ │       1232156250000x y  - 14757374609375x y  + 1232156250000x y  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -173,26 +173,26 @@
│ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time secondaryInvariants(P1,C4xC2)
│ │ │ - -- used 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ + -- used 0.10863s (cpu); 0.0419354s (thread); 0s (gc)
│ │ │  
│ │ │            3       3
│ │ │  o6 = {1, x y - x*y }
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ + -- used 1.13054s (cpu); 0.868848s (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 0.487583s (cpu); 0.324563s (thread); 0s (gc)
│ │ │ │ + -- used 0.449998s (cpu); 0.327006s (thread); 0s (gc)
│ │ │ │  
│ │ │ │ -       2   2    2   4    4
│ │ │ │ -o4 = {z , x  + y , x  + y }
│ │ │ │ +       2   2    2   2 2
│ │ │ │ +o4 = {z , x  + y , x y }
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : time P2=primaryInvariants(C4xC2,Dade=>true)
│ │ │ │ - -- used 0.444139s (cpu); 0.375345s (thread); 0s (gc)
│ │ │ │ + -- used 0.386015s (cpu); 0.316236s (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 0.137804s (cpu); 0.0658453s (thread); 0s (gc)
│ │ │ │ + -- used 0.10863s (cpu); 0.0419354s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            3       3
│ │ │ │  o6 = {1, x y - x*y }
│ │ │ │  
│ │ │ │  o6 : List
│ │ │ │  i7 : time secondaryInvariants(P2,C4xC2)
│ │ │ │ - -- used 1.41802s (cpu); 1.13291s (thread); 0s (gc)
│ │ │ │ + -- used 1.13054s (cpu); 0.868848s (thread); 0s (gc)
│ │ │ │  
│ │ │ │            2   2    2   4   2 2    2 2   2 2   3       3   4    4   6   2 4
│ │ │ │  o7 = {1, z , x  + y , z , x z  + y z , x y , x y - x*y , x  + y , z , x z  +
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        2 4   2 2 2   3   2      3 2   4 2    4 2   4 2    2 4   5       5   6
│ │ │ │       y z , x y z , x y*z  - x*y z , x z  + y z , x y  + x y , x y - x*y , x
│ │ │ │       ------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Degree__Bound_eq_gt..._rp.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .422203s elapsed
│ │ │ + -- .341975s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -122,15 +122,15 @@
│ │ │
    		│ │ │ 
    i5 : elapsedTime invariants(S4, DegreeBound => 4)
│ │ │  
│ │ │  Warning: stopping condition not met!
│ │ │  Output may not generate the entire ring of invariants.
│ │ │  Increase value of DegreeBound.
│ │ │  
│ │ │ - -- .339469s elapsed
│ │ │ + -- .241384s 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
│ │ │ │ - -- .422203s elapsed
│ │ │ │ + -- .341975s 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.
│ │ │ │  
│ │ │ │ - -- .339469s elapsed
│ │ │ │ + -- .241384s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o5 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_invariants_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : elapsedTime invariants S4
│ │ │ - -- .735714s elapsed
│ │ │ + -- .384335s elapsed
│ │ │  
│ │ │                            2    2    2    2   3    3    3    3   4    4    4  
│ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3  
│ │ │       ------------------------------------------------------------------------
│ │ │        4
│ │ │       x }
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ - -- .126449s elapsed
│ │ │ + -- .0655101s elapsed
│ │ │  
│ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,27 +26,27 @@
│ │ │ │  o3 = R <- <| 0 1 0 0 |, | 0 0 0 1 |>
│ │ │ │             | 1 0 0 0 |  | 1 0 0 0 |
│ │ │ │             | 0 0 1 0 |  | 0 1 0 0 |
│ │ │ │             | 0 0 0 1 |  | 0 0 1 0 |
│ │ │ │  
│ │ │ │  o3 : FiniteGroupAction
│ │ │ │  i4 : elapsedTime invariants S4
│ │ │ │ - -- .735714s elapsed
│ │ │ │ + -- .384335s elapsed
│ │ │ │  
│ │ │ │                            2    2    2    2   3    3    3    3   4    4    4
│ │ │ │  o4 = {x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  + x , x  + x  + x  +
│ │ │ │         1    2    3    4   1    2    3    4   1    2    3    4   1    2    3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │        4
│ │ │ │       x }
│ │ │ │        4
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : elapsedTime invariants(S4, Strategy => "LinearAlgebra")
│ │ │ │ - -- .126449s elapsed
│ │ │ │ + -- .0655101s elapsed
│ │ │ │  
│ │ │ │  o5 = {x  + x  + x  + x , x x  + x x  + x x  + x x  + x x  + x x , x x x  +
│ │ │ │         1    2    3    4   1 2    1 3    2 3    1 4    2 4    3 4   1 2 3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x x x  + x x x  + x x x , x x x x }
│ │ │ │        1 2 4    1 3 4    2 3 4   1 2 3 4
│ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants.html
│ │ │ @@ -106,16 +106,16 @@
│ │ │  o3 : FiniteGroupAction
    │ │ │
    │ │ │ 
    i4 : primaryInvariants S3
│ │ │  
│ │ │ -                                   3    3    3
│ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │ +                  2    2    2   2       2    2     2       2      2
│ │ │ +o4 = {x + y + z, x  + y  + z , x y + x*y  + x z + y z + x*z  + y*z }
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,16 +38,16 @@ │ │ │ │ o3 = QQ[x..z] <- <| 0 1 0 |, | 0 1 0 |> │ │ │ │ | 0 0 1 | | 1 0 0 | │ │ │ │ | 1 0 0 | | 0 0 1 | │ │ │ │ │ │ │ │ o3 : FiniteGroupAction │ │ │ │ i4 : primaryInvariants S3 │ │ │ │ │ │ │ │ - 3 3 3 │ │ │ │ -o4 = {x + y + z, x*y + x*z + y*z, x + y + z } │ │ │ │ + 2 2 2 2 2 2 2 2 2 │ │ │ │ +o4 = {x + y + z, x + y + z , x y + x*y + x z + y z + x*z + y*z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with │ │ │ │ 101 elements. │ │ │ │ i5 : K=GF(101) │ │ │ │ │ │ │ │ o5 = K │ │ ├── ./usr/share/doc/Macaulay2/InvariantRing/html/_primary__Invariants_lp..._cm__Dade_eq_gt..._rp.html │ │ │ @@ -169,16 +169,16 @@ │ │ │
    │ │ │

    Compare this result to the hsop output when Dade is left to its default value false.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : primaryInvariants(S3)
│ │ │  
│ │ │ -                  2    2    2
│ │ │ -o5 = {x + y + z, x  + y  + z , x*y*z}
│ │ │ +                                   3    3    3
│ │ │ +o5 = {x + y + z, x*y + x*z + y*z, x  + y  + z }
│ │ │  
│ │ │  o5 : List
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -94,16 +94,16 @@ │ │ │ │ 69234300y z + 13824000x*z + 13824000y*z + 810000z } │ │ │ │ │ │ │ │ o4 : List │ │ │ │ Compare this result to the hsop output when Dade is left to its default value │ │ │ │ _f_a_l_s_e. │ │ │ │ i5 : primaryInvariants(S3) │ │ │ │ │ │ │ │ - 2 2 2 │ │ │ │ -o5 = {x + y + z, x + y + z , x*y*z} │ │ │ │ + 3 3 3 │ │ │ │ +o5 = {x + y + z, x*y + x*z + y*z, x + y + z } │ │ │ │ │ │ │ │ o5 : List │ │ │ │ Below, the invariant ring QQ[x,y,z]S3 is calculated with K being the field with │ │ │ │ 101 elements. │ │ │ │ i6 : K=GF(101) │ │ │ │ │ │ │ │ o6 = K │ │ ├── ./usr/share/doc/Macaulay2/InverseSystems/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ ZnJvbUR1YWwoTWF0cml4KQ== │ │ │ #:len=249 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzMyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmcm9tRHVhbCxNYXRyaXgpLCJmcm9tRHVhbChNYXRy │ │ ├── ./usr/share/doc/Macaulay2/InvolutiveBases/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=38 │ │ │ aW52UmVkdWNlKFJpbmdFbGVtZW50LEludm9sdXRpdmVCYXNpcyk= │ │ │ #:len=299 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaW52UmVkdWNlLFJpbmdFbGVtZW50LEludm9sdXRp │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ aXNJc29tb3JwaGljKE1hdHJpeCxNYXRyaXgp │ │ │ #:len=270 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTMxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0lzb21vcnBoaWMsTWF0cml4LE1hdHJpeCksImlz │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/example-output/_is__Isomorphic.out │ │ │ @@ -156,20 +156,20 @@ │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_2 x_1 x_0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_0^2 -x_2 x_1 | │ │ │ {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | {-1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x_0 -x_2 x_1 -x_3 x_2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -x_1^2 -x_3 x_2 | │ │ │ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ i23 : elapsedTime isIsomorphic(T1, T2) │ │ │ - -- 1.69407s elapsed │ │ │ + -- 1.43641s elapsed │ │ │ │ │ │ o23 = true │ │ │ │ │ │ i24 : elapsedTime isomorphism(T1, T2) │ │ │ - -- .000029024s elapsed │ │ │ + -- .000020646s elapsed │ │ │ │ │ │ o24 = {-1} | 1 -3976 -13490 13495 -2886 2577 14757 -881 7677 │ │ │ {-1} | -2527 -13566 2778 -6934 -14806 4619 -13099 6022 -10907 │ │ │ {-1} | -15420 5642 1489 1354 4591 11881 -5253 7296 -1098 │ │ │ {-1} | 7909 -12428 -2260 -8465 12113 -6893 8411 4186 -9393 │ │ │ {-1} | -9615 2934 10440 5015 8145 -5585 1360 3295 12851 │ │ │ {-1} | -4881 -7984 12700 -10391 -10009 -14538 13207 262 -6500 │ │ ├── ./usr/share/doc/Macaulay2/Isomorphism/html/_is__Isomorphic.html │ │ │ @@ -333,23 +333,23 @@ │ │ │ 40 │ │ │ o22 : S-module, subquotient of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ - -- 1.69407s elapsed
│ │ │ + -- 1.43641s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │
    │ │ │ 
    i24 : elapsedTime isomorphism(T1, T2)
│ │ │ - -- .000029024s elapsed
│ │ │ + -- .000020646s elapsed
│ │ │  
│ │ │  o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
│ │ │        {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
│ │ │        {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
│ │ │        {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
│ │ │        {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
│ │ │        {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500
│ │ │ ├── html2text {}
│ │ │ │ @@ -724,19 +724,19 @@
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   0      0    0
│ │ │ │  0      0    0   0      0    0   0      0    0   0      0    0   -x_1^2 -x_3 x_2
│ │ │ │  |
│ │ │ │  
│ │ │ │                                  40
│ │ │ │  o22 : S-module, subquotient of S
│ │ │ │  i23 : elapsedTime isIsomorphic(T1, T2)
│ │ │ │ - -- 1.69407s elapsed
│ │ │ │ + -- 1.43641s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isomorphism(T1, T2)
│ │ │ │ - -- .000029024s elapsed
│ │ │ │ + -- .000020646s 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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=13
│ │ │  TmFtZVNlcGFyYXRvcg==
│ │ │  #:len=210
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzQzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJOYW1lU2VwYXJhdG9yIiwiTmFtZVNlcGFyYXRvciIs
│ │ ├── ./usr/share/doc/Macaulay2/JSON/example-output/_from__J__S__O__N.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  
│ │ │  o8 = {1, 2, 3}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-35451-0/0.json
│ │ │ +o9 = /tmp/M2-48780-0/0.json
│ │ │  
│ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ +o10 = /tmp/M2-48780-0/0.json
│ │ │  
│ │ │  o10 : File
│ │ │  
│ │ │  i11 : fromJSON openIn jsonFile
│ │ │  
│ │ │  o11 = {1, 2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/JSON/html/_from__J__S__O__N.html
│ │ │ @@ -170,22 +170,22 @@
│ │ │            
    The input may also be a file containing JSON data.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i9 : jsonFile = temporaryFileName() | ".json"
│ │ │  
│ │ │ -o9 = /tmp/M2-35451-0/0.json
    
│ │ │ +o9 = /tmp/M2-48780-0/0.json
│ │ │              
    
│ │ │                
    i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │  
│ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ +o10 = /tmp/M2-48780-0/0.json
│ │ │  
│ │ │  o10 : File
    
│ │ │              
    
│ │ │                
    i11 : fromJSON openIn jsonFile
│ │ │ ├── html2text {}
│ │ │ │ @@ -49,18 +49,18 @@
│ │ │ │  
│ │ │ │  o8 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  The input may also be a file containing JSON data.
│ │ │ │  i9 : jsonFile = temporaryFileName() | ".json"
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-35451-0/0.json
│ │ │ │ +o9 = /tmp/M2-48780-0/0.json
│ │ │ │  i10 : jsonFile << "[1, 2, 3]" << endl << close
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-35451-0/0.json
│ │ │ │ +o10 = /tmp/M2-48780-0/0.json
│ │ │ │  
│ │ │ │  o10 : File
│ │ │ │  i11 : fromJSON openIn jsonFile
│ │ │ │  
│ │ │ │  o11 = {1, 2, 3}
│ │ │ │  
│ │ │ │  o11 : List
│ │ ├── ./usr/share/doc/Macaulay2/JSONRPC/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  bWFrZVJlcXVlc3QoTGlzdCk=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUmVxdWVzdCxMaXN0KSwibWFrZVJlcXVlc3Qo
│ │ ├── ./usr/share/doc/Macaulay2/Jets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  amV0cyhaWixBZmZpbmVWYXJpZXR5KQ==
│ │ │  #:len=2469
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGpldHMgb2YgYW4gYWZmaW5lIHZh
│ │ │  cmlldHkiLCAibGluZW51bSIgPT4gMTcxMCwgSW5wdXRzID0+IHtTUEFOe1RUeyJuIn0sIiwgIixT
│ │ ├── ./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)
│ │ │ - -- .0023111s elapsed
│ │ │ + -- .00240081s 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
│ │ │ - -- .424366s elapsed
│ │ │ + -- .209693s 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)
│ │ │ - -- .00766552s elapsed
│ │ │ + -- .00720051s 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)
│ │ │ - -- .0025072s elapsed
│ │ │ + -- .0027853s 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)
│ │ │ - -- .00226291s elapsed
│ │ │ + -- .00249627s 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)
│ │ │ - -- .0113717s elapsed
│ │ │ + -- .0115217s 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)
│ │ │ - -- .000732911s elapsed
│ │ │ + -- .000771632s elapsed
│ │ │  
│ │ │                                     QQ[x0, y0][x1, y1][x2, y2]                          2                    2
│ │ │  o27 = map (QQ[t0][t1][t2], ------------------------------------------, {t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                              2                 2
│ │ │                             (2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                           QQ[x0, y0][x1, y1][x2, y2]
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Example_sp1.html
│ │ │ @@ -92,27 +92,27 @@
│ │ │          
    
│ │ │            
    However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -27,23 +27,23 @@
│ │ │ │  However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial
│ │ │ │  ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is
│ │ │ │  generated by the individual terms in the generators of the ideal of jets. This
│ │ │ │  observation provides an alternative algorithm for computing radicals of jets of
│ │ │ │  monomial ideals, which can be faster than the default radical computation in
│ │ │ │  Macaulay2.
│ │ │ │  i4 : elapsedTime jetsRadical(2,I)
│ │ │ │ - -- .0023111s elapsed
│ │ │ │ + -- .00240081s 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
│ │ │ │ - -- .424366s elapsed
│ │ │ │ + -- .209693s elapsed
│ │ │ │  
│ │ │ │  o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2)
│ │ │ │  
│ │ │ │  o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]
│ │ │ │  For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes
│ │ ├── ./usr/share/doc/Macaulay2/Jets/html/___Storing_sp__Computations.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │  
│ │ │  o7 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -151,26 +151,26 @@
│ │ │                                 | x0^2-y0        |
│ │ │                   jetsMaxOrder => 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -295,15 +295,15 @@
│ │ │  
│ │ │  o23 = false
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : elapsedTime jetsRadical(2,I)
│ │ │ - -- .0023111s elapsed
│ │ │ + -- .00240081s 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
│ │ │ - -- .424366s elapsed
│ │ │ + -- .209693s 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)
│ │ │ - -- .00766552s elapsed
│ │ │ + -- .00720051s 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)
│ │ │ - -- .0025072s elapsed
│ │ │ + -- .0027853s 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)
│ │ │ - -- .00226291s elapsed
│ │ │ + -- .00249627s 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)
│ │ │ - -- .0113717s elapsed
│ │ │ + -- .0115217s elapsed
│ │ │  
│ │ │                                                QQ[x0, y0][x1, y1][x2, y2][x3, y3]                                                      2                    2
│ │ │  o24 = map (QQ[t0][t1][t2][t3], ----------------------------------------------------------------, {t3, 2t0*t3 + 2t1*t2, t2, 2t0*t2 + t1 , t1, 2t0*t1, t0, t0 })
│ │ │                                                                        2                 2
│ │ │                                 (2x0*x3 - y3 + 2x1*x2, 2x0*x2 - y2 + x1 , 2x0*x1 - y1, x0  - y0)
│ │ │  
│ │ │                                                      QQ[x0, y0][x1, y1][x2, y2][x3, y3]
│ │ │ @@ -329,15 +329,15 @@
│ │ │                                 | t0 t0^2        |
│ │ │                   jetsMaxOrder => 3
    
│ │ │              
    
│ │ │                
    i27 : elapsedTime jets(2,f)
│ │ │ - -- .000732911s elapsed
│ │ │ + -- .000771632s 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)
│ │ │ │ - -- .00766552s elapsed
│ │ │ │ + -- .00720051s 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)
│ │ │ │ - -- .0025072s elapsed
│ │ │ │ + -- .0027853s 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)
│ │ │ │ - -- .00226291s elapsed
│ │ │ │ + -- .00249627s 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)
│ │ │ │ - -- .0113717s elapsed
│ │ │ │ + -- .0115217s 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)
│ │ │ │ - -- .000732911s elapsed
│ │ │ │ + -- .000771632s 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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  Y2Fub25pY2FsSG9tb3RvcGllcyguLi4sRmluZUdyYWRpbmc9Pi4uLik=
│ │ │  #:len=298
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTU5Niwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2Fub25pY2FsSG9tb3RvcGllcyxGaW5lR3JhZGlu
│ │ ├── ./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
│ │ │ - -- .0249269s elapsed
│ │ │ + -- .0234806s 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)
│ │ │ - -- .00229463s elapsed
│ │ │ - -- .00936212s elapsed
│ │ │ - -- .0239474s elapsed
│ │ │ - -- .011302s elapsed
│ │ │ - -- .00360161s elapsed
│ │ │ + -- .00256122s elapsed
│ │ │ + -- .00617979s elapsed
│ │ │ + -- .0243346s elapsed
│ │ │ + -- .00902719s elapsed
│ │ │ + -- .00372951s 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)
│ │ │ - -- .00231006s elapsed
│ │ │ - -- .0217836s elapsed
│ │ │ - -- .0230895s elapsed
│ │ │ - -- .0101838s elapsed
│ │ │ - -- .00353337s elapsed
│ │ │ - -- .294917s elapsed
│ │ │ + -- .00217748s elapsed
│ │ │ + -- .00620783s elapsed
│ │ │ + -- .0233805s elapsed
│ │ │ + -- .00996756s elapsed
│ │ │ + -- .00375936s elapsed
│ │ │ + -- .282847s 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
│ │ │ - -- .125196s elapsed
│ │ │ + -- .163269s 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);
│ │ │ - -- .00475633s elapsed
│ │ │ - -- .0968306s elapsed
│ │ │ - -- .111433s elapsed
│ │ │ - -- 1.47579s elapsed
│ │ │ - -- .551722s elapsed
│ │ │ - -- .0398804s elapsed
│ │ │ - -- .00727613s elapsed
│ │ │ - -- 6.05064s elapsed
│ │ │ + -- .00473581s elapsed
│ │ │ + -- .0168751s elapsed
│ │ │ + -- .0931932s elapsed
│ │ │ + -- .809624s elapsed
│ │ │ + -- .394797s elapsed
│ │ │ + -- .0393118s elapsed
│ │ │ + -- .00686754s elapsed
│ │ │ + -- 4.43102s 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)
│ │ │ - -- .00224642s elapsed
│ │ │ - -- .010053s elapsed
│ │ │ - -- .0365563s elapsed
│ │ │ - -- .0100498s elapsed
│ │ │ - -- .00353941s elapsed
│ │ │ + -- .00234641s elapsed
│ │ │ + -- .00564509s elapsed
│ │ │ + -- .0232305s elapsed
│ │ │ + -- .00891118s elapsed
│ │ │ + -- .00361696s 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
│ │ │ - -- .196215s elapsed
│ │ │ + -- .174358s 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);
│ │ │ - -- .00490087s elapsed
│ │ │ - -- .0189491s elapsed
│ │ │ - -- .0966489s elapsed
│ │ │ - -- 1.19159s elapsed
│ │ │ - -- .524549s elapsed
│ │ │ - -- .0405632s elapsed
│ │ │ - -- .00680951s elapsed
│ │ │ - -- 5.37805s elapsed
│ │ │ + -- .00447441s elapsed
│ │ │ + -- .0165111s elapsed
│ │ │ + -- .0970322s elapsed
│ │ │ + -- .843674s elapsed
│ │ │ + -- .39112s elapsed
│ │ │ + -- .0361504s elapsed
│ │ │ + -- .00681437s elapsed
│ │ │ + -- 4.49428s 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)
│ │ │ - -- .00631474s elapsed
│ │ │ - -- .0125312s elapsed
│ │ │ + -- .00684823s elapsed
│ │ │ + -- .0119311s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299471s elapsed
│ │ │ + -- .000231598s elapsed
│ │ │  1
│ │ │ - -- .000146984s elapsed
│ │ │ + -- .000187525s elapsed
│ │ │  1
│ │ │ - -- .00012821s elapsed
│ │ │ + -- .000197935s elapsed
│ │ │  1
│ │ │ - -- .000124793s elapsed
│ │ │ + -- .000144851s elapsed
│ │ │  1
│ │ │ - -- .000129423s elapsed
│ │ │ + -- .000153222s elapsed
│ │ │  2
│ │ │ - -- .000136796s elapsed
│ │ │ + -- .000147206s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000137637s elapsed
│ │ │ + -- .000153237s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000148369s elapsed
│ │ │ + -- .000161622s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144691s elapsed
│ │ │ + -- .000156548s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132719s elapsed
│ │ │ + -- .000203872s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000125866s elapsed
│ │ │ + -- .000145923s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132297s elapsed
│ │ │ + -- .000151677s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140242s elapsed
│ │ │ + -- .000136144s elapsed
│ │ │  2
│ │ │ - -- .000122429s elapsed
│ │ │ + -- .000138329s elapsed
│ │ │  2
│ │ │ - -- .000133088s elapsed
│ │ │ + -- .000146644s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120466s elapsed
│ │ │ + -- .000141976s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136004s elapsed
│ │ │ + -- .000155456s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000131376s elapsed
│ │ │ + -- .000169861s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000208069s elapsed
│ │ │ + -- .000171294s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140834s elapsed
│ │ │ + -- .000158089s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000134101s elapsed
│ │ │ + -- .000164858s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126856s elapsed
│ │ │ + -- .000140399s elapsed
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000136645s elapsed
│ │ │  1
│ │ │ - -- .000130714s elapsed
│ │ │ + -- .00014147s elapsed
│ │ │  1
│ │ │ - -- .000156823s elapsed
│ │ │ + -- .000143442s elapsed
│ │ │  1
│ │ │ - -- .000123991s elapsed
│ │ │ + -- .000140393s 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
│ │ │ - -- .268927s elapsed
│ │ │ - -- .33298s elapsed
│ │ │ - -- .23855s elapsed
│ │ │ - -- .286255s elapsed
│ │ │ - -- .382732s elapsed
│ │ │ - -- .298869s elapsed
│ │ │ + -- .122356s elapsed
│ │ │ + -- .144495s elapsed
│ │ │ + -- .149958s elapsed
│ │ │ + -- .147845s elapsed
│ │ │ + -- .15113s elapsed
│ │ │ + -- .146368s 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)
│ │ │ - -- .0030227s elapsed
│ │ │ - -- .00794913s elapsed
│ │ │ - -- .023886s elapsed
│ │ │ - -- .0114692s elapsed
│ │ │ - -- .00368452s elapsed
│ │ │ + -- .00274193s elapsed
│ │ │ + -- .00532833s elapsed
│ │ │ + -- .0304834s elapsed
│ │ │ + -- .0115025s elapsed
│ │ │ + -- .00412674s 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)
│ │ │ - -- .147997s elapsed
│ │ │ + -- .202849s 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)
│ │ │ - -- .00447939s elapsed
│ │ │ - -- .0118282s elapsed
│ │ │ - -- .043588s elapsed
│ │ │ - -- .0167169s elapsed
│ │ │ - -- .00606616s elapsed
│ │ │ + -- .00247049s elapsed
│ │ │ + -- .00564632s elapsed
│ │ │ + -- .0296563s elapsed
│ │ │ + -- .00869968s elapsed
│ │ │ + -- .00323616s 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)
│ │ │ - -- .0201004s elapsed
│ │ │ + -- .0202117s 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
│ │ │ - -- .000054162s elapsed
│ │ │ + -- .000054588s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000094566s elapsed
│ │ │ + -- .000089767s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071484s elapsed
│ │ │ + -- .000070984s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081081s elapsed
│ │ │ + -- .000085117s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000092442s elapsed
│ │ │ + -- .000089993s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000092503s elapsed
│ │ │ + -- .00009516s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024937s elapsed
│ │ │ + -- .000026169s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000097393s elapsed
│ │ │ + -- .000071233s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000083987s elapsed
│ │ │ + -- .000080604s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000091982s elapsed
│ │ │ + -- .000102697s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000094778s elapsed
│ │ │ + -- .000085352s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000082534s elapsed
│ │ │ + -- .000097074s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000077535s elapsed
│ │ │ + -- .000075507s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000072095s elapsed
│ │ │ + -- .000071508s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067977s elapsed
│ │ │ + -- .000066617s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000022111s elapsed
│ │ │ + -- .000067669s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071755s elapsed
│ │ │ + -- .00006663s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000026329s elapsed
│ │ │ + -- .000026249s elapsed
│ │ │  (e )
│ │ │    1
│ │ │  
│ │ │         6      32    32
│ │ │  o2 = (3 , (e )  (e )  )
│ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_analyze__Strand.html
│ │ │ @@ -107,15 +107,15 @@
│ │ │  
│ │ │  o3 : Complex
    
│ │ │              
    
│ │ │                
    i4 : L = analyzeStrand(F,a); #L
│ │ │ - -- .0249269s elapsed
│ │ │ + -- .0234806s elapsed
│ │ │  
│ │ │  o5 = 350
    
│ │ │              
    
│ │ │                
    i6 : betti F_a, betti F
│ │ │ @@ -146,19 +146,19 @@
│ │ │  
│ │ │  o9 = 14
    
│ │ │              
    
│ │ │                
    i10 : carpetBettiTable(a,b,3)
│ │ │ - -- .00229463s elapsed
│ │ │ - -- .00936212s elapsed
│ │ │ - -- .0239474s elapsed
│ │ │ - -- .011302s elapsed
│ │ │ - -- .00360161s elapsed
│ │ │ + -- .00256122s elapsed
│ │ │ + -- .00617979s elapsed
│ │ │ + -- .0243346s elapsed
│ │ │ + -- .00902719s elapsed
│ │ │ + -- .00372951s 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
│ │ │ │ - -- .0249269s elapsed
│ │ │ │ + -- .0234806s 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)
│ │ │ │ - -- .00229463s elapsed
│ │ │ │ - -- .00936212s elapsed
│ │ │ │ - -- .0239474s elapsed
│ │ │ │ - -- .011302s elapsed
│ │ │ │ - -- .00360161s elapsed
│ │ │ │ + -- .00256122s elapsed
│ │ │ │ + -- .00617979s elapsed
│ │ │ │ + -- .0243346s elapsed
│ │ │ │ + -- .00902719s elapsed
│ │ │ │ + -- .00372951s elapsed
│ │ │ │  
│ │ │ │               0  1   2   3   4   5   6   7  8 9
│ │ │ │  o10 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │ │            0: 1  .   .   .   .   .   .   .  . .
│ │ │ │            1: . 36 160 315 288  14   .   .  . .
│ │ │ │            2: .  .   .   .  14 288 315 160 36 .
│ │ │ │            3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Table.html
│ │ │ @@ -88,20 +88,20 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : elapsedTime T=carpetBettiTable(a,b,3)
│ │ │ - -- .00231006s elapsed
│ │ │ - -- .0217836s elapsed
│ │ │ - -- .0230895s elapsed
│ │ │ - -- .0101838s elapsed
│ │ │ - -- .00353337s elapsed
│ │ │ - -- .294917s elapsed
│ │ │ + -- .00217748s elapsed
│ │ │ + -- .00620783s elapsed
│ │ │ + -- .0233805s elapsed
│ │ │ + -- .00996756s elapsed
│ │ │ + -- .00375936s elapsed
│ │ │ + -- .282847s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -117,15 +117,15 @@
│ │ │  o3 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime T'=minimalBetti J
│ │ │ - -- .125196s elapsed
│ │ │ + -- .163269s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o4 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -145,22 +145,22 @@
│ │ │  
│ │ │  o5 : BettiTally
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00475633s elapsed
│ │ │ - -- .0968306s elapsed
│ │ │ - -- .111433s elapsed
│ │ │ - -- 1.47579s elapsed
│ │ │ - -- .551722s elapsed
│ │ │ - -- .0398804s elapsed
│ │ │ - -- .00727613s elapsed
│ │ │ - -- 6.05064s elapsed
    
│ │ │ + -- .00473581s elapsed
│ │ │ + -- .0168751s elapsed
│ │ │ + -- .0931932s elapsed
│ │ │ + -- .809624s elapsed
│ │ │ + -- .394797s elapsed
│ │ │ + -- .0393118s elapsed
│ │ │ + -- .00686754s elapsed
│ │ │ + -- 4.43102s 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)
│ │ │ │ - -- .00231006s elapsed
│ │ │ │ - -- .0217836s elapsed
│ │ │ │ - -- .0230895s elapsed
│ │ │ │ - -- .0101838s elapsed
│ │ │ │ - -- .00353337s elapsed
│ │ │ │ - -- .294917s elapsed
│ │ │ │ + -- .00217748s elapsed
│ │ │ │ + -- .00620783s elapsed
│ │ │ │ + -- .0233805s elapsed
│ │ │ │ + -- .00996756s elapsed
│ │ │ │ + -- .00375936s elapsed
│ │ │ │ + -- .282847s 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
│ │ │ │ - -- .125196s elapsed
│ │ │ │ + -- .163269s 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);
│ │ │ │ - -- .00475633s elapsed
│ │ │ │ - -- .0968306s elapsed
│ │ │ │ - -- .111433s elapsed
│ │ │ │ - -- 1.47579s elapsed
│ │ │ │ - -- .551722s elapsed
│ │ │ │ - -- .0398804s elapsed
│ │ │ │ - -- .00727613s elapsed
│ │ │ │ - -- 6.05064s elapsed
│ │ │ │ + -- .00473581s elapsed
│ │ │ │ + -- .0168751s elapsed
│ │ │ │ + -- .0931932s elapsed
│ │ │ │ + -- .809624s elapsed
│ │ │ │ + -- .394797s elapsed
│ │ │ │ + -- .0393118s elapsed
│ │ │ │ + -- .00686754s elapsed
│ │ │ │ + -- 4.43102s elapsed
│ │ │ │  i7 : carpetBettiTable(h,7)
│ │ │ │  
│ │ │ │              0  1   2   3    4    5    6    7   8   9 10 11
│ │ │ │  o7 = total: 1 55 320 891 1408 1155 1155 1408 891 320 55  1
│ │ │ │           0: 1  .   .   .    .    .    .    .   .   .  .  .
│ │ │ │           1: . 55 320 891 1408 1155    .    .   .   .  .  .
│ │ │ │           2: .  .   .   .    .    . 1155 1408 891 320 55  .
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Betti__Tables.html
│ │ │ @@ -85,19 +85,19 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : h=carpetBettiTables(a,b)
│ │ │ - -- .00224642s elapsed
│ │ │ - -- .010053s elapsed
│ │ │ - -- .0365563s elapsed
│ │ │ - -- .0100498s elapsed
│ │ │ - -- .00353941s elapsed
│ │ │ + -- .00234641s elapsed
│ │ │ + -- .00564509s elapsed
│ │ │ + -- .0232305s elapsed
│ │ │ + -- .00891118s elapsed
│ │ │ + -- .00361696s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o2 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -139,15 +139,15 @@
│ │ │  o4 : Ideal of --[x ..x , y ..y ]
│ │ │                 3  0   5   0   5
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T'=minimalBetti J
│ │ │ - -- .196215s elapsed
│ │ │ + -- .174358s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 160 315 302 302 315 160 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 315 288  14   .   .  . .
│ │ │           2: .  .   .   .  14 288 315 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -167,22 +167,22 @@
│ │ │  
│ │ │  o6 : BettiTally
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime h=carpetBettiTables(6,6);
│ │ │ - -- .00490087s elapsed
│ │ │ - -- .0189491s elapsed
│ │ │ - -- .0966489s elapsed
│ │ │ - -- 1.19159s elapsed
│ │ │ - -- .524549s elapsed
│ │ │ - -- .0405632s elapsed
│ │ │ - -- .00680951s elapsed
│ │ │ - -- 5.37805s elapsed
    
│ │ │ + -- .00447441s elapsed
│ │ │ + -- .0165111s elapsed
│ │ │ + -- .0970322s elapsed
│ │ │ + -- .843674s elapsed
│ │ │ + -- .39112s elapsed
│ │ │ + -- .0361504s elapsed
│ │ │ + -- .00681437s elapsed
│ │ │ + -- 4.49428s 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)
│ │ │ │ - -- .00224642s elapsed
│ │ │ │ - -- .010053s elapsed
│ │ │ │ - -- .0365563s elapsed
│ │ │ │ - -- .0100498s elapsed
│ │ │ │ - -- .00353941s elapsed
│ │ │ │ + -- .00234641s elapsed
│ │ │ │ + -- .00564509s elapsed
│ │ │ │ + -- .0232305s elapsed
│ │ │ │ + -- .00891118s elapsed
│ │ │ │ + -- .00361696s 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
│ │ │ │ - -- .196215s elapsed
│ │ │ │ + -- .174358s 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);
│ │ │ │ - -- .00490087s elapsed
│ │ │ │ - -- .0189491s elapsed
│ │ │ │ - -- .0966489s elapsed
│ │ │ │ - -- 1.19159s elapsed
│ │ │ │ - -- .524549s elapsed
│ │ │ │ - -- .0405632s elapsed
│ │ │ │ - -- .00680951s elapsed
│ │ │ │ - -- 5.37805s elapsed
│ │ │ │ + -- .00447441s elapsed
│ │ │ │ + -- .0165111s elapsed
│ │ │ │ + -- .0970322s elapsed
│ │ │ │ + -- .843674s elapsed
│ │ │ │ + -- .39112s elapsed
│ │ │ │ + -- .0361504s elapsed
│ │ │ │ + -- .00681437s elapsed
│ │ │ │ + -- 4.49428s elapsed
│ │ │ │  i8 : keys h
│ │ │ │  
│ │ │ │  o8 = {0, 2, 3, 5}
│ │ │ │  
│ │ │ │  o8 : List
│ │ │ │  i9 : carpetBettiTable(h,7)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_carpet__Det.html
│ │ │ @@ -85,82 +85,82 @@
│ │ │  
│ │ │  o1 : Sequence
    
│ │ │              
    
│ │ │                
    i2 : d=carpetDet(a,b)
│ │ │ - -- .00631474s elapsed
│ │ │ - -- .0125312s elapsed
│ │ │ + -- .00684823s elapsed
│ │ │ + -- .0119311s elapsed
│ │ │  (number Of blocks, 26)
│ │ │ - -- .000299471s elapsed
│ │ │ + -- .000231598s elapsed
│ │ │  1
│ │ │ - -- .000146984s elapsed
│ │ │ + -- .000187525s elapsed
│ │ │  1
│ │ │ - -- .00012821s elapsed
│ │ │ + -- .000197935s elapsed
│ │ │  1
│ │ │ - -- .000124793s elapsed
│ │ │ + -- .000144851s elapsed
│ │ │  1
│ │ │ - -- .000129423s elapsed
│ │ │ + -- .000153222s elapsed
│ │ │  2
│ │ │ - -- .000136796s elapsed
│ │ │ + -- .000147206s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000137637s elapsed
│ │ │ + -- .000153237s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000148369s elapsed
│ │ │ + -- .000161622s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000144691s elapsed
│ │ │ + -- .000156548s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000132719s elapsed
│ │ │ + -- .000203872s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000125866s elapsed
│ │ │ + -- .000145923s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000132297s elapsed
│ │ │ + -- .000151677s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000140242s elapsed
│ │ │ + -- .000136144s elapsed
│ │ │  2
│ │ │ - -- .000122429s elapsed
│ │ │ + -- .000138329s elapsed
│ │ │  2
│ │ │ - -- .000133088s elapsed
│ │ │ + -- .000146644s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000120466s elapsed
│ │ │ + -- .000141976s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000136004s elapsed
│ │ │ + -- .000155456s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000131376s elapsed
│ │ │ + -- .000169861s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000208069s elapsed
│ │ │ + -- .000171294s elapsed
│ │ │   2
│ │ │  2 3
│ │ │ - -- .000140834s elapsed
│ │ │ + -- .000158089s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000134101s elapsed
│ │ │ + -- .000164858s elapsed
│ │ │   2
│ │ │  2
│ │ │ - -- .000126856s elapsed
│ │ │ + -- .000140399s elapsed
│ │ │  2
│ │ │ - -- .000129172s elapsed
│ │ │ + -- .000136645s elapsed
│ │ │  1
│ │ │ - -- .000130714s elapsed
│ │ │ + -- .00014147s elapsed
│ │ │  1
│ │ │ - -- .000156823s elapsed
│ │ │ + -- .000143442s elapsed
│ │ │  1
│ │ │ - -- .000123991s elapsed
│ │ │ + -- .000140393s 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)
│ │ │ │ - -- .00631474s elapsed
│ │ │ │ - -- .0125312s elapsed
│ │ │ │ + -- .00684823s elapsed
│ │ │ │ + -- .0119311s elapsed
│ │ │ │  (number Of blocks, 26)
│ │ │ │ - -- .000299471s elapsed
│ │ │ │ + -- .000231598s elapsed
│ │ │ │  1
│ │ │ │ - -- .000146984s elapsed
│ │ │ │ + -- .000187525s elapsed
│ │ │ │  1
│ │ │ │ - -- .00012821s elapsed
│ │ │ │ + -- .000197935s elapsed
│ │ │ │  1
│ │ │ │ - -- .000124793s elapsed
│ │ │ │ + -- .000144851s elapsed
│ │ │ │  1
│ │ │ │ - -- .000129423s elapsed
│ │ │ │ + -- .000153222s elapsed
│ │ │ │  2
│ │ │ │ - -- .000136796s elapsed
│ │ │ │ + -- .000147206s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000137637s elapsed
│ │ │ │ + -- .000153237s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000148369s elapsed
│ │ │ │ + -- .000161622s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000144691s elapsed
│ │ │ │ + -- .000156548s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000132719s elapsed
│ │ │ │ + -- .000203872s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000125866s elapsed
│ │ │ │ + -- .000145923s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000132297s elapsed
│ │ │ │ + -- .000151677s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000140242s elapsed
│ │ │ │ + -- .000136144s elapsed
│ │ │ │  2
│ │ │ │ - -- .000122429s elapsed
│ │ │ │ + -- .000138329s elapsed
│ │ │ │  2
│ │ │ │ - -- .000133088s elapsed
│ │ │ │ + -- .000146644s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000120466s elapsed
│ │ │ │ + -- .000141976s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000136004s elapsed
│ │ │ │ + -- .000155456s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000131376s elapsed
│ │ │ │ + -- .000169861s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000208069s elapsed
│ │ │ │ + -- .000171294s elapsed
│ │ │ │   2
│ │ │ │  2 3
│ │ │ │ - -- .000140834s elapsed
│ │ │ │ + -- .000158089s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000134101s elapsed
│ │ │ │ + -- .000164858s elapsed
│ │ │ │   2
│ │ │ │  2
│ │ │ │ - -- .000126856s elapsed
│ │ │ │ + -- .000140399s elapsed
│ │ │ │  2
│ │ │ │ - -- .000129172s elapsed
│ │ │ │ + -- .000136645s elapsed
│ │ │ │  1
│ │ │ │ - -- .000130714s elapsed
│ │ │ │ + -- .00014147s elapsed
│ │ │ │  1
│ │ │ │ - -- .000156823s elapsed
│ │ │ │ + -- .000143442s elapsed
│ │ │ │  1
│ │ │ │ - -- .000123991s elapsed
│ │ │ │ + -- .000140393s elapsed
│ │ │ │  1
│ │ │ │  
│ │ │ │  o2 = 3131031158784
│ │ │ │  i3 : factor d
│ │ │ │  
│ │ │ │        32 6
│ │ │ │  o3 = 2  3
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_compute__Bound.html
│ │ │ @@ -90,20 +90,20 @@
│ │ │  
│ │ │  o1 : Sequence
│ │ │              
    
│ │ │                
    i2 : computeBound 3
│ │ │ - -- .268927s elapsed
│ │ │ - -- .33298s elapsed
│ │ │ - -- .23855s elapsed
│ │ │ - -- .286255s elapsed
│ │ │ - -- .382732s elapsed
│ │ │ - -- .298869s elapsed
│ │ │ + -- .122356s elapsed
│ │ │ + -- .144495s elapsed
│ │ │ + -- .149958s elapsed
│ │ │ + -- .147845s elapsed
│ │ │ + -- .15113s elapsed
│ │ │ + -- .146368s 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
│ │ │ │ - -- .268927s elapsed
│ │ │ │ - -- .33298s elapsed
│ │ │ │ - -- .23855s elapsed
│ │ │ │ - -- .286255s elapsed
│ │ │ │ - -- .382732s elapsed
│ │ │ │ - -- .298869s elapsed
│ │ │ │ + -- .122356s elapsed
│ │ │ │ + -- .144495s elapsed
│ │ │ │ + -- .149958s elapsed
│ │ │ │ + -- .147845s elapsed
│ │ │ │ + -- .15113s elapsed
│ │ │ │ + -- .146368s elapsed
│ │ │ │  
│ │ │ │  o2 = 6
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_l_a_t_i_v_e_E_q_u_a_t_i_o_n_s -- compute the relative quadrics
│ │ │ │  ********** WWaayyss ttoo uussee ccoommppuutteeBBoouunndd:: **********
│ │ │ │      * computeBound(ZZ)
│ │ │ │      * computeBound(ZZ,ZZ,ZZ)
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_degenerate__K3__Betti__Tables.html
│ │ │ @@ -95,19 +95,19 @@
│ │ │  
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .0030227s elapsed
│ │ │ - -- .00794913s elapsed
│ │ │ - -- .023886s elapsed
│ │ │ - -- .0114692s elapsed
│ │ │ - -- .00368452s elapsed
│ │ │ + -- .00274193s elapsed
│ │ │ + -- .00532833s elapsed
│ │ │ + -- .0304834s elapsed
│ │ │ + -- .0115025s elapsed
│ │ │ + -- .00412674s elapsed
│ │ │  
│ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │  o3 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1}
│ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -141,15 +141,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime T= minimalBetti degenerateK3(a,b,e,Characteristic=>5)
│ │ │ - -- .147997s elapsed
│ │ │ + -- .202849s elapsed
│ │ │  
│ │ │              0  1   2   3   4   5   6   7  8 9
│ │ │  o5 = total: 1 36 167 370 476 476 370 167 36 1
│ │ │           0: 1  .   .   .   .   .   .   .  . .
│ │ │           1: . 36 160 322 336 140  48   7  . .
│ │ │           2: .  .   7  48 140 336 322 160 36 .
│ │ │           3: .  .   .   .   .   .   .   .  . 1
│ │ │ @@ -183,19 +183,19 @@
│ │ │  
│ │ │  o7 : Sequence
    
│ │ │              
    
│ │ │                
    i8 : h=degenerateK3BettiTables(a,b,e)
│ │ │ - -- .00447939s elapsed
│ │ │ - -- .0118282s elapsed
│ │ │ - -- .043588s elapsed
│ │ │ - -- .0167169s elapsed
│ │ │ - -- .00606616s elapsed
│ │ │ + -- .00247049s elapsed
│ │ │ + -- .00564632s elapsed
│ │ │ + -- .0296563s elapsed
│ │ │ + -- .00869968s elapsed
│ │ │ + -- .00323616s 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)
│ │ │ │ - -- .0030227s elapsed
│ │ │ │ - -- .00794913s elapsed
│ │ │ │ - -- .023886s elapsed
│ │ │ │ - -- .0114692s elapsed
│ │ │ │ - -- .00368452s elapsed
│ │ │ │ + -- .00274193s elapsed
│ │ │ │ + -- .00532833s elapsed
│ │ │ │ + -- .0304834s elapsed
│ │ │ │ + -- .0115025s elapsed
│ │ │ │ + -- .00412674s 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)
│ │ │ │ - -- .147997s elapsed
│ │ │ │ + -- .202849s 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)
│ │ │ │ - -- .00447939s elapsed
│ │ │ │ - -- .0118282s elapsed
│ │ │ │ - -- .043588s elapsed
│ │ │ │ - -- .0167169s elapsed
│ │ │ │ - -- .00606616s elapsed
│ │ │ │ + -- .00247049s elapsed
│ │ │ │ + -- .00564632s elapsed
│ │ │ │ + -- .0296563s elapsed
│ │ │ │ + -- .00869968s elapsed
│ │ │ │ + -- .00323616s elapsed
│ │ │ │  
│ │ │ │                             0  1   2   3   4   5   6   7  8 9
│ │ │ │  o8 = HashTable{0 => total: 1 36 160 315 288 288 315 160 36 1     }
│ │ │ │                          0: 1  .   .   .   .   .   .   .  . .
│ │ │ │                          1: . 36 160 315 288   .   .   .  . .
│ │ │ │                          2: .  .   .   .   . 288 315 160 36 .
│ │ │ │                          3: .  .   .   .   .   .   .   .  . 1
│ │ ├── ./usr/share/doc/Macaulay2/K3Carpets/html/_resonance__Det.html
│ │ │ @@ -83,172 +83,172 @@
│ │ │  
│ │ │  o1 = 4
    
│ │ │              
    
│ │ │                
    i2 : (d1,d2)=resonanceDet(a)
│ │ │ - -- .0201004s elapsed
│ │ │ + -- .0202117s 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
│ │ │ - -- .000054162s elapsed
│ │ │ + -- .000054588s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . 2
│ │ │ - -- .000094566s elapsed
│ │ │ + -- .000089767s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      7: 2 .
│ │ │      8: . .
│ │ │      9: . 2
│ │ │ - -- .000071484s elapsed
│ │ │ + -- .000070984s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │      7: 2 .
│ │ │      8: 1 .
│ │ │      9: . 1
│ │ │     10: . 2
│ │ │ - -- .000081081s elapsed
│ │ │ + -- .000085117s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (-3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      7: 1 .
│ │ │      8: 1 .
│ │ │      9: 2 2
│ │ │     10: . 1
│ │ │     11: . 1
│ │ │ - -- .000092442s elapsed
│ │ │ + -- .000089993s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      8: 1 .
│ │ │      9: 2 1
│ │ │     10: 1 2
│ │ │     11: . 1
│ │ │ - -- .000092503s elapsed
│ │ │ + -- .00009516s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │      9: 1 1
│ │ │ - -- .000024937s elapsed
│ │ │ + -- .000026169s elapsed
│ │ │  (e )(-1)
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │      9: 1 1
│ │ │     10: 1 1
│ │ │ - -- .000097393s elapsed
│ │ │ + -- .000071233s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000083987s elapsed
│ │ │ + -- .000080604s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 2 1
│ │ │     11: 1 2
│ │ │     12: . 1
│ │ │ - -- .000091982s elapsed
│ │ │ + -- .000102697s elapsed
│ │ │      2    3
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 1 .
│ │ │     10: 1 .
│ │ │     11: 2 2
│ │ │     12: . 1
│ │ │     13: . 1
│ │ │ - -- .000094778s elapsed
│ │ │ + -- .000085352s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 4 4
│ │ │      9: 2 1
│ │ │     10: 1 1
│ │ │     11: 1 2
│ │ │ - -- .000082534s elapsed
│ │ │ + -- .000097074s elapsed
│ │ │      2    2
│ │ │  (e ) (e ) (-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 3 3
│ │ │     10: 2 .
│ │ │     11: 1 .
│ │ │     12: . 1
│ │ │     13: . 2
│ │ │ - -- .000077535s elapsed
│ │ │ + -- .000075507s elapsed
│ │ │      2    4
│ │ │  (e ) (e ) (3)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     10: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000072095s elapsed
│ │ │ + -- .000071508s elapsed
│ │ │      2
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     11: 2 .
│ │ │     12: . .
│ │ │     13: . 2
│ │ │ - -- .000067977s elapsed
│ │ │ + -- .000066617s elapsed
│ │ │      2    2
│ │ │  (e ) (e )
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     11: 1 1
│ │ │ - -- .000022111s elapsed
│ │ │ + -- .000067669s elapsed
│ │ │  (e )
│ │ │    1
│ │ │         0 1
│ │ │  total: 2 2
│ │ │     12: 2 .
│ │ │     13: . 2
│ │ │ - -- .000071755s elapsed
│ │ │ + -- .00006663s elapsed
│ │ │      2
│ │ │  (e ) (e )(-1)
│ │ │    1    2
│ │ │         0 1
│ │ │  total: 1 1
│ │ │     13: 1 1
│ │ │ - -- .000026329s elapsed
│ │ │ + -- .000026249s 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)
│ │ │ │ - -- .0201004s elapsed
│ │ │ │ + -- .0202117s 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
│ │ │ │ - -- .000054162s elapsed
│ │ │ │ + -- .000054588s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . 2
│ │ │ │ - -- .000094566s elapsed
│ │ │ │ + -- .000089767s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      7: 2 .
│ │ │ │      8: . .
│ │ │ │      9: . 2
│ │ │ │ - -- .000071484s elapsed
│ │ │ │ + -- .000070984s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │      7: 2 .
│ │ │ │      8: 1 .
│ │ │ │      9: . 1
│ │ │ │     10: . 2
│ │ │ │ - -- .000081081s elapsed
│ │ │ │ + -- .000085117s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (-3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      7: 1 .
│ │ │ │      8: 1 .
│ │ │ │      9: 2 2
│ │ │ │     10: . 1
│ │ │ │     11: . 1
│ │ │ │ - -- .000092442s elapsed
│ │ │ │ + -- .000089993s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      8: 1 .
│ │ │ │      9: 2 1
│ │ │ │     10: 1 2
│ │ │ │     11: . 1
│ │ │ │ - -- .000092503s elapsed
│ │ │ │ + -- .00009516s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │      9: 1 1
│ │ │ │ - -- .000024937s elapsed
│ │ │ │ + -- .000026169s elapsed
│ │ │ │  (e )(-1)
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │      9: 1 1
│ │ │ │     10: 1 1
│ │ │ │ - -- .000097393s elapsed
│ │ │ │ + -- .000071233s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000083987s elapsed
│ │ │ │ + -- .000080604s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 2 1
│ │ │ │     11: 1 2
│ │ │ │     12: . 1
│ │ │ │ - -- .000091982s elapsed
│ │ │ │ + -- .000102697s elapsed
│ │ │ │      2    3
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 1 .
│ │ │ │     10: 1 .
│ │ │ │     11: 2 2
│ │ │ │     12: . 1
│ │ │ │     13: . 1
│ │ │ │ - -- .000094778s elapsed
│ │ │ │ + -- .000085352s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 4 4
│ │ │ │      9: 2 1
│ │ │ │     10: 1 1
│ │ │ │     11: 1 2
│ │ │ │ - -- .000082534s elapsed
│ │ │ │ + -- .000097074s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e ) (-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 3 3
│ │ │ │     10: 2 .
│ │ │ │     11: 1 .
│ │ │ │     12: . 1
│ │ │ │     13: . 2
│ │ │ │ - -- .000077535s elapsed
│ │ │ │ + -- .000075507s elapsed
│ │ │ │      2    4
│ │ │ │  (e ) (e ) (3)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     10: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000072095s elapsed
│ │ │ │ + -- .000071508s elapsed
│ │ │ │      2
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     11: 2 .
│ │ │ │     12: . .
│ │ │ │     13: . 2
│ │ │ │ - -- .000067977s elapsed
│ │ │ │ + -- .000066617s elapsed
│ │ │ │      2    2
│ │ │ │  (e ) (e )
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     11: 1 1
│ │ │ │ - -- .000022111s elapsed
│ │ │ │ + -- .000067669s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │         0 1
│ │ │ │  total: 2 2
│ │ │ │     12: 2 .
│ │ │ │     13: . 2
│ │ │ │ - -- .000071755s elapsed
│ │ │ │ + -- .00006663s elapsed
│ │ │ │      2
│ │ │ │  (e ) (e )(-1)
│ │ │ │    1    2
│ │ │ │         0 1
│ │ │ │  total: 1 1
│ │ │ │     13: 1 1
│ │ │ │ - -- .000026329s elapsed
│ │ │ │ + -- .000026249s elapsed
│ │ │ │  (e )
│ │ │ │    1
│ │ │ │  
│ │ │ │         6      32    32
│ │ │ │  o2 = (3 , (e )  (e )  )
│ │ │ │              1     2
│ │ ├── ./usr/share/doc/Macaulay2/K3Surfaces/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  cHJvamVjdChWaXNpYmxlTGlzdCxFbWJlZGRlZEszc3VyZmFjZSk=
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQ4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwcm9qZWN0LFZpc2libGVMaXN0LEVtYmVkZGVkSzNz
│ │ ├── ./usr/share/doc/Macaulay2/Kronecker/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  a3JvbmVja2VyTm9ybWFsRm9ybShNYXRyaXgp
│ │ │  #:len=279
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIwNiwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoa3JvbmVja2VyTm9ybWFsRm9ybSxNYXRyaXgpLCJr
│ │ ├── ./usr/share/doc/Macaulay2/KustinMiller/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=37
│ │ │  a3VzdGluTWlsbGVyQ29tcGxleCguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=577
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHRvIHByaW50IGludGVybWVk
│ │ │  aWF0ZSBkYXRhIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VE97bmV3IERvY3VtZW50VGFn
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  UmVhbFhE
│ │ │  #:len=499
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidXNlIGV4dGVuZGVkIGV4cG9uZW50IHJl
│ │ │  YWwgbnVtYmVycyIsIERlc2NyaXB0aW9uID0+IChUVHsiUmVhbFhEIn0sIiAtLSBhIHN0cmF0ZWd5
│ │ ├── ./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 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ + -- used 0.00801682s (cpu); 0.00800211s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ + -- used 0.0229312s (cpu); 0.0229354s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ + -- used 0.0912361s (cpu); 0.0912418s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ + -- used 0.0113616s (cpu); 0.0113663s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ + -- used 0.0507505s (cpu); 0.050755s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ + -- used 0.0530692s (cpu); 0.0530742s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ + -- used 0.311557s (cpu); 0.311423s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ + -- used 0.126s (cpu); 0.126006s (thread); 0s (gc)
│ │ │  
│ │ │                 50       47
│ │ │  o10 : Matrix ZZ   <-- ZZ
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/LLLBases/html/___L__L__L_lp..._cm__Strategy_eq_gt..._rp.html
│ │ │ @@ -144,78 +144,78 @@
│ │ │                50       47
│ │ │  o2 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i3 : time LLL m;
│ │ │ - -- used 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ + -- used 0.00801682s (cpu); 0.00800211s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o3 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ + -- used 0.0229312s (cpu); 0.0229354s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o4 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ + -- used 0.0912361s (cpu); 0.0912418s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o5 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ + -- used 0.0113616s (cpu); 0.0113663s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o6 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ + -- used 0.0507505s (cpu); 0.050755s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o7 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ + -- used 0.0530692s (cpu); 0.0530742s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o8 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ + -- used 0.311557s (cpu); 0.311423s (thread); 0s (gc)
│ │ │  
│ │ │                50       47
│ │ │  o9 : Matrix ZZ   <-- ZZ
    
│ │ │              
    
│ │ │                
    i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ + -- used 0.126s (cpu); 0.126006s (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 0.00867687s (cpu); 0.00867114s (thread); 0s (gc)
│ │ │ │ + -- used 0.00801682s (cpu); 0.00800211s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o3 : Matrix ZZ   <-- ZZ
│ │ │ │  i4 : time LLL(m, Strategy=>CohenEngine);
│ │ │ │ - -- used 0.0275296s (cpu); 0.0275336s (thread); 0s (gc)
│ │ │ │ + -- used 0.0229312s (cpu); 0.0229354s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o4 : Matrix ZZ   <-- ZZ
│ │ │ │  i5 : time LLL(m, Strategy=>CohenTopLevel);
│ │ │ │ - -- used 0.104159s (cpu); 0.104163s (thread); 0s (gc)
│ │ │ │ + -- used 0.0912361s (cpu); 0.0912418s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o5 : Matrix ZZ   <-- ZZ
│ │ │ │  i6 : time LLL(m, Strategy=>{Givens,RealFP});
│ │ │ │ - -- used 0.0107558s (cpu); 0.010756s (thread); 0s (gc)
│ │ │ │ + -- used 0.0113616s (cpu); 0.0113663s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o6 : Matrix ZZ   <-- ZZ
│ │ │ │  i7 : time LLL(m, Strategy=>{Givens,RealQP});
│ │ │ │ - -- used 0.0473029s (cpu); 0.0473041s (thread); 0s (gc)
│ │ │ │ + -- used 0.0507505s (cpu); 0.050755s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o7 : Matrix ZZ   <-- ZZ
│ │ │ │  i8 : time LLL(m, Strategy=>{Givens,RealXD});
│ │ │ │ - -- used 0.0557251s (cpu); 0.0557271s (thread); 0s (gc)
│ │ │ │ + -- used 0.0530692s (cpu); 0.0530742s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o8 : Matrix ZZ   <-- ZZ
│ │ │ │  i9 : time LLL(m, Strategy=>{Givens,RealRR});
│ │ │ │ - -- used 0.317152s (cpu); 0.317156s (thread); 0s (gc)
│ │ │ │ + -- used 0.311557s (cpu); 0.311423s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                50       47
│ │ │ │  o9 : Matrix ZZ   <-- ZZ
│ │ │ │  i10 : time LLL(m, Strategy=>{BKZ,Givens,RealQP});
│ │ │ │ - -- used 0.11309s (cpu); 0.113092s (thread); 0s (gc)
│ │ │ │ + -- used 0.126s (cpu); 0.126006s (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/LanguageServer/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  c3RhcnQoTFNQU2VydmVyKQ==
│ │ │  #:len=246
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjYsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KHN0YXJ0LExTUFNlcnZlciksInN0YXJ0KExTUFNlcnZl
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  YXJlSXNvbW9ycGhpYyhNYXRyaXgsTWF0cml4KQ==
│ │ │  #:len=289
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhcmVJc29tb3JwaGljLE1hdHJpeCxNYXRyaXgpLCJh
│ │ ├── ./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 0.74582s (cpu); 0.506968s (thread); 0s (gc)
│ │ │ + -- used 0.799637s (cpu); 0.467246s (thread); 0s (gc)
│ │ │  
│ │ │  i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc)
│ │ │ + -- used 0.468929s (cpu); 0.26553s (thread); 0s (gc)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/LatticePolytopes/html/_are__Isomorphic.html
│ │ │ @@ -125,21 +125,21 @@
│ │ │
    		│ │ │ 
    i5 : P = convexHull(M);
    │ │ │
    │ │ │ 
    i6 : time areIsomorphic(P,P);
│ │ │ - -- used 0.74582s (cpu); 0.506968s (thread); 0s (gc)
    │ │ │ + -- used 0.799637s (cpu); 0.467246s (thread); 0s (gc) │ │ │
    │ │ │ 
    i7 : time areIsomorphic(P,P,smoothTest=>false);
│ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc)
    │ │ │ + -- used 0.468929s (cpu); 0.26553s (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 0.74582s (cpu); 0.506968s (thread); 0s (gc) │ │ │ │ + -- used 0.799637s (cpu); 0.467246s (thread); 0s (gc) │ │ │ │ i7 : time areIsomorphic(P,P,smoothTest=>false); │ │ │ │ - -- used 0.361052s (cpu); 0.277771s (thread); 0s (gc) │ │ │ │ + -- used 0.468929s (cpu); 0.26553s (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/LexIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ TGV4SWRlYWxz │ │ │ #:len=470 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciB3b3JraW5nIHdp │ │ │ dGggbGV4IGlkZWFscyIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsiIixFTXsiTGV4 │ │ ├── ./usr/share/doc/Macaulay2/LieAlgebraRepresentations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ c3ltbWV0cmljUG93ZXIoWlosTGllQWxnZWJyYVJlcHJlc2VudGF0aW9uKQ== │ │ │ #:len=1614 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIGV4cGxpY2l0IGFj │ │ │ dGlvbiBvbiAkXFxvcGVyYXRvcm5hbWV7U3ltfV5kIFYkIGZvciBhICRcXG1hdGhmcmFre2d9JC1t │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c3VwcG9ydE9mVG9yKENvbXBsZXgp │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE0Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3VwcG9ydE9mVG9yLENvbXBsZXgpLCJzdXBwb3J0 │ │ ├── ./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) │ │ │ - -- .134906s elapsed │ │ │ + -- .0698071s 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}}) │ │ │ - -- .0127315s elapsed │ │ │ + -- .0139574s 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) │ │ │ - -- 3.72413s elapsed │ │ │ + -- 2.58842s elapsed │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o6 : List │ │ │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ - -- .115325s elapsed │ │ │ + -- .0482154s elapsed │ │ │ │ │ │ o7 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_find__Region.html │ │ │ @@ -130,25 +130,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime findRegion({{0,0},{4,4}},M,f)
│ │ │ - -- .134906s elapsed
│ │ │ + -- .0698071s 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}})
│ │ │ - -- .0127315s elapsed
│ │ │ + -- .0139574s 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) │ │ │ │ - -- .134906s elapsed │ │ │ │ + -- .0698071s 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}}) │ │ │ │ - -- .0127315s elapsed │ │ │ │ + -- .0139574s elapsed │ │ │ │ │ │ │ │ o7 = {{1, 2}, {3, 1}} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** CCoonnttrriibbuuttoorrss ********** │ │ │ │ Mahrud Sayrafi contributed to the code for this function. │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/LinearTruncations/html/_linear__Truncations__Bound.html │ │ │ @@ -128,25 +128,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime linearTruncations({{2,2,2},{4,4,4}}, M)
│ │ │ - -- 3.72413s elapsed
│ │ │ + -- 2.58842s elapsed
│ │ │  
│ │ │  o6 = {{4, 3, 3}, {4, 4, 2}}
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : elapsedTime linearTruncationsBound M
│ │ │ - -- .115325s elapsed
│ │ │ + -- .0482154s 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) │ │ │ │ - -- 3.72413s elapsed │ │ │ │ + -- 2.58842s elapsed │ │ │ │ │ │ │ │ o6 = {{4, 3, 3}, {4, 4, 2}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ i7 : elapsedTime linearTruncationsBound M │ │ │ │ - -- .115325s elapsed │ │ │ │ + -- .0482154s 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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ TG9jYWxSaW5ncw== │ │ │ #:len=5280 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTG9jYWxpemF0aW9ucyBvZiBwb2x5bm9t │ │ │ aWFsIHJpbmdzIGF0IHByaW1lIGlkZWFscyIsICJsaW5lbnVtIiA9PiA2MiwgImZpbGVuYW1lIiA9 │ │ ├── ./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) │ │ │ - -- .192462s elapsed │ │ │ + -- .157891s 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 │ │ │ - -- .0128718s elapsed │ │ │ + -- .0149572s elapsed │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ o11 : List │ │ │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ - -- .376491s elapsed │ │ │ + -- .228343s elapsed │ │ │ │ │ │ o12 = {6, 12, 18, 24, 30, 36} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/LocalRings/html/_hilbert__Samuel__Function.html │ │ │ @@ -116,15 +116,15 @@ │ │ │ 1 │ │ │ o4 : RP-module, quotient of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : elapsedTime hilbertSamuelFunction(M, 0, 6)
│ │ │ - -- .192462s elapsed
│ │ │ + -- .157891s elapsed
│ │ │  
│ │ │  o5 = {1, 3, 6, 7, 6, 3, 1}
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -168,25 +168,25 @@ │ │ │ │ │ │ o10 : Ideal of RP │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime hilbertSamuelFunction(N, 0, 5) -- n+1 -- 0.02 seconds
│ │ │ - -- .0128718s elapsed
│ │ │ + -- .0149572s elapsed
│ │ │  
│ │ │  o11 = {1, 2, 3, 4, 5, 6}
│ │ │  
│ │ │  o11 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds
│ │ │ - -- .376491s elapsed
│ │ │ + -- .228343s 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) │ │ │ │ - -- .192462s elapsed │ │ │ │ + -- .157891s 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 │ │ │ │ - -- .0128718s elapsed │ │ │ │ + -- .0149572s elapsed │ │ │ │ │ │ │ │ o11 = {1, 2, 3, 4, 5, 6} │ │ │ │ │ │ │ │ o11 : List │ │ │ │ i12 : elapsedTime hilbertSamuelFunction(q, N, 0, 5) -- 6(n+1) -- 0.32 seconds │ │ │ │ - -- .376491s elapsed │ │ │ │ + -- .228343s 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/M0nbar/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ dGV4KEN1cnZlQ2xhc3NSZXByZXNlbnRhdGl2ZU0wbmJhcik= │ │ │ #:len=923 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydCB0byBUZVggZm9ybWF0Iiwg │ │ │ ImxpbmVudW0iID0+IDE1ODQsIElucHV0cyA9PiB7U1BBTntUVHsiQyJ9LCIsICIsU1BBTnsiYW4g │ │ ├── ./usr/share/doc/Macaulay2/MCMApproximations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ Y29BcHByb3hpbWF0aW9uKC4uLixUb3RhbD0+Li4uKQ== │ │ │ #:len=293 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tjb0FwcHJveGltYXRpb24sVG90YWxdLCJjb0FwcHJv │ │ ├── ./usr/share/doc/Macaulay2/MRDI/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ c2F2ZU1SREkoLi4uLEZpbGVOYW1lPT4uLi4p │ │ │ #:len=232 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTc3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzYXZlTVJESSxGaWxlTmFtZV0sInNhdmVNUkRJKC4u │ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_load__M__R__D__I.out │ │ │ @@ -57,15 +57,15 @@ │ │ │ │ │ │ i9 : I === J │ │ │ │ │ │ o9 = true │ │ │ │ │ │ i10 : fn = temporaryFileName() | ".mrdi" │ │ │ │ │ │ -o10 = /tmp/M2-40008-0/0.mrdi │ │ │ +o10 = /tmp/M2-56075-0/0.mrdi │ │ │ │ │ │ i11 : saveMRDI(I, FileName => fn) │ │ │ │ │ │ o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name": │ │ │ "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1", │ │ │ "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1", │ │ │ "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]], │ │ ├── ./usr/share/doc/Macaulay2/MRDI/example-output/_save__M__R__D__I.out │ │ │ @@ -39,15 +39,15 @@ │ │ │ "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.06"]}, │ │ │ "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": │ │ │ {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data": │ │ │ {"variables": ["x", "y", "z"]}}}} │ │ │ │ │ │ i7 : fn = temporaryFileName() | ".mrdi" │ │ │ │ │ │ -o7 = /tmp/M2-40027-0/0.mrdi │ │ │ +o7 = /tmp/M2-56114-0/0.mrdi │ │ │ │ │ │ i8 : saveMRDI(f, FileName => fn) │ │ │ │ │ │ o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name": │ │ │ "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1", │ │ │ "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2": │ │ │ ["https://macaulay2.com", "1.26.06"]}, "_refs": │ │ │ @@ -69,29 +69,29 @@ │ │ │ "_type" => ZZ │ │ │ "data" => 5 │ │ │ │ │ │ o11 : HashTable │ │ │ │ │ │ i12 : methods {"Macaulay2", saveMRDI} │ │ │ │ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, ZZ) } │ │ │ - {1 => ({Macaulay2, saveMRDI}, Matrix) } │ │ │ - {2 => ({Macaulay2, saveMRDI}, Ideal) } │ │ │ - {3 => ({Macaulay2, saveMRDI}, RingElement) } │ │ │ - {4 => ({Macaulay2, saveMRDI}, PolynomialRing)} │ │ │ - {5 => ({Macaulay2, saveMRDI}, Ring) } │ │ │ - {6 => ({Macaulay2, saveMRDI}, QuotientRing) } │ │ │ - {7 => ({Macaulay2, saveMRDI}, GaloisField) } │ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, GaloisField) } │ │ │ + {1 => ({Macaulay2, saveMRDI}, ZZ) } │ │ │ + {2 => ({Macaulay2, saveMRDI}, Matrix) } │ │ │ + {3 => ({Macaulay2, saveMRDI}, Ideal) } │ │ │ + {4 => ({Macaulay2, saveMRDI}, RingElement) } │ │ │ + {5 => ({Macaulay2, saveMRDI}, PolynomialRing)} │ │ │ + {6 => ({Macaulay2, saveMRDI}, Ring) } │ │ │ + {7 => ({Macaulay2, saveMRDI}, QuotientRing) } │ │ │ │ │ │ o12 : NumberedVerticalList │ │ │ │ │ │ i13 : methods {"Oscar", saveMRDI} │ │ │ │ │ │ -o13 = {0 => ({Oscar, saveMRDI}, RingElement) } │ │ │ - {1 => ({Oscar, saveMRDI}, Ring) } │ │ │ - {2 => ({Oscar, saveMRDI}, PolynomialRing)} │ │ │ - {3 => ({Oscar, saveMRDI}, QQ) } │ │ │ - {4 => ({Oscar, saveMRDI}, ZZ) } │ │ │ +o13 = {0 => ({Oscar, saveMRDI}, QQ) } │ │ │ + {1 => ({Oscar, saveMRDI}, ZZ) } │ │ │ + {2 => ({Oscar, saveMRDI}, RingElement) } │ │ │ + {3 => ({Oscar, saveMRDI}, Ring) } │ │ │ + {4 => ({Oscar, saveMRDI}, PolynomialRing)} │ │ │ │ │ │ o13 : NumberedVerticalList │ │ │ │ │ │ i14 : │ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_load__M__R__D__I.html │ │ │ @@ -176,15 +176,15 @@ │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
    │ │ │ +o10 = /tmp/M2-56075-0/0.mrdi │ │ │
    │ │ │ 
    i11 : saveMRDI(I, FileName => fn)
│ │ │  
│ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ ├── html2text {}
│ │ │ │ @@ -69,15 +69,15 @@
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : I === J
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  Objects can be loaded from a file as well using _g_e_t.
│ │ │ │  i10 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-40008-0/0.mrdi
│ │ │ │ +o10 = /tmp/M2-56075-0/0.mrdi
│ │ │ │  i11 : saveMRDI(I, FileName => fn)
│ │ │ │  
│ │ │ │  o11 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │        "Ideal"}, "data": [[[["0", "0", "2", "0"], ["1", "1"]], [["0", "1",
│ │ │ │        "0", "1"], ["-1", "1"]]], [[["0", "1", "1", "0"], ["1", "1"]], [["1",
│ │ │ │        "0", "0", "1"], ["-1", "1"]]], [[["0", "2", "0", "0"], ["1", "1"]],
│ │ │ │        [["1", "0", "1", "0"], ["-1", "1"]]]], "_ns": {"Macaulay2":
│ │ ├── ./usr/share/doc/Macaulay2/MRDI/html/_save__M__R__D__I.html
│ │ │ @@ -160,15 +160,15 @@
│ │ │            
    The output can be written directly to a file using the FileName option.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn = temporaryFileName() | ".mrdi"
│ │ │  
│ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
    
│ │ │ +o7 = /tmp/M2-56114-0/0.mrdi
│ │ │              
    
│ │ │                
    i8 : saveMRDI(f, FileName => fn)
│ │ │  
│ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ @@ -220,35 +220,35 @@
│ │ │            
    To see which types have built-in save methods for a given namespace, call methods as follows.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i12 : methods {"Macaulay2", saveMRDI}
│ │ │  
│ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ -      {1 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ -      {2 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ -      {3 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ -      {4 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ -      {5 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ -      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ -      {7 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ +      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ +      {3 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ +      {4 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ +      {6 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ +      {7 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │  
│ │ │  o12 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i13 : methods {"Oscar", saveMRDI}
│ │ │  
│ │ │ -o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ -      {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ -      {2 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │ -      {3 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ -      {4 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ +o13 = {0 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ +      {1 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ +      {2 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ +      {3 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ +      {4 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │  
│ │ │  o13 : NumberedVerticalList
    
│ │ │              
    
│ │ │          
    
│ │ │            
    Additional types can be supported by calling addSaveMethod.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -66,15 +66,15 @@
│ │ │ │       "1"]]]], "_ns": {"Macaulay2": ["https://macaulay2.com", "1.26.06"]},
│ │ │ │       "_refs": {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params":
│ │ │ │       {"_type": "Ring", "data": "QQ"}, "name": "PolynomialRing"}, "data":
│ │ │ │       {"variables": ["x", "y", "z"]}}}}
│ │ │ │  The output can be written directly to a file using the _F_i_l_e_N_a_m_e option.
│ │ │ │  i7 : fn = temporaryFileName() | ".mrdi"
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-40027-0/0.mrdi
│ │ │ │ +o7 = /tmp/M2-56114-0/0.mrdi
│ │ │ │  i8 : saveMRDI(f, FileName => fn)
│ │ │ │  
│ │ │ │  o8 = {"_type": {"params": "6caf806b-9118-4741-bec7-217a71096848", "name":
│ │ │ │       "RingElement"}, "data": [[["2", "0", "0"], ["1", "1"]], [["0", "1",
│ │ │ │       "1"], ["1", "1"]], [["1", "0", "0"], ["-3", "1"]]], "_ns": {"Macaulay2":
│ │ │ │       ["https://macaulay2.com", "1.26.06"]}, "_refs":
│ │ │ │       {"6caf806b-9118-4741-bec7-217a71096848": {"_type": {"params": {"_type":
│ │ │ │ @@ -98,31 +98,31 @@
│ │ │ │                  "data" => 5
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  To see which types have built-in save methods for a given namespace, call
│ │ │ │  _m_e_t_h_o_d_s as follows.
│ │ │ │  i12 : methods {"Macaulay2", saveMRDI}
│ │ │ │  
│ │ │ │ -o12 = {0 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ │ -      {1 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ │ -      {2 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ │ -      {3 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ │ -      {4 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ │ -      {5 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │ -      {6 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │ -      {7 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │ +o12 = {0 => ({Macaulay2, saveMRDI}, GaloisField)   }
│ │ │ │ +      {1 => ({Macaulay2, saveMRDI}, ZZ)            }
│ │ │ │ +      {2 => ({Macaulay2, saveMRDI}, Matrix)        }
│ │ │ │ +      {3 => ({Macaulay2, saveMRDI}, Ideal)         }
│ │ │ │ +      {4 => ({Macaulay2, saveMRDI}, RingElement)   }
│ │ │ │ +      {5 => ({Macaulay2, saveMRDI}, PolynomialRing)}
│ │ │ │ +      {6 => ({Macaulay2, saveMRDI}, Ring)          }
│ │ │ │ +      {7 => ({Macaulay2, saveMRDI}, QuotientRing)  }
│ │ │ │  
│ │ │ │  o12 : NumberedVerticalList
│ │ │ │  i13 : methods {"Oscar", saveMRDI}
│ │ │ │  
│ │ │ │ -o13 = {0 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ │ -      {1 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ │ -      {2 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │ │ -      {3 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ │ -      {4 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ │ +o13 = {0 => ({Oscar, saveMRDI}, QQ)            }
│ │ │ │ +      {1 => ({Oscar, saveMRDI}, ZZ)            }
│ │ │ │ +      {2 => ({Oscar, saveMRDI}, RingElement)   }
│ │ │ │ +      {3 => ({Oscar, saveMRDI}, Ring)          }
│ │ │ │ +      {4 => ({Oscar, saveMRDI}, PolynomialRing)}
│ │ │ │  
│ │ │ │  o13 : NumberedVerticalList
│ │ │ │  Additional types can be supported by calling _a_d_d_S_a_v_e_M_e_t_h_o_d.
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Not all Macaulay2 types have save methods defined. Attempting to serialize an
│ │ │ │  unsupported type will produce an error. Quotient rings other than finite prime
│ │ │ │  fields are not yet supported.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  dW5pbnN0YWxsUGFja2FnZShTdHJpbmcp
│ │ │  #:len=297
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjg5LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh1bmluc3RhbGxQYWNrYWdlLFN0cmluZyksInVuaW5z
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Command.out
│ │ │ @@ -5,12 +5,12 @@
│ │ │  i2 : f
│ │ │  
│ │ │  o2 = 1073741824
│ │ │  
│ │ │  i3 : (c = Command "date";)
│ │ │  
│ │ │  i4 : c
│ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ +Sat Oct 24 09:31:24 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/M2-12569-0/0.dbm
│ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │  
│ │ │  i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ +o2 = /tmp/M2-14359-0/0.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" => 69379379194        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 69264345690        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 219217920
│ │ │ -               "numGCs" => 1481
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 255078400
│ │ │ +               "numGCs" => 1447
│ │ │ +               "numGCThreads" => 16
│ │ │  
│ │ │  o1 : HashTable
│ │ │  
│ │ │  i2 : s#"heapSize"
│ │ │  
│ │ │ -o2 = 219217920
│ │ │ +o2 = 255078400
│ │ │  
│ │ │  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 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ + -- used 0.00473147s (cpu); 0.00472981s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
│ │ │  
│ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ + -- used 0.0392479s (cpu); 0.0392539s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
│ │ │  
│ │ │  i10 : numgens J
│ │ │  
│ │ │  o10 = 1067
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Monomial__Ideal.out
│ │ │ @@ -77,15 +77,15 @@
│ │ │  o11 = monomialIdeal (a*b*c, a d, b*c*d)
│ │ │  
│ │ │  o11 : MonomialIdeal of R
│ │ │  
│ │ │  i12 : standardPairs I
│ │ │  
│ │ │                                                                             
│ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │        -----------------------------------------------------------------------
│ │ │                             2
│ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │  
│ │ │  o12 : List
│ │ │  
│ │ │  i13 : independentSets I
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutable__List.out
│ │ │ @@ -37,19 +37,19 @@
│ │ │  i10 : s = new MutableList
│ │ │  
│ │ │  o10 = MutableList{}
│ │ │  
│ │ │  o10 : MutableList
│ │ │  
│ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00605048s elapsed
│ │ │ + -- .00423495s elapsed
│ │ │  
│ │ │  i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
│ │ │  
│ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000712171s elapsed
│ │ │ + -- .000305035s elapsed
│ │ │  
│ │ │  i14 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Mutex.out
│ │ │ @@ -8,60 +8,51 @@
│ │ │  
│ │ │  o2 = sayhello
│ │ │  
│ │ │  o2 : FunctionClosure
│ │ │  
│ │ │  i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │  
│ │ │ -o3 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     task done>>, <>, <>, <>, <>, <>,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     <>}
│ │ │ +     <>, <>, <>}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : while not all(T, isReady) do null
│ │ │  
│ │ │  i5 : stack sort lines msgs
│ │ │  
│ │ │  o5 = hello from thread #0
│ │ │       hello from thread #1
│ │ │       hello from thread #2
│ │ │ -     hello from thread #3
│ │ │       hello from thread #4
│ │ │ -     hello from thread #5
│ │ │       hello from thread #6
│ │ │ -     hello from thread #7
│ │ │ +     hello from thread #8
│ │ │ +     hello from thread #9
│ │ │  
│ │ │  i6 : m = new Mutex
│ │ │  
│ │ │  o6 = m
│ │ │  
│ │ │  o6 : Mutex
│ │ │  
│ │ │  i7 : msgs = ""
│ │ │  
│ │ │  o7 = 
│ │ │  
│ │ │  i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m)))
│ │ │  
│ │ │ -o8 = {<>, <>, <>, <>, <>, <>,
│ │ │ +o8 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     created>>}
│ │ │ +     <>, <>, <>}
│ │ │  
│ │ │  o8 : List
│ │ │  
│ │ │  i9 : while not all(T, isReady) do null
│ │ │  
│ │ │  i10 : stack sort lines msgs
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.out
│ │ │ @@ -3,13 +3,13 @@
│ │ │  i1 : M = random(RR^200, RR^200);
│ │ │  
│ │ │                  200         200
│ │ │  o1 : Matrix RR      <-- RR
│ │ │                53          53
│ │ │  
│ │ │  i2 : time SVD(M);
│ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
│ │ │ + -- used 0.0309803s (cpu); 0.0309795s (thread); 0s (gc)
│ │ │  
│ │ │  i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
│ │ │ + -- used 0.0299955s (cpu); 0.030004s (thread); 0s (gc)
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/___Set.out
│ │ │ @@ -2,23 +2,23 @@
│ │ │  
│ │ │  i1 : A = set {1,2};
│ │ │  
│ │ │  i2 : R = QQ[a..d];
│ │ │  
│ │ │  i3 : B = set{a^2-b*c,b*d}
│ │ │  
│ │ │ -           2
│ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ +                2
│ │ │ +o3 = set {b*d, a  - b*c}
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 : toList B
│ │ │  
│ │ │ -       2
│ │ │ -o4 = {a  - b*c, b*d}
│ │ │ +            2
│ │ │ +o4 = {b*d, a  - b*c}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : member(1,A)
│ │ │  
│ │ │  o5 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_a_spfirst_sp__Macaulay2_spsession.out
│ │ │ @@ -347,15 +347,15 @@
│ │ │                 | b e h k n q |
│ │ │                 | c f i l o r |
│ │ │  
│ │ │                               3
│ │ │  o58 : R-module, quotient of R
│ │ │  
│ │ │  i59 : time C = resolution M
│ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ + -- used 0.00302468s (cpu); 0.00301751s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_at__End__Of__File_lp__File_rp.out
│ │ │ @@ -14,10 +14,10 @@
│ │ │  
│ │ │  i4 : peek read f
│ │ │  
│ │ │  o4 = "hi there"
│ │ │  
│ │ │  i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
│ │ │ +o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_benchmark.out
│ │ │ @@ -1,9 +1,9 @@
│ │ │  -- -*- M2-comint -*- hash: 1330379359420
│ │ │  
│ │ │  i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0003750952360409743
│ │ │ +o1 = .0003024443739726021
│ │ │  
│ │ │  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'
│ │ │ - -- .0035172s elapsed
│ │ │ + -- .00355487s elapsed
│ │ │  
│ │ │  o4 = 3
│ │ │  
│ │ │  i5 : elapsedTime pdim' M
│ │ │ - -- 9.22e-7s elapsed
│ │ │ + -- .000002484s elapsed
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : peek M.cache
│ │ │  
│ │ │  o6 = CacheTable{cache => MutableHashTable{}                                              }
│ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_can__Use__Hilbert__Hint.out
│ │ │ @@ -211,15 +211,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │             5
│ │ │        + 12w )
│ │ │  
│ │ │  o17 : Ideal of Rlex
│ │ │  
│ │ │  i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ - -- 1.68769s elapsed
│ │ │ + -- 1.789s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o18 : Matrix Rlex  <-- Rlex
│ │ │  
│ │ │  i19 : Ilex = ideal(Ilex_*) -- clear out the previous Groebner basis
│ │ │  
│ │ │                  5      4       4       4       3 2      3         3     
│ │ │ @@ -293,15 +293,15 @@
│ │ │        -----------------------------------------------------------------------
│ │ │             5
│ │ │        + 12w )
│ │ │  
│ │ │  o19 : Ideal of Rlex
│ │ │  
│ │ │  i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ - -- .850458s elapsed
│ │ │ + -- .80331s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o20 : Matrix Rlex  <-- Rlex
│ │ │  
│ │ │  i21 : g1 == g2
│ │ │  
│ │ │  o21 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_cancel__Task_lp__Task_rp.out
│ │ │ @@ -18,29 +18,29 @@
│ │ │  
│ │ │  o4 = <>
│ │ │  
│ │ │  o4 : Task
│ │ │  
│ │ │  i5 : n
│ │ │  
│ │ │ -o5 = 336394
│ │ │ +o5 = 721956
│ │ │  
│ │ │  i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
│ │ │  
│ │ │  i7 : t
│ │ │  
│ │ │  o7 = <>
│ │ │  
│ │ │  o7 : Task
│ │ │  
│ │ │  i8 : n
│ │ │  
│ │ │ -o8 = 929987
│ │ │ +o8 = 1699995
│ │ │  
│ │ │  i9 : isReady t
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 : cancelTask t
│ │ │  
│ │ │ @@ -53,22 +53,22 @@
│ │ │  
│ │ │  o12 = <>
│ │ │  
│ │ │  o12 : Task
│ │ │  
│ │ │  i13 : n
│ │ │  
│ │ │ -o13 = 930227
│ │ │ +o13 = 1700180
│ │ │  
│ │ │  i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
│ │ │  
│ │ │  i15 : n
│ │ │  
│ │ │ -o15 = 930227
│ │ │ +o15 = 1700180
│ │ │  
│ │ │  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/M2-11296-0/0
│ │ │ +o1 = /tmp/M2-11766-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11296-0/0
│ │ │ +o2 = /tmp/M2-11766-0/0
│ │ │  
│ │ │  i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11296-0/0/
│ │ │ +o3 = /tmp/M2-11766-0/0/
│ │ │  
│ │ │  i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11296-0/0/
│ │ │ +o4 = /tmp/M2-11766-0/0/
│ │ │  
│ │ │  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")        -- .313321s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .167999s elapsed
│ │ │  
│ │ │  i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .146472s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .141847s elapsed
│ │ │  
│ │ │  i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .146792s elapsed
│ │ │  
│ │ │  i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .140652s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .141035s 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")        -- .302726s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .15447s 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")        -- .28407s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
│ │ │ + -- capturing check(0, "FirstPackage")        -- .142307s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .141349s 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")        -- .283414s elapsed
│ │ │ + -- capturing check(1, "FirstPackage")        -- .143194s elapsed
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_communicating_spwith_spprograms.out
│ │ │ @@ -1,26 +1,26 @@
│ │ │  -- -*- M2-comint -*- hash: 10365735446967377456
│ │ │  
│ │ │  i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) 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 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
│ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ +     6.12.100-1 (2026-07-30) 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 0x7f57805dd000
│ │ │ +   -- registering gb 5 at 0x7f6c00181000
│ │ │  
│ │ │     -- [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 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ + -- used 0.17595s (cpu); 0.173459s (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 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ + -- used 0.00788594s (cpu); 0.00650792s (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/M2-12075-0/0/
│ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12075-0/1/
│ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12075-0/0/a/
│ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12075-0/0/b/
│ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ -     /tmp/M2-12075-0/0/a/f
│ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │  
│ │ │  i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ -      /tmp/M2-12075-0/1/b/c/g
│ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ +      /tmp/M2-13345-0/1/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/M2-11841-0/0
│ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11841-0/1
│ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
│ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │  
│ │ │  i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │  
│ │ │  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 = 611.770723009
│ │ │ +o1 = 495.5602200340001
│ │ │  
│ │ │  o1 : RR (of precision 53)
│ │ │  
│ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │  
│ │ │  i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 613.313432871
│ │ │ +o3 = 496.629935466
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.542709862000038
│ │ │ +o4 = 1.069715431999953
│ │ │  
│ │ │  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 = 1792811657
│ │ │ +o1 = 1792834376
│ │ │  
│ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.81193979063632
│ │ │ +o2 = 56.81265972708645
│ │ │  
│ │ │  o2 : RR (of precision 53)
│ │ │  
│ │ │  i3 : 12 * (oo - floor oo)
│ │ │  
│ │ │ -o3 = 9.743277487635794
│ │ │ +o3 = 9.751916725037375
│ │ │  
│ │ │  o3 : RR (of precision 53)
│ │ │  
│ │ │  i4 : run "date"
│ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ +Sat Oct 24 09:32:56 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
│ │ │ + -- 1.00019s 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 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ + -- used 0.347347s (cpu); 0.148903s (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 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ + -- used 0.0843781s (cpu); 0.0843817s (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 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ + -- used 0.0259684s (cpu); 0.0259673s (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 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ + -- used 0.32765s (cpu); 0.126762s (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 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ + -- used 0.00152028s (cpu); 0.0015189s (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 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ + -- used 0.0285218s (cpu); 0.0283111s (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/M2-11022-0/96-rundir/
│ │ │ +             source directory => /tmp/M2-11242-0/96-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/M2-11391-0/0
│ │ │ +o1 = /tmp/M2-11961-0/0
│ │ │  
│ │ │  i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ +o3 = /tmp/M2-11961-0/0
│ │ │  
│ │ │  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/M2-13078-0/0
│ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o1 : File
│ │ │  
│ │ │  i2 : fileLength f
│ │ │  
│ │ │  o2 = 8
│ │ │  
│ │ │  i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-13078-0/0
│ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  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/M2-12284-0/0
│ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : fileMode f
│ │ │  
│ │ │  o3 = 420
│ │ │  
│ │ │  i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  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/M2-11860-0/0
│ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  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/M2-11725-0/0
│ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  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/M2-11725-0/0
│ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  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/M2-12905-0/0
│ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  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 = 111
│ │ │ +o1 = 79
│ │ │  
│ │ │  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 0x7f466ed708c0
│ │ │ +   -- registering gb 0 at 0x7f690de108c0
│ │ │  
│ │ │     -- [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 = Sat Oct 24 03:12:48 UTC 2026
│ │ │ +o4 = Sat Oct 24 09:31:57 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/M2-11213-0/0
│ │ │ +o2 = /tmp/M2-11603-0/0
│ │ │  
│ │ │  i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ +o3 = /tmp/M2-11603-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : isDirectory fn
│ │ │  
│ │ │  o4 = false
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_is__Pseudoprime_lp__Z__Z_rp.out
│ │ │ @@ -75,15 +75,15 @@
│ │ │  o17 = false
│ │ │  
│ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │  
│ │ │  o18 = false
│ │ │  
│ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.67323s elapsed
│ │ │ + -- 4.34611s 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
│ │ │ - -- .05718s elapsed
│ │ │ + -- .0502503s elapsed
│ │ │  
│ │ │  o23 = true
│ │ │  
│ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000110026s elapsed
│ │ │ + -- .000102701s 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/M2-13116-0/0
│ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  o2 : File
│ │ │  
│ │ │  i3 : isRegularFile fn
│ │ │  
│ │ │  o3 = true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_lock_lp__Mutex_cm__Function_rp.out
│ │ │ @@ -6,15 +6,15 @@
│ │ │  
│ │ │  i2 : f = i -> x += 1;
│ │ │  
│ │ │  i3 : parallelApply(1..1000, f);
│ │ │  
│ │ │  i4 : x
│ │ │  
│ │ │ -o4 = 333
│ │ │ +o4 = 337
│ │ │  
│ │ │  i5 : x = 0
│ │ │  
│ │ │  o5 = 0
│ │ │  
│ │ │  i6 : f = lock f;
│ │ ├── ./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/M2-11593-0/0
│ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │  
│ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
│ │ │ +o2 = /tmp/M2-12363-0/0/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 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ + -- used 0.885041s (cpu); 0.504981s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
│ │ │  
│ │ │  i3 : fib = memoize fib
│ │ │  
│ │ │  o3 = fib
│ │ │  
│ │ │  o3 : FunctionClosure
│ │ │  
│ │ │  i4 : time fib 28
│ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ + -- used 5.8387e-05s (cpu); 5.5927e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
│ │ │  
│ │ │  i5 : time fib 28
│ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ + -- used 2.988e-06s (cpu); 2.779e-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
│ │ │ @@ -4,33 +4,33 @@
│ │ │  
│ │ │  o1 = {0 => (==, BettiTally, BettiTally)                           }
│ │ │       {1 => (++, BettiTally, BettiTally)                           }
│ │ │       {2 => (**, BettiTally, BettiTally)                           }
│ │ │       {3 => (SPACE, BettiTally, Array)                             }
│ │ │       {4 => (SPACE, BettiTally, ZZ)                                }
│ │ │       {5 => (lift, BettiTally, ZZ)                                 }
│ │ │ -     {6 => (*, ZZ, BettiTally)                                    }
│ │ │ -     {7 => (*, QQ, BettiTally)                                    }
│ │ │ +     {6 => (*, QQ, BettiTally)                                    }
│ │ │ +     {7 => (*, ZZ, BettiTally)                                    }
│ │ │       {8 => (multigraded, BettiTally)                              }
│ │ │       {9 => (net, BettiTally)                                      }
│ │ │       {10 => (texMath, BettiTally)                                 }
│ │ │       {11 => (betti, BettiTally)                                   }
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ +     {19 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {21 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {22 => (codim, BettiTally)                                   }
│ │ │ +     {23 => (dual, BettiTally)                                    }
│ │ │ +     {24 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │  
│ │ │  i2 : methods resolution
│ │ │  
│ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ @@ -60,20 +60,20 @@
│ │ │  
│ │ │  o4 = {0 => (++, Module, Module)}
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ +     {1 => (+, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract', Matrix, Matrix)                           }
│ │ │ +     {3 => (diff', Matrix, Matrix)                               }
│ │ │ +     {4 => (-, 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)                        }
│ │ │ @@ -84,22 +84,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │       {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {31 => (pullback, 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
│ │ │ - -- 1.20053s elapsed
│ │ │ + -- 1.43525s 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)
│ │ │ - -- .517063s elapsed
│ │ │ + -- .621932s 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)
│ │ │ - -- .0258497s elapsed
│ │ │ + -- .030078s 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)
│ │ │ - -- .826979s elapsed
│ │ │ + -- .982488s 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/M2-11612-0/0/
│ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │  
│ │ │  i2 : mkdir p
│ │ │  
│ │ │  i3 : isDirectory p
│ │ │  
│ │ │  o3 = true
│ │ │  
│ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ +o4 = /tmp/M2-12402-0/0/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/M2-11448-0/0
│ │ │ +o1 = /tmp/M2-12078-0/0
│ │ │  
│ │ │  i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/1
│ │ │ +o2 = /tmp/M2-12078-0/1
│ │ │  
│ │ │  i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ +o3 = /tmp/M2-12078-0/0
│ │ │  
│ │ │  o3 : File
│ │ │  
│ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
│ │ │ +--moving: /tmp/M2-12078-0/0 -> /tmp/M2-12078-0/1
│ │ │  
│ │ │  i5 : get dst
│ │ │  
│ │ │  o5 = hi there
│ │ │  
│ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ +--backup file created: /tmp/M2-12078-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11448-0/1.bak
│ │ │ +o6 = /tmp/M2-12078-0/1.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
│ │ │ - -- .500147s elapsed
│ │ │ + -- .500914s elapsed
│ │ │  
│ │ │  o1 = 0
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_options_lp__Function_rp.out
│ │ │ @@ -20,49 +20,49 @@
│ │ │  
│ │ │  o3 = OptionTable{Generic => false}
│ │ │  
│ │ │  o3 : OptionTable
│ │ │  
│ │ │  i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ -     {1 => (codim, Ideal)         }
│ │ │ -     {2 => (codim, PolynomialRing)}
│ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ -     {5 => (codim, Module)        }
│ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ -     {7 => (codim, BettiTally)    }
│ │ │ +o4 = {0 => (codim, BettiTally)    }
│ │ │ +     {1 => (codim, Variety)       }
│ │ │ +     {2 => (codim, Ideal)         }
│ │ │ +     {3 => (codim, PolynomialRing)}
│ │ │ +     {4 => (codim, QuotientRing)  }
│ │ │ +     {5 => (codim, MonomialIdeal) }
│ │ │ +     {6 => (codim, Module)        }
│ │ │ +     {7 => (codim, CoherentSheaf) }
│ │ │  
│ │ │  o4 : NumberedVerticalList
│ │ │  
│ │ │  i5 : options oo
│ │ │  
│ │ │ -o5 = {0 => (OptionTable{Generic => false})}
│ │ │ +o5 = {0 => (OptionTable{})                }
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │       {2 => (OptionTable{Generic => false})}
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ -     {7 => (OptionTable{})                }
│ │ │ +     {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
│ │ │  
│ │ │  i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                          }
│ │ │       {1 => (intersect, Module)                        }
│ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ -     {6 => (intersect, Set)                           }
│ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {8 => (intersect, Set)                           }
│ │ │       {9 => (intersect, Ideal)                         }
│ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │       {11 => (intersect, CCi)                          }
│ │ │       {12 => (intersect, RRi)                          }
│ │ │       {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │       {14 => (intersect, Cone, Cone)                   }
│ │ │       {15 => (intersect, Matrix)                       }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallel_spprogramming_spwith_spthreads_spand_sptasks.out
│ │ │ @@ -5,26 +5,26 @@
│ │ │  o1 = {1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800}
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 : L = shuffle toList (1..10000);
│ │ │  
│ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .659735s elapsed
│ │ │ + -- .625351s elapsed
│ │ │  
│ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .211134s elapsed
│ │ │ + -- .167328s 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)
│ │ │ @@ -74,15 +74,15 @@
│ │ │  
│ │ │  o16 : Task
│ │ │  
│ │ │  i17 : schedule t';
│ │ │  
│ │ │  i18 : t'
│ │ │  
│ │ │ -o18 = <>
│ │ │ +o18 = <>
│ │ │  
│ │ │  o18 : Task
│ │ │  
│ │ │  i19 : taskResult t'
│ │ │  
│ │ │  o19 = | 980z2-18y-201z+13 35yz-4y+2z-1 10y2-y-12z+1 5x2-4y+2z-1 |
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_parallelism_spin_spengine_spcomputations.out
│ │ │ @@ -67,15 +67,15 @@
│ │ │  i3 : S = ring I
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.15969s elapsed
│ │ │ + -- 1.49412s 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)
│ │ │ - -- 1.14895s elapsed
│ │ │ + -- 1.38782s 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)
│ │ │ - -- 1.62269s elapsed
│ │ │ + -- 1.48296s 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)
│ │ │ - -- 2.37571s elapsed
│ │ │ + -- 1.76019s 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)
│ │ │ - -- 1.51803s elapsed
│ │ │ + -- 1.78929s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
│ │ │ @@ -174,43 +174,43 @@
│ │ │  o18 : PolynomialRing
│ │ │  
│ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │  
│ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.09157s elapsed
│ │ │ + -- 3.04216s 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");
│ │ │ - -- 7.21927s elapsed
│ │ │ + -- 6.99351s 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");
│ │ │ - -- 4.08357s elapsed
│ │ │ + -- 3.13269s 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);
│ │ │ - -- .823549s elapsed
│ │ │ + -- .748632s 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);
│ │ │ - -- 1.45319s elapsed
│ │ │ + -- 1.2163s 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 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00270475s (cpu); 9.833e-06s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
│ │ │  
│ │ │  i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ +   -- registering gb 16 at 0x7f9895d5ac40
│ │ │  
│ │ │     -- [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 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00931577s (cpu); 0.00985886s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
│ │ │  
│ │ │  i30 : R = QQ[a..d];
│ │ │  
│ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ +   -- registering gb 17 at 0x7f9895d5aa80
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │     -- nsaved = 0
│ │ │     -- 
│ │ │  o31 : Ideal of R
│ │ │  
│ │ │  i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ +   -- registering gb 18 at 0x7f9895d5a8c0
│ │ │  
│ │ │     -- [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 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00321219s (cpu); 0.00352901s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
│ │ │  
│ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ @@ -254,27 +254,27 @@
│ │ │  
│ │ │  i36 : gbTrace = 3
│ │ │  
│ │ │  o36 = 3
│ │ │  
│ │ │  i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ +   -- registering gb 19 at 0x7f9895d5a700
│ │ │  
│ │ │     -- [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 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0439623s (cpu); 0.0445398s (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/M2-11803-0/0
│ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  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/M2-11803-0/0
│ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  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/M2-11803-0/0
│ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  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 = 11022
│ │ │ +o1 = 11242
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_profile.out
│ │ │ @@ -9,35 +9,35 @@
│ │ │  
│ │ │                4       5
│ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0041s  elapsed total                    
│ │ │ +     1     93.86   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     91.14   ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     42.26   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     29.1    ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     27.95   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     23.28   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.94   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     19.76   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     3.55    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.29    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     2.19    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.63    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.49    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     1     1.36    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.27    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.12    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     1     1.12    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     20    .92     ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     20    .72     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     20    .66     ../../m2/matrix1.m2:180:17-180:29
│ │ │ +     1     .0032s  elapsed total                    
│ │ │  
│ │ │  i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │       ../../m2/lists.m2:148:24-148:32
│ │ │       ../../m2/lists.m2:148:34-148:58
│ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_random__K__Rational__Point.out
│ │ │ @@ -13,15 +13,15 @@
│ │ │  i5 : codim I, degree I
│ │ │  
│ │ │  o5 = (2, 10)
│ │ │  
│ │ │  o5 : Sequence
│ │ │  
│ │ │  i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ + -- used 0.448087s (cpu); 0.119215s (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 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ + -- used 0.311606s (cpu); 0.196472s (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 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ + -- used 4.7779s (cpu); 1.86455s (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/M2-12474-0/0
│ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12474-0/0
│ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │  
│ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ +o3 = /tmp/M2-14164-0/0/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/M2-11978-0/0
│ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  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/M2-11978-0/0
│ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  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/M2-11978-0/0
│ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  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/M2-12734-0/0
│ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │  
│ │ │  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/M2-11022-0/91-rundir/
│ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │  
│ │ │  i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12753-0/0
│ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12753-0/1
│ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │  
│ │ │  i4 : symlinkFile(p,q)
│ │ │  
│ │ │  i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  o5 : File
│ │ │  
│ │ │  i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12753-0/0
│ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12753-0/0
│ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  i8 : removeFile p
│ │ │  
│ │ │  i9 : removeFile q
│ │ │  
│ │ │  i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
│ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_register__Finalizer.out
│ │ │ @@ -1,17 +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)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (1)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (2)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (7)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (8)[7]: -- finalizing sequence #8 --
│ │ │  --finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_remove__Directory.out
│ │ │ @@ -1,19 +1,19 @@
│ │ │  -- -*- M2-comint -*- hash: 8532980310097060089
│ │ │  
│ │ │  i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11650-0/0
│ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │  
│ │ │  i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11650-0/0
│ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │  
│ │ │  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/M2-11116-0/0
│ │ │ +o1 = /tmp/M2-11406-0/0
│ │ │  
│ │ │  i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-11116-0/0
│ │ │ +o2 = /tmp/M2-11406-0/0
│ │ │  
│ │ │  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/M2-12417-0/0
│ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │  
│ │ │  i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12417-0/0
│ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │  
│ │ │  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/M2-12265-0/0
│ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  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 = 1367325
│ │ │ +o1 = 1367343
│ │ │  
│ │ │  i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1367327
│ │ │ +o2 = 1367345
│ │ │  
│ │ │  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 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
│ │ │ + -- used 0.00020554s (cpu); 0.000199105s (thread); 0s (gc)
│ │ │  
│ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
│ │ │ + -- used 8.6518e-05s (cpu); 8.6233e-05s (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 0.232067s (cpu); 0.232067s (thread); 0s (gc)
│ │ │ + -- used 0.117585s (cpu); 0.117586s (thread); 0s (gc)
│ │ │  
│ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
│ │ │ + -- used 0.118674s (cpu); 0.118681s (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/M2-12037-0/0/
│ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │  
│ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12037-0/1/
│ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │  
│ │ │  i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12037-0/0/a/
│ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │  
│ │ │  i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12037-0/0/b/
│ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │  
│ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
│ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │  
│ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │  
│ │ │  o6 : File
│ │ │  
│ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │  
│ │ │  o7 : File
│ │ │  
│ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
│ │ │  
│ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/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/M2-12037-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/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/M2-12094-0/0
│ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │  
│ │ │  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/M2-13097-0/0.tex
│ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │  
│ │ │  i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-13097-0/1.html
│ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_time.out
│ │ │ @@ -1,8 +1,8 @@
│ │ │  -- -*- M2-comint -*- hash: 1332435500723
│ │ │  
│ │ │  i1 : time 3^30
│ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ + -- used 1.6199e-05s (cpu); 5.786e-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
│ │ │ -     -- .000013355 seconds
│ │ │ +     -- .000013321 seconds
│ │ │  
│ │ │  o1 : Time
│ │ │  
│ │ │  i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000013355, 205891132094649}
│ │ │ +o2 = Time{.000013321, 205891132094649}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.out
│ │ │ @@ -14,26 +14,26 @@
│ │ │  
│ │ │            4     5    6     9     10     11    14     15    16    20
│ │ │  o3 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │  
│ │ │  o3 : ZZ[T]
│ │ │  
│ │ │  i4 : elapsedTime Mgb = gb M
│ │ │ - -- 1.4495s elapsed
│ │ │ + -- 1.73571s elapsed
│ │ │  
│ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o4 : GroebnerBasis
│ │ │  
│ │ │  i5 : M = ideal M_*;
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │  
│ │ │  i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .415018s elapsed
│ │ │ + -- .463688s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o6 : GroebnerBasis
│ │ │  
│ │ │  i7 : S = degreesRing R
│ │ │  
│ │ │ @@ -55,14 +55,14 @@
│ │ │  o9 : S
│ │ │  
│ │ │  i10 : M = ideal M_*;
│ │ │  
│ │ │  o10 : Ideal of R
│ │ │  
│ │ │  i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .517879s elapsed
│ │ │ + -- .548971s elapsed
│ │ │  
│ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o11 : GroebnerBasis
│ │ │  
│ │ │  i12 :
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/example-output/_version.out
│ │ │ @@ -36,15 +36,15 @@
│ │ │                 "memtailor version" => 1.4
│ │ │                 "mpfi version" => 1.5.5
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism DirectSummands CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic WeierstrassSemigroups ResultantComplexes EuclideanDistanceDegree NeuralIdeals TestAudit LanguageServer
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.14.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2023.1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Command.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │              
    		
│ │ │                
    i3 : (c = Command "date";)
    
│ │ │              
    
│ │ │                
    i4 : c
│ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ +Sat Oct 24 09:31:24 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
│ │ │ │ -Sat Oct 24 03:11:57 UTC 2026
│ │ │ │ +Sat Oct 24 09:31:24 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_u_n -- run an external command
│ │ │ │      * _A_f_t_e_r_E_v_a_l -- top level method applied after evaluation
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa ccoommmmaanndd:: **********
│ │ │ │      * code(Command) -- see _c_o_d_e -- display source code
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Database.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  A database file is just like a hash table, except both the keys and values have to be strings.  In this example we create a database file, store a few entries, remove an entry with remove, close the file, and then remove the file.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -86,15 +86,15 @@
│ │ │ │                               2
│ │ │ │  o11 = monomialIdeal (a*b*c, a d, b*c*d)
│ │ │ │  
│ │ │ │  o11 : MonomialIdeal of R
│ │ │ │  i12 : standardPairs I
│ │ │ │  
│ │ │ │  
│ │ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │                             2
│ │ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │ │  
│ │ │ │  o12 : List
│ │ │ │  i13 : independentSets I
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutable__List.html
│ │ │ @@ -145,30 +145,30 @@
│ │ │  
│ │ │  o10 : MutableList
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : filename = temporaryFileName () | ".dbm"
│ │ │  
│ │ │ -o1 = /tmp/M2-12569-0/0.dbm
    
│ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │              
    
│ │ │                
    i2 : x = openDatabaseOut filename
│ │ │  
│ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ +o2 = /tmp/M2-14359-0/0.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/M2-12569-0/0.dbm
│ │ │ │ +o1 = /tmp/M2-14359-0/0.dbm
│ │ │ │  i2 : x = openDatabaseOut filename
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12569-0/0.dbm
│ │ │ │ +o2 = /tmp/M2-14359-0/0.dbm
│ │ │ │  
│ │ │ │  o2 : Database
│ │ │ │  i3 : x#"first" = "hi there"
│ │ │ │  
│ │ │ │  o3 = hi there
│ │ │ │  i4 : x#"first"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___G__Cstats.html
│ │ │ @@ -58,33 +58,33 @@
│ │ │          
    Description
    
│ │ │          
    Macaulay2 uses the Hans Boehm garbage collector to reclaim unused memory.  The function GCstats provides information about its status, such as the total number of bytes allocated, the current heap size, the number of garbage collections done, the number of threads used in each collection, the total cpu time spent in garbage collection, etc.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : s = GCstats()
│ │ │  
│ │ │ -o1 = HashTable{"bytesAlloc" => 69379379194        }
│ │ │ +o1 = HashTable{"bytesAlloc" => 69264345690        }
│ │ │                 "GC_free_space_divisor" => 3
│ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │                 "gcCpuTimeSecs" => 0
│ │ │ -               "heapSize" => 219217920
│ │ │ -               "numGCs" => 1481
│ │ │ -               "numGCThreads" => 6
│ │ │ +               "heapSize" => 255078400
│ │ │ +               "numGCs" => 1447
│ │ │ +               "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 = 219217920
    
│ │ │ +o2 = 255078400
│ │ │              
    
│ │ │          
    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" => 69379379194        }
│ │ │ │ +o1 = HashTable{"bytesAlloc" => 69264345690        }
│ │ │ │                 "GC_free_space_divisor" => 3
│ │ │ │                 "GC_LARGE_ALLOC_WARN_INTERVAL" => 1
│ │ │ │                 "gcCpuTimeSecs" => 0
│ │ │ │ -               "heapSize" => 219217920
│ │ │ │ -               "numGCs" => 1481
│ │ │ │ -               "numGCThreads" => 6
│ │ │ │ +               "heapSize" => 255078400
│ │ │ │ +               "numGCs" => 1447
│ │ │ │ +               "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 = 219217920
│ │ │ │ +o2 = 255078400
│ │ │ │  Any entries whose keys are all upper case give the values of environment
│ │ │ │  variables affecting the operation of the garbage collector that have been
│ │ │ │  specified by the user.
│ │ │ │  For further information about the individual items in the table, we refer the
│ │ │ │  user to the source code and documentation of the garbage collector.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _G_C_ _g_a_r_b_a_g_e_ _c_o_l_l_e_c_t_o_r
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Minimal__Generators.html
│ │ │ @@ -133,23 +133,23 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time J = truncate(8, I, MinimalGenerators => false);
│ │ │ - -- used 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ + -- used 0.00473147s (cpu); 0.00472981s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
    
│ │ │                
    i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ + -- used 0.0392479s (cpu); 0.0392539s (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 0.00860229s (cpu); 0.00859565s (thread); 0s (gc)
│ │ │ │ + -- used 0.00473147s (cpu); 0.00472981s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time K = truncate(8, I, MinimalGenerators => true);
│ │ │ │ - -- used 0.0658401s (cpu); 0.0658504s (thread); 0s (gc)
│ │ │ │ + -- used 0.0392479s (cpu); 0.0392539s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 : Ideal of R
│ │ │ │  i10 : numgens J
│ │ │ │  
│ │ │ │  o10 = 1067
│ │ │ │  i11 : numgens K
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Monomial__Ideal.html
│ │ │ @@ -181,15 +181,15 @@
│ │ │              
    
│ │ │                
    i12 : standardPairs I
│ │ │  
│ │ │                                                                             
│ │ │ -o12 = {{1, {c, d}}, {a, {c, d}}, {1, {b, d}}, {a, {b, d}}, {1, {a, c}}, {1,
│ │ │ +o12 = {{1, {c, d}}, {a, {c, d}}, {1, {d, b}}, {a, {d, b}}, {1, {c, a}}, {1,
│ │ │        -----------------------------------------------------------------------
│ │ │                             2
│ │ │        {b, a}}, {b, {c}}, {b , {c}}}
│ │ │  
│ │ │  o12 : List
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at each step
│ │ │ - -- .00605048s elapsed
    
│ │ │ + -- .00423495s elapsed
│ │ │              
    
│ │ │                
    i12 : t = new MutableList from 1000
│ │ │  
│ │ │  o12 = MutableList{...1000...}
│ │ │  
│ │ │  o12 : MutableList
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ - -- .000712171s elapsed
    
│ │ │ + -- .000305035s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -43,22 +43,22 @@
│ │ │ │  i10 : s = new MutableList
│ │ │ │  
│ │ │ │  o10 = MutableList{}
│ │ │ │  
│ │ │ │  o10 : MutableList
│ │ │ │  i11 : elapsedTime scan(1000, i -> s#i = i^2) -- quadratic, since we grow s at
│ │ │ │  each step
│ │ │ │ - -- .00605048s elapsed
│ │ │ │ + -- .00423495s elapsed
│ │ │ │  i12 : t = new MutableList from 1000
│ │ │ │  
│ │ │ │  o12 = MutableList{...1000...}
│ │ │ │  
│ │ │ │  o12 : MutableList
│ │ │ │  i13 : elapsedTime scan(1000, i -> t#i = i^2) -- linear
│ │ │ │ - -- .000712171s elapsed
│ │ │ │ + -- .000305035s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _B_a_s_i_c_L_i_s_t -- the class of all basic lists
│ │ │ │  ******** MMeennuu ********
│ │ │ │      * _B_a_g -- the class of all bags
│ │ │ │  ********** TTyyppeess ooff mmuuttaabbllee lliisstt:: **********
│ │ │ │      * _B_a_g -- the class of all bags
│ │ │ │  ********** MMeetthhooddss tthhaatt uussee aa mmuuttaabbllee lliisstt:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___Mutex.html
│ │ │ @@ -78,25 +78,19 @@
│ │ │  o2 : FunctionClosure
    │ │ │
    │ │ │ 
    i3 : T = apply(10, i -> schedule(() -> sayhello i))
│ │ │  
│ │ │ -o3 = {<<task, result available, task done>>, <<task, result available, task
│ │ │ +o3 = {<<task, created>>, <<task, created>>, <<task, created>>, <<task,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     done>>, <<task, result available, task done>>, <<task, result available,
│ │ │ +     created>>, <<task, created>>, <<task, created>>, <<task, created>>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     task done>>, <<task, result available, task done>>, <<task, result
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     available, task done>>, <<task, result available, task done>>, <<task,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     result available, task done>>, <<task, result available, task done>>,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     <<task, result available, task done>>}
│ │ │ +     <<task, created>>, <<task, created>>, <<task, created>>}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : while not all(T, isReady) do null
    │ │ │ @@ -105,19 +99,18 @@ │ │ │
    │ │ │ 
    i5 : stack sort lines msgs
│ │ │  
│ │ │  o5 = hello from thread #0
│ │ │       hello from thread #1
│ │ │       hello from thread #2
│ │ │ -     hello from thread #3
│ │ │       hello from thread #4
│ │ │ -     hello from thread #5
│ │ │       hello from thread #6
│ │ │ -     hello from thread #7
    │ │ │ + hello from thread #8 │ │ │ + hello from thread #9 │ │ │
    │ │ │
    │ │ │

    We likely ended up with fewer than the expected number of 10 messages. We can get around this issue by using a mutex to lock the string so that only one thread can modify it at a time.

    │ │ │
    │ │ │ │ │ │ @@ -137,21 +130,19 @@ │ │ │ o7 = │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m)))
│ │ │  
│ │ │ -o8 = {<<task, result available, task done>>, <<task, running>>, <<task,
│ │ │ -     ------------------------------------------------------------------------
│ │ │ -     running>>, <<task, running>>, <<task, running>>, <<task, created>>,
│ │ │ +o8 = {<<task, running>>, <<task, running>>, <<task, running>>, <<task,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     <<task, created>>, <<task, created>>, <<task, created>>, <<task,
│ │ │ +     running>>, <<task, running>>, <<task, created>>, <<task, created>>,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     created>>}
│ │ │ +     <<task, created>>, <<task, created>>, <<task, created>>}
│ │ │  
│ │ │  o8 : List
    │ │ │
    │ │ │ 
    i9 : while not all(T, isReady) do null
    │ │ │ ├── html2text {} │ │ │ │ @@ -23,58 +23,49 @@ │ │ │ │ i2 : sayhello = i -> msgs |= "hello from thread #" | toString i | newline │ │ │ │ │ │ │ │ o2 = sayhello │ │ │ │ │ │ │ │ o2 : FunctionClosure │ │ │ │ i3 : T = apply(10, i -> schedule(() -> sayhello i)) │ │ │ │ │ │ │ │ -o3 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - task done>>, <>, <>, <>, <>, <>, │ │ │ │ - ------------------------------------------------------------------------ │ │ │ │ - <>} │ │ │ │ + <>, <>, <>} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : while not all(T, isReady) do null │ │ │ │ i5 : stack sort lines msgs │ │ │ │ │ │ │ │ o5 = hello from thread #0 │ │ │ │ hello from thread #1 │ │ │ │ hello from thread #2 │ │ │ │ - hello from thread #3 │ │ │ │ hello from thread #4 │ │ │ │ - hello from thread #5 │ │ │ │ hello from thread #6 │ │ │ │ - hello from thread #7 │ │ │ │ + hello from thread #8 │ │ │ │ + hello from thread #9 │ │ │ │ We likely ended up with fewer than the expected number of 10 messages. We can │ │ │ │ get around this issue by using a mutex to lock the string so that only one │ │ │ │ thread can modify it at a time. │ │ │ │ i6 : m = new Mutex │ │ │ │ │ │ │ │ o6 = m │ │ │ │ │ │ │ │ o6 : Mutex │ │ │ │ i7 : msgs = "" │ │ │ │ │ │ │ │ o7 = │ │ │ │ i8 : T = apply(10, i -> schedule(() -> (lock m; sayhello i; unlock m))) │ │ │ │ │ │ │ │ -o8 = {<>, <>, <>, <>, <>, <>, │ │ │ │ +o8 = {<>, <>, <>, <>, <>, <>, <>, <>, <>, <>, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - created>>} │ │ │ │ + <>, <>, <>} │ │ │ │ │ │ │ │ o8 : List │ │ │ │ i9 : while not all(T, isReady) do null │ │ │ │ i10 : stack sort lines msgs │ │ │ │ │ │ │ │ o10 = hello from thread #0 │ │ │ │ hello from thread #1 │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/___S__V__D_lp..._cm__Divide__Conquer_eq_gt..._rp.html │ │ │ @@ -73,21 +73,21 @@ │ │ │ o1 : Matrix RR <-- RR │ │ │ 53 53 │ │ │
    │ │ │ 
    i2 : time SVD(M);
│ │ │ - -- used 0.0228821s (cpu); 0.0228783s (thread); 0s (gc)
    │ │ │ + -- used 0.0309803s (cpu); 0.0309795s (thread); 0s (gc) │ │ │
    │ │ │ 
    i3 : time SVD(M, DivideConquer=>true);
│ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc)
    │ │ │ + -- used 0.0299955s (cpu); 0.030004s (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 0.0228821s (cpu); 0.0228783s (thread); 0s (gc) │ │ │ │ + -- used 0.0309803s (cpu); 0.0309795s (thread); 0s (gc) │ │ │ │ i3 : time SVD(M, DivideConquer=>true); │ │ │ │ - -- used 0.0228804s (cpu); 0.0228876s (thread); 0s (gc) │ │ │ │ + -- used 0.0299955s (cpu); 0.030004s (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/___Set.html │ │ │ @@ -67,28 +67,28 @@ │ │ │ 
    i2 : R = QQ[a..d];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : B = set{a^2-b*c,b*d}
│ │ │  
│ │ │ -           2
│ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ +                2
│ │ │ +o3 = set {b*d, a  - b*c}
│ │ │  
│ │ │  o3 : Set
    │ │ │ │ │ │ │ │ │ │ │ │ Set operations, such as membership, union, intersection, difference, Cartesian product, Cartesian power, and subset are available. For example, │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : toList B
│ │ │  
│ │ │ -       2
│ │ │ -o4 = {a  - b*c, b*d}
│ │ │ +            2
│ │ │ +o4 = {b*d, a  - b*c}
│ │ │  
│ │ │  o4 : List
    │ │ │
    │ │ │ 
    i5 : member(1,A)
│ │ │ ├── html2text {}
│ │ │ │ @@ -7,24 +7,24 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Elements of sets may be any immutable object, such as integers, ring elements
│ │ │ │  and lists of such. Ideals may also be elements of sets.
│ │ │ │  i1 : A = set {1,2};
│ │ │ │  i2 : R = QQ[a..d];
│ │ │ │  i3 : B = set{a^2-b*c,b*d}
│ │ │ │  
│ │ │ │ -           2
│ │ │ │ -o3 = set {a  - b*c, b*d}
│ │ │ │ +                2
│ │ │ │ +o3 = set {b*d, a  - b*c}
│ │ │ │  
│ │ │ │  o3 : Set
│ │ │ │  Set operations, such as _m_e_m_b_e_r_s_h_i_p, _u_n_i_o_n, _i_n_t_e_r_s_e_c_t_i_o_n, _d_i_f_f_e_r_e_n_c_e, _C_a_r_t_e_s_i_a_n
│ │ │ │  _p_r_o_d_u_c_t, _C_a_r_t_e_s_i_a_n_ _p_o_w_e_r, and _s_u_b_s_e_t are available. For example,
│ │ │ │  i4 : toList B
│ │ │ │  
│ │ │ │ -       2
│ │ │ │ -o4 = {a  - b*c, b*d}
│ │ │ │ +            2
│ │ │ │ +o4 = {b*d, a  - b*c}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : member(1,A)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : member(-b*c+a^2,B)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_a_spfirst_sp__Macaulay2_spsession.html
│ │ │ @@ -827,15 +827,15 @@
│ │ │          
    
│ │ │            
    We may use resolution to produce a projective resolution of it, and time to report the time required.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i59 : time C = resolution M
│ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ + -- used 0.00302468s (cpu); 0.00301751s (thread); 0s (gc)
│ │ │  
│ │ │         3      6      15      18      6
│ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o59 : Complex
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -386,15 +386,15 @@
│ │ │ │                 | c f i l o r |
│ │ │ │  
│ │ │ │                               3
│ │ │ │  o58 : R-module, quotient of R
│ │ │ │  We may use _r_e_s_o_l_u_t_i_o_n to produce a projective resolution of it, and _t_i_m_e to
│ │ │ │  report the time required.
│ │ │ │  i59 : time C = resolution M
│ │ │ │ - -- used 0.00458017s (cpu); 0.00457468s (thread); 0s (gc)
│ │ │ │ + -- used 0.00302468s (cpu); 0.00301751s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         3      6      15      18      6
│ │ │ │  o59 = R  <-- R  <-- R   <-- R   <-- R
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o59 : Complex
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_at__End__Of__File_lp__File_rp.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  o4 = "hi there"
│ │ │              
    
│ │ │                
    i5 : atEndOfFile f
│ │ │  
│ │ │ -o5 = false
    
│ │ │ +o5 = true
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    
│ │ │            
    Ways to use this method:
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,13 +23,13 @@
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : peek read f
│ │ │ │  
│ │ │ │  o4 = "hi there"
│ │ │ │  i5 : atEndOfFile f
│ │ │ │  
│ │ │ │ -o5 = false
│ │ │ │ +o5 = true
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _a_t_E_n_d_O_f_F_i_l_e_(_F_i_l_e_) -- test for end of file
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_files.m2:374:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_benchmark.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  Produces an accurate timing for the code contained in the string s.  The value returned is the number of seconds.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : benchmark "sqrt 2p100000"
│ │ │  
│ │ │ -o1 = .0003750952360409743
│ │ │ +o1 = .0003024443739726021
│ │ │  
│ │ │  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 = .0003750952360409743
│ │ │ │ +o1 = .0003024443739726021
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  The snippet of code provided will be run enough times to register meaningfully
│ │ │ │  on the clock, and the garbage collector will be called beforehand.
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _b_e_n_c_h_m_a_r_k is a _f_u_n_c_t_i_o_n_ _c_l_o_s_u_r_e.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_caching_spcomputation_spresults.html
│ │ │ @@ -74,23 +74,23 @@
│ │ │                
    i3 : M = coker vars R;
    
│ │ │
    │ │ │ 
    i4 : elapsedTime pdim' M
│ │ │   -- computing pdim'
│ │ │ - -- .0035172s elapsed
│ │ │ + -- .00355487s elapsed
│ │ │  
│ │ │  o4 = 3
    │ │ │
    │ │ │ 
    i5 : elapsedTime pdim' M
│ │ │ - -- 9.22e-7s elapsed
│ │ │ + -- .000002484s 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'
│ │ │ │ - -- .0035172s elapsed
│ │ │ │ + -- .00355487s elapsed
│ │ │ │  
│ │ │ │  o4 = 3
│ │ │ │  i5 : elapsedTime pdim' M
│ │ │ │ - -- 9.22e-7s elapsed
│ │ │ │ + -- .000002484s elapsed
│ │ │ │  
│ │ │ │  o5 = 3
│ │ │ │  i6 : peek M.cache
│ │ │ │  
│ │ │ │  o6 = CacheTable{cache => MutableHashTable{}
│ │ │ │  }
│ │ │ │                  isHomogeneous => true
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_can__Use__Hilbert__Hint.html
│ │ │ @@ -347,15 +347,15 @@
│ │ │  
│ │ │  o17 : Ideal of Rlex
    │ │ │
    │ │ │ 
    i18 : elapsedTime g1 = gens gb Ilex;
│ │ │ - -- 1.68769s elapsed
│ │ │ + -- 1.789s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o18 : Matrix Rlex  <-- Rlex
    │ │ │
    │ │ │ @@ -435,15 +435,15 @@ │ │ │ │ │ │ o19 : Ideal of Rlex │ │ │
    │ │ │ 
    i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf);
│ │ │ - -- .850458s elapsed
│ │ │ + -- .80331s elapsed
│ │ │  
│ │ │                   1         372
│ │ │  o20 : Matrix Rlex  <-- Rlex
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -221,15 +221,15 @@ │ │ │ │ 18y*z w + 42y*z*w + 23y*w - 28z + 15z w + 18z w - 16z w - 46z*w │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 5 │ │ │ │ + 12w ) │ │ │ │ │ │ │ │ o17 : Ideal of Rlex │ │ │ │ i18 : elapsedTime g1 = gens gb Ilex; │ │ │ │ - -- 1.68769s elapsed │ │ │ │ + -- 1.789s elapsed │ │ │ │ │ │ │ │ 1 372 │ │ │ │ o18 : Matrix Rlex <-- Rlex │ │ │ │ i19 : Ilex = ideal(Ilex_*) -- clear out the previous Groebner basis │ │ │ │ │ │ │ │ 5 4 4 4 3 2 3 3 │ │ │ │ o19 = ideal (24x - 36x y - 30x z - 29x w + 19x y + 19x y*z - 10x y*w - │ │ │ │ @@ -301,15 +301,15 @@ │ │ │ │ 18y*z w + 42y*z*w + 23y*w - 28z + 15z w + 18z w - 16z w - 46z*w │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ 5 │ │ │ │ + 12w ) │ │ │ │ │ │ │ │ o19 : Ideal of Rlex │ │ │ │ i20 : elapsedTime g2 = gens gb(Ilex, Hilbert => hf); │ │ │ │ - -- .850458s elapsed │ │ │ │ + -- .80331s elapsed │ │ │ │ │ │ │ │ 1 372 │ │ │ │ o20 : Matrix Rlex <-- Rlex │ │ │ │ i21 : g1 == g2 │ │ │ │ │ │ │ │ o21 = true │ │ │ │ ********** CCaavveeaatt ********** │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cancel__Task_lp__Task_rp.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ o4 : Task │ │ │
    │ │ │ 
    i5 : n
│ │ │  
│ │ │ -o5 = 336394
    │ │ │ +o5 = 721956 │ │ │
    │ │ │ 
    i6 : sleep 1
│ │ │  
│ │ │  o6 = 0
    │ │ │ @@ -132,15 +132,15 @@ │ │ │ o7 : Task │ │ │
    │ │ │ 
    i8 : n
│ │ │  
│ │ │ -o8 = 929987
    │ │ │ +o8 = 1699995 │ │ │
    │ │ │ 
    i9 : isReady t
│ │ │  
│ │ │  o9 = false
    │ │ │ @@ -168,29 +168,29 @@ │ │ │ o12 : Task │ │ │
    │ │ │ 
    i13 : n
│ │ │  
│ │ │ -o13 = 930227
    │ │ │ +o13 = 1700180 │ │ │
    │ │ │ 
    i14 : sleep 1
│ │ │  
│ │ │  o14 = 0
    │ │ │
    │ │ │ 
    i15 : n
│ │ │  
│ │ │ -o15 = 930227
    │ │ │ +o15 = 1700180 │ │ │
    │ │ │ 
    i16 : isReady t
│ │ │  
│ │ │  o16 = false
    │ │ │ ├── html2text {} │ │ │ │ @@ -28,26 +28,26 @@ │ │ │ │ i4 : t │ │ │ │ │ │ │ │ o4 = <> │ │ │ │ │ │ │ │ o4 : Task │ │ │ │ i5 : n │ │ │ │ │ │ │ │ -o5 = 336394 │ │ │ │ +o5 = 721956 │ │ │ │ i6 : sleep 1 │ │ │ │ │ │ │ │ o6 = 0 │ │ │ │ i7 : t │ │ │ │ │ │ │ │ o7 = <> │ │ │ │ │ │ │ │ o7 : Task │ │ │ │ i8 : n │ │ │ │ │ │ │ │ -o8 = 929987 │ │ │ │ +o8 = 1699995 │ │ │ │ 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 = 930227 │ │ │ │ +o13 = 1700180 │ │ │ │ i14 : sleep 1 │ │ │ │ │ │ │ │ o14 = 0 │ │ │ │ i15 : n │ │ │ │ │ │ │ │ -o15 = 930227 │ │ │ │ +o15 = 1700180 │ │ │ │ i16 : isReady t │ │ │ │ │ │ │ │ o16 = false │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _c_a_n_c_e_l_T_a_s_k_(_T_a_s_k_) -- stop a task │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_change__Directory.html │ │ │ @@ -76,36 +76,36 @@ │ │ │

    Change the current working directory to dir.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11296-0/0
    │ │ │ +o1 = /tmp/M2-11766-0/0 │ │ │
    │ │ │ 
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11296-0/0
    │ │ │ +o2 = /tmp/M2-11766-0/0 │ │ │
    │ │ │ 
    i3 : changeDirectory dir
│ │ │  
│ │ │ -o3 = /tmp/M2-11296-0/0/
    │ │ │ +o3 = /tmp/M2-11766-0/0/ │ │ │
    │ │ │ 
    i4 : currentDirectory()
│ │ │  
│ │ │ -o4 = /tmp/M2-11296-0/0/
    │ │ │ +o4 = /tmp/M2-11766-0/0/ │ │ │
    │ │ │
    │ │ │

    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/M2-11296-0/0 │ │ │ │ +o1 = /tmp/M2-11766-0/0 │ │ │ │ i2 : makeDirectory dir │ │ │ │ │ │ │ │ -o2 = /tmp/M2-11296-0/0 │ │ │ │ +o2 = /tmp/M2-11766-0/0 │ │ │ │ i3 : changeDirectory dir │ │ │ │ │ │ │ │ -o3 = /tmp/M2-11296-0/0/ │ │ │ │ +o3 = /tmp/M2-11766-0/0/ │ │ │ │ i4 : currentDirectory() │ │ │ │ │ │ │ │ -o4 = /tmp/M2-11296-0/0/ │ │ │ │ +o4 = /tmp/M2-11766-0/0/ │ │ │ │ If dir is omitted, then the current working directory is changed to the user's │ │ │ │ home directory. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_u_r_r_e_n_t_D_i_r_e_c_t_o_r_y -- current working directory │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_a_n_g_e_D_i_r_e_c_t_o_r_y is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_check.html │ │ │ @@ -100,40 +100,40 @@ │ │ │ o1 : Package │ │ │
    │ │ │ 
    i2 : check_1 FirstPackage
│ │ │   -- warning: reloading FirstPackage; recreate instances of types from this package
│ │ │ - -- capturing check(1, "FirstPackage")        -- .313321s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .167999s elapsed │ │ │
    │ │ │ 
    i3 : check FirstPackage
│ │ │ - -- capturing check(0, "FirstPackage")        -- .282764s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283285s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .146472s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .141847s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : check_1 "FirstPackage"
│ │ │ - -- capturing check(1, "FirstPackage")        -- .275704s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .146792s elapsed │ │ │
    │ │ │ 
    i5 : check "FirstPackage"
│ │ │ - -- capturing check(0, "FirstPackage")        -- .246295s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .21342s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .140652s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .141035s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ o6 : TestInput │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : check oo
│ │ │ - -- capturing check(1, "FirstPackage")        -- .302726s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .15447s elapsed │ │ │
    │ │ │ 
    i8 : tests "FirstPackage"
│ │ │  
│ │ │  o8 = {0 => TestInput[/usr/share/Macaulay2/FirstPackage.m2:54:5-56:3]}
│ │ │ @@ -161,16 +161,16 @@
│ │ │  
│ │ │  o8 : NumberedVerticalList
    │ │ │
    │ │ │ 
    i9 : check oo
│ │ │ - -- capturing check(0, "FirstPackage")        -- .28407s elapsed
│ │ │ - -- capturing check(1, "FirstPackage")        -- .282385s elapsed
    │ │ │ + -- capturing check(0, "FirstPackage") -- .142307s elapsed │ │ │ + -- capturing check(1, "FirstPackage") -- .141349s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -183,15 +183,15 @@ │ │ │ │ │ │ o10 : NumberedVerticalList │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : check 1
│ │ │ - -- capturing check(1, "FirstPackage")        -- .283414s elapsed
    │ │ │ + -- capturing check(1, "FirstPackage") -- .143194s 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") -- .313321s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .167999s elapsed │ │ │ │ i3 : check FirstPackage │ │ │ │ - -- capturing check(0, "FirstPackage") -- .282764s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .283285s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .146472s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .141847s elapsed │ │ │ │ Alternatively, if the package is installed somewhere accessible, one can do the │ │ │ │ following. │ │ │ │ i4 : check_1 "FirstPackage" │ │ │ │ - -- capturing check(1, "FirstPackage") -- .275704s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .146792s elapsed │ │ │ │ i5 : check "FirstPackage" │ │ │ │ - -- capturing check(0, "FirstPackage") -- .246295s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .21342s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .140652s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .141035s 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") -- .302726s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .15447s 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") -- .28407s elapsed │ │ │ │ - -- capturing check(1, "FirstPackage") -- .282385s elapsed │ │ │ │ + -- capturing check(0, "FirstPackage") -- .142307s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .141349s 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") -- .283414s elapsed │ │ │ │ + -- capturing check(1, "FirstPackage") -- .143194s elapsed │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ Currently, if the package was only partially loaded because the documentation │ │ │ │ was obtainable from a database (see _b_e_g_i_n_D_o_c_u_m_e_n_t_a_t_i_o_n), then the package will │ │ │ │ be reloaded, this time completely, to ensure that all tests are considered; │ │ │ │ this may affect user objects of types declared by the package, as they may be │ │ │ │ not usable by the new instance of the package. In a future version, either the │ │ │ │ tests and the documentation will both be cached, or neither will. │ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_communicating_spwith_spprograms.html │ │ │ @@ -55,15 +55,15 @@ │ │ │
    │ │ │

    communicating with programs

    │ │ │
    │ │ │ The most naive way to interact with another program is simply to run it, let it communicate directly with the user, and wait for it to finish. This is done with the run command. │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : run "uname -a"
│ │ │ -Linux sbuild 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1 (2026-10-15) x86_64 GNU/Linux
│ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-1 (2026-07-30) x86_64 GNU/Linux
│ │ │  
│ │ │  o1 = 0
    │ │ │
    │ │ │ To run a program and provide it with input, one way is use the operator <<, with a file name whose first character is an exclamation point; the rest of the file name will be taken as the command to run, as in the following example. │ │ │ │ │ │ @@ -79,16 +79,16 @@ │ │ │ │ │ │
    │ │ │ 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 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
    │ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian │ │ │ + 6.12.100-1 (2026-07-30) 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 {} │ │ │ │ @@ -31,17 +31,17 @@ │ │ │ │ o1 : PolynomialRing │ │ │ │ i2 : I = ideal(x, y, z) │ │ │ │ │ │ │ │ o2 = ideal (x, y, z) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : msolveGB(I, Verbosity => 2, Threads => 6) │ │ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/ │ │ │ │ -M2-38527-0/0-out.ms │ │ │ │ -Initial seed for pseudo-random number generator is 1792814436 │ │ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54074-0/0-in.ms -o /tmp/ │ │ │ │ +M2-54074-0/0-out.ms │ │ │ │ +Initial seed for pseudo-random number generator is 1792836101 │ │ │ │ │ │ │ │ --------------- INPUT DATA --------------- │ │ │ │ #variables 3 │ │ │ │ #equations 3 │ │ │ │ #invalid equations 0 │ │ │ │ field characteristic 0 │ │ │ │ homogeneous input? 1 │ │ │ │ @@ -52,15 +52,15 @@ │ │ │ │ initial hash table size 131072 (2^17) │ │ │ │ max pair selection ALL │ │ │ │ reduce gb 1 │ │ │ │ #threads 6 │ │ │ │ info level 2 │ │ │ │ generate pbm files 0 │ │ │ │ ------------------------------------------ │ │ │ │ -Initial prime = 1099986659 │ │ │ │ +Initial prime = 1118389211 │ │ │ │ │ │ │ │ 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) │ │ │ │ @@ -82,18 +82,18 @@ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----------------------- │ │ │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ │ overall(elapsed) 0.00 sec │ │ │ │ overall(cpu) 0.00 sec │ │ │ │ select 0.00 sec 0.0% │ │ │ │ -symbolic prep. 0.00 sec 0.0% │ │ │ │ -update 0.00 sec 96.6% │ │ │ │ -convert 0.00 sec 0.6% │ │ │ │ -linear algebra 0.00 sec 0.2% │ │ │ │ +symbolic prep. 0.00 sec 0.5% │ │ │ │ +update 0.00 sec 69.7% │ │ │ │ +convert 0.00 sec 3.0% │ │ │ │ +linear algebra 0.00 sec 1.3% │ │ │ │ reduce gb 0.00 sec 0.0% │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ │ size of basis 3 │ │ │ │ #terms in basis 3 │ │ │ │ #pairs reduced 0 │ │ │ │ @@ -107,18 +107,18 @@ │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ │ │ │ │ ---------- COMPUTATIONAL DATA ----------- │ │ │ │ [3] │ │ │ │ #polynomials to lift 3 │ │ │ │ ----------------------------------------- │ │ │ │ -New prime = 1120193111 │ │ │ │ +New prime = 1116865681 │ │ │ │ │ │ │ │ ---------------- TIMINGS ---------------- │ │ │ │ -multi-mod overall(elapsed) 0.02 sec │ │ │ │ +multi-mod overall(elapsed) 0.01 sec │ │ │ │ learning phase 0.00 Gops/sec │ │ │ │ application phase 0.00 Gops/sec │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ multi-modular steps │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----------------------- │ │ │ │ @@ -137,15 +137,15 @@ │ │ │ │ CRT (elapsed) 0.00 sec │ │ │ │ ratrecon(elapsed) 0.00 sec │ │ │ │ ----------------------------------------- │ │ │ │ │ │ │ │ │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----- │ │ │ │ -msolve overall time 0.08 sec (elapsed) / 0.27 sec (cpu) │ │ │ │ +msolve overall time 0.01 sec (elapsed) / 0.05 sec (cpu) │ │ │ │ ------------------------------------------------------------------------------- │ │ │ │ ----- │ │ │ │ │ │ │ │ o3 = | z y x | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix R <-- R │ │ ├── ./usr/share/doc/Macaulay2/MultiGradedRationalMap/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ ZGVncmVlT2ZNYXAoSWRlYWwp │ │ │ #:len=283 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjk3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZWdyZWVPZk1hcCxJZGVhbCksImRlZ3JlZU9mTWFw │ │ ├── ./usr/share/doc/Macaulay2/MultigradedBGG/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ ZGVncmVlKERpZmZlcmVudGlhbE1vZHVsZSk= │ │ │ #:len=1348 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgZGVncmVlIG9mIHRo │ │ │ ZSBkaWZmZXJlbnRpYWwiLCAibGluZW51bSIgPT4gNDgzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkQi │ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ TXVsdGlncmFkZWRJbXBsaWNpdGl6YXRpb24= │ │ │ #:len=1401 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUGFja2FnZSBmb3IgbGV2YXJhZ2luZyBt │ │ │ dWx0aWdyYWRpbmdzIHRvIHNvbHZlIGltcGxpY2l0aXphdGlvbiBwcm9ibGVtcyIsIERlc2NyaXB0 │ │ ├── ./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 │ │ │ - -- .00225028s elapsed │ │ │ + -- .00218476s elapsed │ │ │ computing total degree: 2 │ │ │ number of monomials = 21 │ │ │ number of distinct multidegrees = 18 │ │ │ - -- .00849938s elapsed │ │ │ + -- .00877331s elapsed │ │ │ │ │ │ o5 = MutableHashTable{{0, 1, 0, 0, 1} => {} } │ │ │ {0, 1, 0, 1, 0} => {} │ │ │ {0, 1, 1, 0, 0} => {} │ │ │ {0, 2, 0, 0, 2} => {} │ │ │ {0, 2, 0, 1, 1} => {} │ │ │ {0, 2, 0, 2, 0} => {} │ │ ├── ./usr/share/doc/Macaulay2/MultigradedImplicitization/html/_components__Of__Kernel.html │ │ │ @@ -122,19 +122,19 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -20,23 +20,23 @@ │ │ │ │ o1 : PolynomialRing │ │ │ │ i2 : I = ideal"xy,yz,zx" │ │ │ │ │ │ │ │ o2 = ideal (x*y, y*z, x*z) │ │ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ i3 : elapsedTime jMult I │ │ │ │ - -- .0242277s elapsed │ │ │ │ + -- .0225676s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ i4 : elapsedTime monjMult I │ │ │ │ - -- .159695s elapsed │ │ │ │ + -- .0664976s elapsed │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : elapsedTime multiplicitySequence I │ │ │ │ - -- .270321s elapsed │ │ │ │ + -- .120257s elapsed │ │ │ │ │ │ │ │ o5 = HashTable{2 => 3} │ │ │ │ 3 => 2 │ │ │ │ │ │ │ │ o5 : HashTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_p_l_i_c_i_t_y_S_e_q_u_e_n_c_e -- the multiplicity sequence of an ideal │ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_mon__Analytic__Spread.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : f = openInOut "!grep -E '^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 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.113-1
│ │ │ │ -(2026-10-15) x86_64 GNU/Linux
│ │ │ │ +Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian 6.12.100-
│ │ │ │ +1 (2026-07-30) 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
│ │ │ │ @@ -26,16 +26,16 @@
│ │ │ │  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 6.12.113+deb13-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ -     6.12.113-1 (2026-10-15) x86_64 GNU/Linux\n"
│ │ │ │ +o3 = "Linux sbuild 6.12.100+deb13-cloud-amd64 #1 SMP PREEMPT_DYNAMIC Debian
│ │ │ │ +     6.12.100-1 (2026-07-30) x86_64 GNU/Linux\n"
│ │ │ │  Bidirectional communication with a program is also possible. We use _o_p_e_n_I_n_O_u_t
│ │ │ │  to create a file that serves as a bidirectional connection to a program. That
│ │ │ │  file is called an input output file. In this example we open a connection to
│ │ │ │  the unix utility grep and use it to locate the symbol names in Macaulay2 that
│ │ │ │  begin with in.
│ │ │ │  i4 : f = openInOut "!grep -E '^in'"
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_computing_sp__Groebner_spbases.html
│ │ │ @@ -274,15 +274,15 @@
│ │ │                 1277
    │ │ │
    │ │ │ 
    i24 : gb I
│ │ │  
│ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ +   -- registering gb 5 at 0x7f6c00181000
│ │ │  
│ │ │     -- [gb]{2}(2)mm{3}(1)m{4}(2)om{5}(1)onumber of (nonminimal) gb elements = 4
│ │ │     -- number of monomials                = 8
│ │ │     -- #reduction steps = 2
│ │ │     -- #spairs done = 6
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -378,15 +378,15 @@
│ │ │                1      4
│ │ │  o32 : Matrix R  <-- R
    │ │ │
    │ │ │ 
    i33 : time betti gb f
│ │ │ - -- used 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ + -- used 0.17595s (cpu); 0.173459s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o33 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ @@ -422,15 +422,15 @@
│ │ │  
│ │ │  o35 : ZZ[T]
    │ │ │
    │ │ │ 
    i36 : time betti gb f
│ │ │ - -- used 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ + -- used 0.00788594s (cpu); 0.00650792s (thread); 0s (gc)
│ │ │  
│ │ │               0  1
│ │ │  o36 = total: 1 53
│ │ │            0: 1  .
│ │ │            1: .  .
│ │ │            2: .  2
│ │ │            3: .  1
│ │ │ ├── html2text {}
│ │ │ │ @@ -142,15 +142,15 @@
│ │ │ │  o23 = ideal (x*y - z , y  - w )
│ │ │ │  
│ │ │ │                  ZZ
│ │ │ │  o23 : Ideal of ----[x..z, w]
│ │ │ │                 1277
│ │ │ │  i24 : gb I
│ │ │ │  
│ │ │ │ -   -- registering gb 5 at 0x7f57805dd000
│ │ │ │ +   -- registering gb 5 at 0x7f6c00181000
│ │ │ │  
│ │ │ │     -- [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
│ │ │ │ @@ -215,15 +215,15 @@
│ │ │ │  
│ │ │ │  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 0.266722s (cpu); 0.187826s (thread); 0s (gc)
│ │ │ │ + -- used 0.17595s (cpu); 0.173459s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o33 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ │ │ @@ -247,15 +247,15 @@
│ │ │ │  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 0.00399931s (cpu); 0.00644121s (thread); 0s (gc)
│ │ │ │ + -- used 0.00788594s (cpu); 0.00650792s (thread); 0s (gc)
│ │ │ │  
│ │ │ │               0  1
│ │ │ │  o36 = total: 1 53
│ │ │ │            0: 1  .
│ │ │ │            1: .  .
│ │ │ │            2: .  2
│ │ │ │            3: .  1
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,112 +85,112 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,41 +22,41 @@
│ │ │ │            o npaths, an _i_n_t_e_g_e_r,
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Output is verbose. For other dynamic strategies, see _M_o_n_o_d_r_o_m_y_S_o_l_v_e_r_O_p_t_i_o_n_s.
│ │ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ │ - -- .00616338s elapsed
│ │ │ │ +--backup directory created: /tmp/M2-33141-0/2
│ │ │ │ + -- .00322685s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00900117s elapsed
│ │ │ │ + -- .00334371s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000649635s elapsed
│ │ │ │ + -- .000568627s elapsed
│ │ │ │  number of paths tracked: 2
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00324889s elapsed
│ │ │ │ + -- .00311715s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00435076s elapsed
│ │ │ │ + -- .00353501s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000544598s elapsed
│ │ │ │ + -- .000564146s elapsed
│ │ │ │  number of paths tracked: 4
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .00428387s elapsed
│ │ │ │ + -- .0029798s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00298132s elapsed
│ │ │ │ + -- .00329803s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000466532s elapsed
│ │ │ │ + -- .000517887s elapsed
│ │ │ │  number of paths tracked: 6
│ │ │ │  found 1 points in the fiber so far
│ │ │ │ - -- .0044506s elapsed
│ │ │ │ + -- .00311868s elapsed
│ │ │ │    H01: 1
│ │ │ │ - -- .00309306s elapsed
│ │ │ │ + -- .00297752s elapsed
│ │ │ │    H10: 1
│ │ │ │ - -- .000457295s elapsed
│ │ │ │ + -- .000481962s elapsed
│ │ │ │  number of paths tracked: 8
│ │ │ │  found 1 points in the fiber so far
│ │ │ │  
│ │ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_get__Track__Time.html
│ │ │ @@ -96,22 +96,22 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12075-0/0/
    
│ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12075-0/1/
    
│ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12075-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12075-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : stack findFiles src
│ │ │  
│ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ -     /tmp/M2-12075-0/0/a/f
    
│ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │              
    
│ │ │                
    i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
    
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
    
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │              
    
│ │ │                
    i12 : stack findFiles dst
│ │ │  
│ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ -      /tmp/M2-12075-0/1/b/c/g
    
│ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ +      /tmp/M2-13345-0/1/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/M2-12075-0/0/
│ │ │ │ +o1 = /tmp/M2-13345-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12075-0/1/
│ │ │ │ +o2 = /tmp/M2-13345-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12075-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13345-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12075-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13345-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12075-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13345-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12075-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13345-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12075-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13345-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-12075-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13345-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : stack findFiles src
│ │ │ │  
│ │ │ │ -o9 = /tmp/M2-12075-0/0/
│ │ │ │ -     /tmp/M2-12075-0/0/b/
│ │ │ │ -     /tmp/M2-12075-0/0/b/c/
│ │ │ │ -     /tmp/M2-12075-0/0/b/c/g
│ │ │ │ -     /tmp/M2-12075-0/0/a/
│ │ │ │ -     /tmp/M2-12075-0/0/a/g
│ │ │ │ -     /tmp/M2-12075-0/0/a/f
│ │ │ │ +o9 = /tmp/M2-13345-0/0/
│ │ │ │ +     /tmp/M2-13345-0/0/a/
│ │ │ │ +     /tmp/M2-13345-0/0/a/g
│ │ │ │ +     /tmp/M2-13345-0/0/a/f
│ │ │ │ +     /tmp/M2-13345-0/0/b/
│ │ │ │ +     /tmp/M2-13345-0/0/b/c/
│ │ │ │ +     /tmp/M2-13345-0/0/b/c/g
│ │ │ │  i10 : copyDirectory(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/b/c/g -> /tmp/M2-12075-0/1/b/c/g
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/a/g -> /tmp/M2-12075-0/1/a/g
│ │ │ │ - -- copying: /tmp/M2-12075-0/0/a/f -> /tmp/M2-12075-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/a/g -> /tmp/M2-13345-0/1/a/g
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/a/f -> /tmp/M2-13345-0/1/a/f
│ │ │ │ + -- copying: /tmp/M2-13345-0/0/b/c/g -> /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i11 : copyDirectory(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/b/c/g not newer than /tmp/M2-12075-0/1/b/c/g
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/a/g not newer than /tmp/M2-12075-0/1/a/g
│ │ │ │ - -- skipping: /tmp/M2-12075-0/0/a/f not newer than /tmp/M2-12075-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/a/g not newer than /tmp/M2-13345-0/1/a/g
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/a/f not newer than /tmp/M2-13345-0/1/a/f
│ │ │ │ + -- skipping: /tmp/M2-13345-0/0/b/c/g not newer than /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i12 : stack findFiles dst
│ │ │ │  
│ │ │ │ -o12 = /tmp/M2-12075-0/1/
│ │ │ │ -      /tmp/M2-12075-0/1/a/
│ │ │ │ -      /tmp/M2-12075-0/1/a/f
│ │ │ │ -      /tmp/M2-12075-0/1/a/g
│ │ │ │ -      /tmp/M2-12075-0/1/b/
│ │ │ │ -      /tmp/M2-12075-0/1/b/c/
│ │ │ │ -      /tmp/M2-12075-0/1/b/c/g
│ │ │ │ +o12 = /tmp/M2-13345-0/1/
│ │ │ │ +      /tmp/M2-13345-0/1/a/
│ │ │ │ +      /tmp/M2-13345-0/1/a/g
│ │ │ │ +      /tmp/M2-13345-0/1/a/f
│ │ │ │ +      /tmp/M2-13345-0/1/b/
│ │ │ │ +      /tmp/M2-13345-0/1/b/c/
│ │ │ │ +      /tmp/M2-13345-0/1/b/c/g
│ │ │ │  i13 : get (dst|"b/c/g")
│ │ │ │  
│ │ │ │  o13 = ho there
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i14 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o14 = rm
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_copy__File_lp__String_cm__String_rp.html
│ │ │ @@ -83,65 +83,65 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11841-0/0
    
│ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11841-0/1
    
│ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : copyFile(src,dst,Verbose=>true)
│ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
    
│ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
    
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    i7 : src << "ho there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
    
│ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │              
    
│ │ │                
    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/M2-11841-0/0
│ │ │ │ +o1 = /tmp/M2-12871-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11841-0/1
│ │ │ │ +o2 = /tmp/M2-12871-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11841-0/0
│ │ │ │ +o3 = /tmp/M2-12871-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : copyFile(src,dst,Verbose=>true)
│ │ │ │ - -- copying: /tmp/M2-11841-0/0 -> /tmp/M2-11841-0/1
│ │ │ │ + -- copying: /tmp/M2-12871-0/0 -> /tmp/M2-12871-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │ │  i7 : src << "ho there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-11841-0/0
│ │ │ │ +o7 = /tmp/M2-12871-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : copyFile(src,dst,Verbose=>true,UpdateOnly => true)
│ │ │ │ - -- skipping: /tmp/M2-11841-0/0 not newer than /tmp/M2-11841-0/1
│ │ │ │ + -- skipping: /tmp/M2-12871-0/0 not newer than /tmp/M2-12871-0/1
│ │ │ │  i9 : get dst
│ │ │ │  
│ │ │ │  o9 = hi there
│ │ │ │  i10 : removeFile src
│ │ │ │  i11 : removeFile dst
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_D_i_r_e_c_t_o_r_y
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_cpu__Time.html
│ │ │ @@ -69,38 +69,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : t1 = cpuTime()
│ │ │  
│ │ │ -o1 = 611.770723009
│ │ │ +o1 = 495.5602200340001
│ │ │  
│ │ │  o1 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i2 : for i from 0 to 1000000 do 223131321321*324234324324;
    
│ │ │              
    
│ │ │                
    i3 : t2 = cpuTime()
│ │ │  
│ │ │ -o3 = 613.313432871
│ │ │ +o3 = 496.629935466
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i4 : t2-t1
│ │ │  
│ │ │ -o4 = 1.542709862000038
│ │ │ +o4 = 1.069715431999953
│ │ │  
│ │ │  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 = 611.770723009
│ │ │ │ +o1 = 495.5602200340001
│ │ │ │  
│ │ │ │  o1 : RR (of precision 53)
│ │ │ │  i2 : for i from 0 to 1000000 do 223131321321*324234324324;
│ │ │ │  i3 : t2 = cpuTime()
│ │ │ │  
│ │ │ │ -o3 = 613.313432871
│ │ │ │ +o3 = 496.629935466
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  i4 : t2-t1
│ │ │ │  
│ │ │ │ -o4 = 1.542709862000038
│ │ │ │ +o4 = 1.069715431999953
│ │ │ │  
│ │ │ │  o4 : RR (of precision 53)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_e -- time a computation
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_current__Time.html
│ │ │ @@ -69,48 +69,48 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime()
│ │ │  
│ │ │ -o1 = 1792811657
    
│ │ │ +o1 = 1792834376
│ │ │              
    
│ │ │          
    We can compute, roughly, how many years ago the epoch began as follows.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │  
│ │ │ -o2 = 56.81193979063632
│ │ │ +o2 = 56.81265972708645
│ │ │  
│ │ │  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 = 9.743277487635794
│ │ │ +o3 = 9.751916725037375
│ │ │  
│ │ │  o3 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    Compare that to the current date, available from a standard Unix command.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i4 : run "date"
│ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ +Sat Oct 24 09:32:56 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 = 1792811657
│ │ │ │ +o1 = 1792834376
│ │ │ │  We can compute, roughly, how many years ago the epoch began as follows.
│ │ │ │  i2 : currentTime() /( (365 + 97./400) * 24 * 60 * 60 )
│ │ │ │  
│ │ │ │ -o2 = 56.81193979063632
│ │ │ │ +o2 = 56.81265972708645
│ │ │ │  
│ │ │ │  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 = 9.743277487635794
│ │ │ │ +o3 = 9.751916725037375
│ │ │ │  
│ │ │ │  o3 : RR (of precision 53)
│ │ │ │  Compare that to the current date, available from a standard Unix command.
│ │ │ │  i4 : run "date"
│ │ │ │ -Sat Oct 24 03:14:17 UTC 2026
│ │ │ │ +Sat Oct 24 09:32:56 UTC 2026
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _c_u_r_r_e_n_t_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:1849:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_elapsed__Time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  elapsedTime e evaluates e, prints the amount of time elapsed, and returns the value of e.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime sleep 1
│ │ │ - -- 1.00015s elapsed
│ │ │ + -- 1.00019s 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
│ │ │ │ + -- 1.00019s 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
│ │ │ @@ -71,15 +71,15 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │              
    
│ │ │                
    i3 : time leadTerm gens gb I
│ │ │ - -- used 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ + -- used 0.347347s (cpu); 0.148903s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │       ------------------------------------------------------------------------
│ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │       ------------------------------------------------------------------------
│ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -168,15 +168,15 @@
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
    
│ │ │                
    i8 : time G = eliminate(I,{s,t})
│ │ │ - -- used 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ + -- used 0.0843781s (cpu); 0.0843817s (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
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -251,15 +251,15 @@
│ │ │  
│ │ │  o11 : Ideal of R1
    
│ │ │              
    
│ │ │                
    i12 : time G = eliminate(I1,{s,t})
│ │ │ - -- used 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ + -- used 0.0259684s (cpu); 0.0259673s (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  -
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -343,15 +343,15 @@
│ │ │  
│ │ │  o16 : RingMap A <-- B
    
│ │ │              
    
│ │ │                
    i17 : time G = kernel F
│ │ │ - -- used 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ + -- used 0.32765s (cpu); 0.126762s (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
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -424,26 +424,26 @@
│ │ │  
│ │ │  o19 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ + -- used 0.00152028s (cpu); 0.0015189s (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 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ + -- used 0.0285218s (cpu); 0.0283111s (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 {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  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 0.453514s (cpu); 0.254782s (thread); 0s (gc)
│ │ │ │ + -- used 0.347347s (cpu); 0.148903s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = | x3y9 5148txy3 108729sxy2z2 sy4z 46644741sxy3z 143sy5 6sxy4
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       563515116021sx2y3 4374txy2z3 612704350498473090tx2yz3 217458ty4z2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       267076255345488270sy3z4 5256861933965245618410txyz6
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -94,15 +94,15 @@
│ │ │ │  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 0.436423s (cpu); 0.247727s (thread); 0s (gc)
│ │ │ │ + -- used 0.0843781s (cpu); 0.0843817s (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
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ @@ -161,15 +161,15 @@
│ │ │ │  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 0.0611301s (cpu); 0.0611395s (thread); 0s (gc)
│ │ │ │ + -- used 0.0259684s (cpu); 0.0259673s (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  -
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -232,15 +232,15 @@
│ │ │ │  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 0.450122s (cpu); 0.254702s (thread); 0s (gc)
│ │ │ │ + -- used 0.32765s (cpu); 0.126762s (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
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │ @@ -301,22 +301,22 @@
│ │ │ │  involve the variables s and t.
│ │ │ │  i19 : use ring I
│ │ │ │  
│ │ │ │  o19 = R
│ │ │ │  
│ │ │ │  o19 : PolynomialRing
│ │ │ │  i20 : time f1 = resultant(I_0,I_2,s)
│ │ │ │ - -- used 0.00269518s (cpu); 0.00269678s (thread); 0s (gc)
│ │ │ │ + -- used 0.00152028s (cpu); 0.0015189s (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 0.0712746s (cpu); 0.0712854s (thread); 0s (gc)
│ │ │ │ + -- used 0.0285218s (cpu); 0.0283111s (thread); 0s (gc)
│ │ │ │  
│ │ │ │           3 9     2 9     2 8      2 6 3       9    2 7         8        7 2
│ │ │ │  o21 = - x y  + 3x y  + 6x y z + 3x y z  - 3x*y  + x y z - 12x*y z - 7x*y z  +
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            2 5 3       6 3    7 3        5 4       3 6    9       7      8
│ │ │ │        324x y z  - 6x*y z  + y z  + 15x*y z  - 3x*y z  + y  - 2x*y z + 6y z +
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_end__Package.html
│ │ │ @@ -159,15 +159,15 @@
│ │ │                                      Version => 0.0
│ │ │               package prefix => /usr/
│ │ │               PackageIsLoaded => true
│ │ │               pkgname => Foo
│ │ │               private dictionary => Foo#"private dictionary"
│ │ │               processed documentation => MutableHashTable{}
│ │ │               raw documentation => MutableHashTable{}
│ │ │ -             source directory => /tmp/M2-11022-0/96-rundir/
│ │ │ +             source directory => /tmp/M2-11242-0/96-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/M2-11022-0/96-rundir/
│ │ │ │ +             source directory => /tmp/M2-11242-0/96-rundir/
│ │ │ │               source file => stdio
│ │ │ │               test inputs => MutableList{}
│ │ │ │  i7 : dictionaryPath
│ │ │ │  
│ │ │ │  o7 = {Foo.Dictionary, Varieties.Dictionary, Isomorphism.Dictionary,
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       Truncations.Dictionary, Polyhedra.Dictionary, Saturation.Dictionary,
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Exists.html
│ │ │ @@ -73,29 +73,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -515,27 +515,27 @@
│ │ │  o36 = 3
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11391-0/0
    
│ │ │ +o1 = /tmp/M2-11961-0/0
│ │ │              
    
│ │ │                
    i2 : fileExists fn
│ │ │  
│ │ │  o2 = false
    
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ +o3 = /tmp/M2-11961-0/0
│ │ │  
│ │ │  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/M2-11391-0/0
│ │ │ │ +o1 = /tmp/M2-11961-0/0
│ │ │ │  i2 : fileExists fn
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11391-0/0
│ │ │ │ +o3 = /tmp/M2-11961-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : fileExists fn
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  i5 : removeFile fn
│ │ │ │  If fn refers to a symbolic link, then whether the file exists is determined by
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Length.html
│ │ │ @@ -74,15 +74,15 @@
│ │ │          
    Description
    
│ │ │          
    The length of an open output file is determined from the internal count of the number of bytes written so far.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : f = temporaryFileName() << "hi there"
│ │ │  
│ │ │ -o1 = /tmp/M2-13078-0/0
│ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o1 : File
    
│ │ │              
    
│ │ │                
    i2 : fileLength f
│ │ │ @@ -90,24 +90,24 @@
│ │ │  o2 = 8
    
│ │ │              
    
│ │ │                
    i3 : close f
│ │ │  
│ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : filename = toString f
│ │ │  
│ │ │ -o4 = /tmp/M2-13078-0/0
    
│ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │              
    
│ │ │                
    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/M2-13078-0/0
│ │ │ │ +o1 = /tmp/M2-15398-0/0
│ │ │ │  
│ │ │ │  o1 : File
│ │ │ │  i2 : fileLength f
│ │ │ │  
│ │ │ │  o2 = 8
│ │ │ │  i3 : close f
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-13078-0/0
│ │ │ │ +o3 = /tmp/M2-15398-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : filename = toString f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-13078-0/0
│ │ │ │ +o4 = /tmp/M2-15398-0/0
│ │ │ │  i5 : fileLength filename
│ │ │ │  
│ │ │ │  o5 = 8
│ │ │ │  i6 : get filename
│ │ │ │  
│ │ │ │  o6 = hi there
│ │ │ │  i7 : length oo
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__File_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │              
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -430,15 +430,15 @@
│ │ │  
│ │ │  o33 = 1
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12284-0/0
    
│ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : fileMode f
│ │ │ @@ -97,15 +97,15 @@
│ │ │  o3 = 420
    
│ │ │              
    
│ │ │                
    i4 : close f
│ │ │  
│ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │  
│ │ │  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/M2-12284-0/0
│ │ │ │ +o1 = /tmp/M2-13774-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12284-0/0
│ │ │ │ +o2 = /tmp/M2-13774-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode f
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : close f
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12284-0/0
│ │ │ │ +o4 = /tmp/M2-13774-0/0
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _f_i_l_e_M_o_d_e_(_F_i_l_e_) -- get file mode
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__String_rp.html
│ │ │ @@ -74,22 +74,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11860-0/0
    
│ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │  
│ │ │  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/M2-11860-0/0
│ │ │ │ +o1 = /tmp/M2-12910-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11860-0/0
│ │ │ │ +o2 = /tmp/M2-12910-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__File_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -250,23 +250,23 @@
│ │ │  o22 = 10000000000000000000000000001710000000000000000000000000097470000000000
│ │ │        00000000000000185613
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11725-0/0
    
│ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │              
    
│ │ │                
    i2 : f = fn << "hi there"
│ │ │  
│ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │                
    i3 : m = 7 + 7*8 + 7*64
│ │ │ @@ -113,15 +113,15 @@
│ │ │  o5 = 511
    
│ │ │              
    
│ │ │                
    i6 : close f
│ │ │  
│ │ │ -o6 = /tmp/M2-11725-0/0
│ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │  
│ │ │  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/M2-11725-0/0
│ │ │ │ +o1 = /tmp/M2-12635-0/0
│ │ │ │  i2 : f = fn << "hi there"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11725-0/0
│ │ │ │ +o2 = /tmp/M2-12635-0/0
│ │ │ │  
│ │ │ │  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/M2-11725-0/0
│ │ │ │ +o6 = /tmp/M2-12635-0/0
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : fileMode fn
│ │ │ │  
│ │ │ │  o7 = 511
│ │ │ │  i8 : removeFile fn
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Mode_lp__Z__Z_cm__String_rp.html
│ │ │ @@ -78,22 +78,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12905-0/0
    
│ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │  
│ │ │  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/M2-12905-0/0
│ │ │ │ +o1 = /tmp/M2-15045-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12905-0/0
│ │ │ │ +o2 = /tmp/M2-15045-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : m = fileMode fn
│ │ │ │  
│ │ │ │  o3 = 420
│ │ │ │  i4 : fileMode(m|7,fn)
│ │ │ │  i5 : fileMode fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_file__Time.html
│ │ │ @@ -81,15 +81,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The value is the number of seconds since 00:00:00 1970-01-01 UTC, the beginning of the epoch, so the number of seconds ago a file or directory was modified may be found by using the following code.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : currentTime() - fileTime "."
│ │ │  
│ │ │ -o1 = 111
    
│ │ │ +o1 = 79
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 111
│ │ │ │ +o1 = 79
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_u_r_r_e_n_t_T_i_m_e -- get the current time
│ │ │ │      * _f_i_l_e_ _m_a_n_i_p_u_l_a_t_i_o_n -- Unix file manipulation functions
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _f_i_l_e_T_i_m_e is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_force__G__B_lp..._cm__Syzygy__Matrix_eq_gt..._rp.html
│ │ │ @@ -125,15 +125,15 @@
│ │ │  o6 : Matrix R  <-- R
    
│ │ │              
    
│ │ │                
    i7 : syz f
│ │ │  
│ │ │ -   -- registering gb 0 at 0x7f466ed708c0
│ │ │ +   -- registering gb 0 at 0x7f690de108c0
│ │ │  
│ │ │     -- [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 0x7f466ed708c0
│ │ │ │ +   -- registering gb 0 at 0x7f690de108c0
│ │ │ │  
│ │ │ │     -- [gb]{2}(1)m{3}(1)m{4}(1)m{5}(1)z{6}(1)z{7}(1)znumber of (nonminimal) gb
│ │ │ │  elements = 3
│ │ │ │     -- number of monomials                = 9
│ │ │ │     -- #reduction steps = 6
│ │ │ │     -- #spairs done = 6
│ │ │ │     -- ncalls = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_get.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │                
    i3 : removeFile "test-file"
    
│ │ │              
    
│ │ │                
    i4 : get "!date"
│ │ │  
│ │ │ -o4 = Sat Oct 24 03:12:48 UTC 2026
    
│ │ │ +o4 = Sat Oct 24 09:31:57 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 = Sat Oct 24 03:12:48 UTC 2026
│ │ │ │ +o4 = Sat Oct 24 09:31:57 UTC 2026
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d -- read from a file
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ │ │      * _c_l_o_s_e -- close a file
│ │ │ │      * _F_i_l_e_ _<_<_ _T_h_i_n_g -- print to a file
│ │ │ │  ********** WWaayyss ttoo uussee ggeett:: **********
│ │ │ │      * get(File)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_instances.html
│ │ │ @@ -89,15 +89,15 @@
│ │ │                 defaultPrecision => 53
│ │ │                 engineDebugLevel => 0
│ │ │                 errorDepth => 0
│ │ │                 gbTrace => 0
│ │ │                 interpreterDepth => 1
│ │ │                 lineNumber => 2
│ │ │                 loadDepth => 3
│ │ │ -               maxAllowableThreads => 7
│ │ │ +               maxAllowableThreads => 17
│ │ │                 maxExponent => 1073741823
│ │ │                 minExponent => -1073741824
│ │ │                 numTBBThreads => 0
│ │ │                 o1 => 2432902008176640000
│ │ │                 oo => 2432902008176640000
│ │ │                 printingAccuracy => -1
│ │ │                 printingLeadLimit => 5
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │                 defaultPrecision => 53
│ │ │ │                 engineDebugLevel => 0
│ │ │ │                 errorDepth => 0
│ │ │ │                 gbTrace => 0
│ │ │ │                 interpreterDepth => 1
│ │ │ │                 lineNumber => 2
│ │ │ │                 loadDepth => 3
│ │ │ │ -               maxAllowableThreads => 7
│ │ │ │ +               maxAllowableThreads => 17
│ │ │ │                 maxExponent => 1073741823
│ │ │ │                 minExponent => -1073741824
│ │ │ │                 numTBBThreads => 0
│ │ │ │                 o1 => 2432902008176640000
│ │ │ │                 oo => 2432902008176640000
│ │ │ │                 printingAccuracy => -1
│ │ │ │                 printingLeadLimit => 5
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Directory.html
│ │ │ @@ -80,22 +80,22 @@
│ │ │  o1 = true
    
│ │ │              
    
│ │ │                
    i2 : fn = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11213-0/0
    
│ │ │ +o2 = /tmp/M2-11603-0/0
│ │ │              
    
│ │ │                
    i3 : fn << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ +o3 = /tmp/M2-11603-0/0
│ │ │  
│ │ │  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/M2-11213-0/0
│ │ │ │ +o2 = /tmp/M2-11603-0/0
│ │ │ │  i3 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11213-0/0
│ │ │ │ +o3 = /tmp/M2-11603-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : isDirectory fn
│ │ │ │  
│ │ │ │  o4 = false
│ │ │ │  i5 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Pseudoprime_lp__Z__Z_rp.html
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o18 = false
    
│ │ │              
    
│ │ │                
    i19 : elapsedTime facs = factor(m*m1)
│ │ │ - -- 3.67323s elapsed
│ │ │ + -- 4.34611s elapsed
│ │ │  
│ │ │  o19 = 1000000000000000000000000000057*1000000000000000000010000000083
│ │ │  
│ │ │  o19 : Expression of class Product
    
│ │ │              
    
│ │ │                
    i23 : elapsedTime isPrime m3
│ │ │ - -- .05718s elapsed
│ │ │ + -- .0502503s elapsed
│ │ │  
│ │ │  o23 = true
    
│ │ │              
    
│ │ │                
    i24 : elapsedTime isPseudoprime m3
│ │ │ - -- .000110026s elapsed
│ │ │ + -- .000102701s elapsed
│ │ │  
│ │ │  o24 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │  i17 : isPrime (m*m1)
│ │ │ │  
│ │ │ │  o17 = false
│ │ │ │  i18 : isPrime(m*m*m1*m1*m2^6)
│ │ │ │  
│ │ │ │  o18 = false
│ │ │ │  i19 : elapsedTime facs = factor(m*m1)
│ │ │ │ - -- 3.67323s elapsed
│ │ │ │ + -- 4.34611s 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
│ │ │ │ - -- .05718s elapsed
│ │ │ │ + -- .0502503s elapsed
│ │ │ │  
│ │ │ │  o23 = true
│ │ │ │  i24 : elapsedTime isPseudoprime m3
│ │ │ │ - -- .000110026s elapsed
│ │ │ │ + -- .000102701s elapsed
│ │ │ │  
│ │ │ │  o24 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_P_r_i_m_e_(_Z_Z_) -- whether a integer or polynomial is prime
│ │ │ │      * _f_a_c_t_o_r_(_Z_Z_) -- factor a ring element
│ │ │ │      * _n_e_x_t_P_r_i_m_e_(_N_u_m_b_e_r_) -- compute the smallest prime greater than or equal to
│ │ │ │        a given number
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_is__Regular__File.html
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  In UNIX, a regular file is one that is not special in some way.  Special files include symbolic links and directories.  A regular file is a sequence of bytes stored permanently in a file system.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-13116-0/0
    
│ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │              
    
│ │ │                
    i2 : fn << "hi there" << close
│ │ │  
│ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │  
│ │ │  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/M2-13116-0/0
│ │ │ │ +o1 = /tmp/M2-15476-0/0
│ │ │ │  i2 : fn << "hi there" << close
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-13116-0/0
│ │ │ │ +o2 = /tmp/M2-15476-0/0
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : isRegularFile fn
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : removeFile fn
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_lock_lp__Mutex_cm__Function_rp.html
│ │ │ @@ -98,15 +98,15 @@
│ │ │                
    i3 : parallelApply(1..1000, f);
    
│ │ │              
    
│ │ │                
    i4 : x
│ │ │  
│ │ │ -o4 = 333
    
│ │ │ +o4 = 337
│ │ │              
    
│ │ │          
    
│ │ │            
    You likely see that x ended up less than the expected value of 1000.  This is an example of a race condition, as multiple threads were trying to modify the global variable x at the same time.
    
│ │ │            
    Let's try again using lock.
    
│ │ │          
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  i1 : x = 0
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  i2 : f = i -> x += 1;
│ │ │ │  i3 : parallelApply(1..1000, f);
│ │ │ │  i4 : x
│ │ │ │  
│ │ │ │ -o4 = 333
│ │ │ │ +o4 = 337
│ │ │ │  You likely see that x ended up less than the expected value of 1000. This is an
│ │ │ │  example of a rraaccee ccoonnddiittiioonn, as multiple threads were trying to modify the
│ │ │ │  global variable x at the same time.
│ │ │ │  Let's try again using lock.
│ │ │ │  i5 : x = 0
│ │ │ │  
│ │ │ │  o5 = 0
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_make__Directory_lp__String_rp.html
│ │ │ @@ -81,22 +81,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,18 +13,18 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the name of the newly made directory
│ │ │ │      * Consequences:
│ │ │ │            o the directory is made, with as many new path components as needed
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : dir = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11593-0/0
│ │ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │ │  i2 : makeDirectory (dir|"/a/b/c")
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
│ │ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │ │  i3 : removeDirectory (dir|"/a/b/c")
│ │ │ │  i4 : removeDirectory (dir|"/a/b")
│ │ │ │  i5 : removeDirectory (dir|"/a")
│ │ │ │  A filename starting with ~/ will have the tilde replaced by the home directory.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_k_d_i_r
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_max__Allowable__Threads.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11593-0/0
    
│ │ │ +o1 = /tmp/M2-12363-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory (dir|"/a/b/c")
│ │ │  
│ │ │ -o2 = /tmp/M2-11593-0/0/a/b/c
    
│ │ │ +o2 = /tmp/M2-12363-0/0/a/b/c
│ │ │              
    
│ │ │                
    i3 : removeDirectory (dir|"/a/b/c")
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : maxAllowableThreads
│ │ │  
│ │ │ -o1 = 7
    
│ │ │ +o1 = 17
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -9,15 +9,15 @@
│ │ │ │      *   Usage:
│ │ │ │              maxAllowableThreads
│ │ │ │      * Outputs:
│ │ │ │            o an _i_n_t_e_g_e_r, the maximum number to which _a_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s can be set
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : maxAllowableThreads
│ │ │ │  
│ │ │ │ -o1 = 7
│ │ │ │ +o1 = 17
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_a_r_a_l_l_e_l_ _p_r_o_g_r_a_m_m_i_n_g_ _w_i_t_h_ _t_h_r_e_a_d_s_ _a_n_d_ _t_a_s_k_s
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _m_a_x_A_l_l_o_w_a_b_l_e_T_h_r_e_a_d_s is an _i_n_t_e_g_e_r.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_threads.m2:502:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_memoize.html
│ │ │ @@ -66,15 +66,15 @@
│ │ │  
│ │ │  o1 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i2 : time fib 28
│ │ │ - -- used 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ + -- used 0.885041s (cpu); 0.504981s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 514229
    
│ │ │              
    
│ │ │                
    i3 : fib = memoize fib
│ │ │ @@ -83,23 +83,23 @@
│ │ │  
│ │ │  o3 : FunctionClosure
    
│ │ │              
    
│ │ │                
    i4 : time fib 28
│ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ + -- used 5.8387e-05s (cpu); 5.5927e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 514229
    
│ │ │              
    
│ │ │                
    i5 : time fib 28
│ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ + -- used 2.988e-06s (cpu); 2.779e-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 1.47303s (cpu); 0.790587s (thread); 0s (gc)
│ │ │ │ + -- used 0.885041s (cpu); 0.504981s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o2 = 514229
│ │ │ │  i3 : fib = memoize fib
│ │ │ │  
│ │ │ │  o3 = fib
│ │ │ │  
│ │ │ │  o3 : FunctionClosure
│ │ │ │  i4 : time fib 28
│ │ │ │ - -- used 6.6374e-05s (cpu); 6.5843e-05s (thread); 0s (gc)
│ │ │ │ + -- used 5.8387e-05s (cpu); 5.5927e-05s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = 514229
│ │ │ │  i5 : time fib 28
│ │ │ │ - -- used 4.499e-06s (cpu); 4.078e-06s (thread); 0s (gc)
│ │ │ │ + -- used 2.988e-06s (cpu); 2.779e-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
│ │ │ @@ -81,33 +81,33 @@
│ │ │  
│ │ │  o1 = {0 => (==, BettiTally, BettiTally)                           }
│ │ │       {1 => (++, BettiTally, BettiTally)                           }
│ │ │       {2 => (**, BettiTally, BettiTally)                           }
│ │ │       {3 => (SPACE, BettiTally, Array)                             }
│ │ │       {4 => (SPACE, BettiTally, ZZ)                                }
│ │ │       {5 => (lift, BettiTally, ZZ)                                 }
│ │ │ -     {6 => (*, ZZ, BettiTally)                                    }
│ │ │ -     {7 => (*, QQ, BettiTally)                                    }
│ │ │ +     {6 => (*, QQ, BettiTally)                                    }
│ │ │ +     {7 => (*, ZZ, BettiTally)                                    }
│ │ │       {8 => (multigraded, BettiTally)                              }
│ │ │       {9 => (net, BettiTally)                                      }
│ │ │       {10 => (texMath, BettiTally)                                 }
│ │ │       {11 => (betti, BettiTally)                                   }
│ │ │       {12 => (poincare, BettiTally)                                }
│ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │       {14 => (degree, BettiTally)                                  }
│ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │       {17 => (regularity, BettiTally)                              }
│ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ -     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ +     {19 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ +     {21 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ +     {22 => (codim, BettiTally)                                   }
│ │ │ +     {23 => (dual, BettiTally)                                    }
│ │ │ +     {24 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │  
│ │ │  o1 : NumberedVerticalList
│ │ │                
    
│ │ │ @@ -193,20 +193,20 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                  
    i5 : methods( Matrix, Matrix )
│ │ │  
│ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ +o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ +     {1 => (+, Matrix, Matrix)                                   }
│ │ │ +     {2 => (contract', Matrix, Matrix)                           }
│ │ │ +     {3 => (diff', Matrix, Matrix)                               }
│ │ │ +     {4 => (-, 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)                        }
│ │ │ @@ -217,22 +217,22 @@
│ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ +     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │       {30 => (tensor, Matrix, Matrix)                             }
│ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ +     {31 => (pullback, 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 {}
│ │ │ │ @@ -16,33 +16,33 @@
│ │ │ │  
│ │ │ │  o1 = {0 => (==, BettiTally, BettiTally)                           }
│ │ │ │       {1 => (++, BettiTally, BettiTally)                           }
│ │ │ │       {2 => (**, BettiTally, BettiTally)                           }
│ │ │ │       {3 => (SPACE, BettiTally, Array)                             }
│ │ │ │       {4 => (SPACE, BettiTally, ZZ)                                }
│ │ │ │       {5 => (lift, BettiTally, ZZ)                                 }
│ │ │ │ -     {6 => (*, ZZ, BettiTally)                                    }
│ │ │ │ -     {7 => (*, QQ, BettiTally)                                    }
│ │ │ │ +     {6 => (*, QQ, BettiTally)                                    }
│ │ │ │ +     {7 => (*, ZZ, BettiTally)                                    }
│ │ │ │       {8 => (multigraded, BettiTally)                              }
│ │ │ │       {9 => (net, BettiTally)                                      }
│ │ │ │       {10 => (texMath, BettiTally)                                 }
│ │ │ │       {11 => (betti, BettiTally)                                   }
│ │ │ │       {12 => (poincare, BettiTally)                                }
│ │ │ │       {13 => (hilbertPolynomial, ZZ, BettiTally)                   }
│ │ │ │       {14 => (degree, BettiTally)                                  }
│ │ │ │       {15 => (hilbertSeries, ZZ, BettiTally)                       }
│ │ │ │       {16 => (pdim, BettiTally)                                    }
│ │ │ │       {17 => (regularity, BettiTally)                              }
│ │ │ │       {18 => (mathML, BettiTally)                                  }
│ │ │ │ -     {19 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ -     {20 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ -     {21 => (dual, BettiTally)                                    }
│ │ │ │ -     {22 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ -     {23 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │ -     {24 => (codim, BettiTally)                                   }
│ │ │ │ +     {19 => (truncate, BettiTally, ZZ, InfiniteNumber)            }
│ │ │ │ +     {20 => (truncate, BettiTally, InfiniteNumber, ZZ)            }
│ │ │ │ +     {21 => (truncate, BettiTally, InfiniteNumber, InfiniteNumber)}
│ │ │ │ +     {22 => (codim, BettiTally)                                   }
│ │ │ │ +     {23 => (dual, BettiTally)                                    }
│ │ │ │ +     {24 => (truncate, BettiTally, ZZ, ZZ)                        }
│ │ │ │       {25 => (^, Ring, BettiTally)                                 }
│ │ │ │  
│ │ │ │  o1 : NumberedVerticalList
│ │ │ │  i2 : methods resolution
│ │ │ │  
│ │ │ │  o2 = {0 => (freeResolution, Ideal)        }
│ │ │ │       {1 => (freeResolution, MonomialIdeal)}
│ │ │ │ @@ -85,20 +85,20 @@
│ │ │ │      * Inputs:
│ │ │ │            o X, a _t_y_p_e
│ │ │ │            o Y, a _t_y_p_e
│ │ │ │      * Outputs:
│ │ │ │            o a _v_e_r_t_i_c_a_l_ _l_i_s_t of those methods associated with
│ │ │ │  i5 : methods( Matrix, Matrix )
│ │ │ │  
│ │ │ │ -o5 = {0 => (-, Matrix, Matrix)                                   }
│ │ │ │ -     {1 => (diff', Matrix, Matrix)                               }
│ │ │ │ -     {2 => (+, Matrix, Matrix)                                   }
│ │ │ │ -     {3 => (contract, Matrix, Matrix)                            }
│ │ │ │ -     {4 => (diff, Matrix, Matrix)                                }
│ │ │ │ -     {5 => (contract', Matrix, Matrix)                           }
│ │ │ │ +o5 = {0 => (contract, Matrix, Matrix)                            }
│ │ │ │ +     {1 => (+, Matrix, Matrix)                                   }
│ │ │ │ +     {2 => (contract', Matrix, Matrix)                           }
│ │ │ │ +     {3 => (diff', Matrix, Matrix)                               }
│ │ │ │ +     {4 => (-, 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)                        }
│ │ │ │ @@ -109,22 +109,22 @@
│ │ │ │       {17 => (quotientRemainder', Matrix, Matrix)                 }
│ │ │ │       {18 => (quotientRemainder, Matrix, Matrix)                  }
│ │ │ │       {19 => (//, Matrix, Matrix)                                 }
│ │ │ │       {20 => (\\, Matrix, Matrix)                                 }
│ │ │ │       {21 => (quotient, Matrix, Matrix)                           }
│ │ │ │       {22 => (quotient', Matrix, Matrix)                          }
│ │ │ │       {23 => (remainder', Matrix, Matrix)                         }
│ │ │ │ -     {24 => (%, Matrix, Matrix)                                  }
│ │ │ │ -     {25 => (remainder, Matrix, Matrix)                          }
│ │ │ │ +     {24 => (remainder, Matrix, Matrix)                          }
│ │ │ │ +     {25 => (%, Matrix, Matrix)                                  }
│ │ │ │       {26 => (pushout, Matrix, Matrix)                            }
│ │ │ │       {27 => (solve, Matrix, Matrix)                              }
│ │ │ │ -     {28 => (pullback, Matrix, Matrix)                           }
│ │ │ │ +     {28 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │       {29 => (intersect, Matrix, Matrix)                          }
│ │ │ │       {30 => (tensor, Matrix, Matrix)                             }
│ │ │ │ -     {31 => (intersect, Matrix, Matrix, Matrix, Matrix)          }
│ │ │ │ +     {31 => (pullback, Matrix, Matrix)                           }
│ │ │ │       {32 => (substitute, Matrix, Matrix)                         }
│ │ │ │       {33 => (yonedaProduct, Matrix, Matrix)                      }
│ │ │ │       {34 => (isShortExactSequence, Matrix, Matrix)               }
│ │ │ │       {35 => (horseshoeResolution, Matrix, Matrix)                }
│ │ │ │       {36 => (connectingExtMap, Module, Matrix, Matrix)           }
│ │ │ │       {37 => (connectingExtMap, Matrix, Matrix, Module)           }
│ │ │ │       {38 => (connectingTorMap, Module, Matrix, Matrix)           }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_minimal__Betti.html
│ │ │ @@ -102,15 +102,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime C = minimalBetti I
│ │ │ - -- 1.20053s elapsed
│ │ │ + -- 1.43525s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o3 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : Ideal of S
    
│ │ │              
    
│ │ │                
    i5 : elapsedTime C = minimalBetti(I, DegreeLimit=>2)
│ │ │ - -- .517063s elapsed
│ │ │ + -- .621932s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7
│ │ │  o5 = total: 1 35 140 385 819 1080 735 196
│ │ │           0: 1  .   .   .   .    .   .   .
│ │ │           1: . 35 140 189  84    .   .   .
│ │ │           2: .  .   . 196 735 1080 735 196
│ │ │  
│ │ │ @@ -151,15 +151,15 @@
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime C = minimalBetti(I, DegreeLimit=>1, LengthLimit=>5)
│ │ │ - -- .0258497s elapsed
│ │ │ + -- .030078s elapsed
│ │ │  
│ │ │              0  1   2   3  4
│ │ │  o7 = total: 1 35 140 189 84
│ │ │           0: 1  .   .   .  .
│ │ │           1: . 35 140 189 84
│ │ │  
│ │ │  o7 : BettiTally
    
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o8 : Ideal of S
│ │ │              
    
│ │ │                
    i9 : elapsedTime C = minimalBetti(I, LengthLimit=>5)
│ │ │ - -- .826979s elapsed
│ │ │ + -- .982488s 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
│ │ │ │ - -- 1.20053s elapsed
│ │ │ │ + -- 1.43525s 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)
│ │ │ │ - -- .517063s elapsed
│ │ │ │ + -- .621932s 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)
│ │ │ │ - -- .0258497s elapsed
│ │ │ │ + -- .030078s 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)
│ │ │ │ - -- .826979s elapsed
│ │ │ │ + -- .982488s elapsed
│ │ │ │  
│ │ │ │              0  1   2   3   4    5
│ │ │ │  o9 = total: 1 35 140 385 819 1080
│ │ │ │           0: 1  .   .   .   .    .
│ │ │ │           1: . 35 140 189  84    .
│ │ │ │           2: .  .   . 196 735 1080
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_mkdir.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │          
    Description
    
│ │ │          
    Only one directory will be made, so the components of the path p other than the last must already exist.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -96,15 +96,15 @@
│ │ │  o3 = true
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-11612-0/0/
    
│ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │              
    
│ │ │                
    i2 : mkdir p
    
│ │ │              
    
│ │ │                
    i4 : (fn = p | "foo") << "hi there" << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ +o4 = /tmp/M2-12402-0/0/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/M2-11612-0/0/
│ │ │ │ +o1 = /tmp/M2-12402-0/0/
│ │ │ │  i2 : mkdir p
│ │ │ │  i3 : isDirectory p
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  i4 : (fn = p | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11612-0/0/foo
│ │ │ │ +o4 = /tmp/M2-12402-0/0/foo
│ │ │ │  
│ │ │ │  o4 : File
│ │ │ │  i5 : get fn
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : removeFile fn
│ │ │ │  i7 : removeDirectory p
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_move__File_lp__String_cm__String_rp.html
│ │ │ @@ -86,52 +86,52 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,32 +20,32 @@
│ │ │ │            o the name of the backup file if one was created, or _n_u_l_l
│ │ │ │      * Consequences:
│ │ │ │            o the file will be moved by creating a new link to the file and
│ │ │ │              removing the old one
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : src = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-11448-0/0
│ │ │ │ +o1 = /tmp/M2-12078-0/0
│ │ │ │  i2 : dst = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11448-0/1
│ │ │ │ +o2 = /tmp/M2-12078-0/1
│ │ │ │  i3 : src << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ │ +o3 = /tmp/M2-12078-0/0
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : moveFile(src,dst,Verbose=>true)
│ │ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
│ │ │ │ +--moving: /tmp/M2-12078-0/0 -> /tmp/M2-12078-0/1
│ │ │ │  i5 : get dst
│ │ │ │  
│ │ │ │  o5 = hi there
│ │ │ │  i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ │ +--backup file created: /tmp/M2-12078-0/1.bak
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-11448-0/1.bak
│ │ │ │ +o6 = /tmp/M2-12078-0/1.bak
│ │ │ │  i7 : removeFile bak
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_o_p_y_F_i_l_e
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * moveFile(String)
│ │ │ │      * _m_o_v_e_F_i_l_e_(_S_t_r_i_n_g_,_S_t_r_i_n_g_)
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_nanosleep.html
│ │ │ @@ -56,15 +56,15 @@
│ │ │        
    nanosleep -- sleep for a given number of nanoseconds
    
│ │ │        
    
│ │ │          
    Description
    
│ │ │  nanosleep n -- sleeps for n nanoseconds.        
    
│ │ │                
    i1 : src = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11448-0/0
    
│ │ │ +o1 = /tmp/M2-12078-0/0
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-11448-0/1
    
│ │ │ +o2 = /tmp/M2-12078-0/1
│ │ │              
    
│ │ │                
    i3 : src << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-11448-0/0
│ │ │ +o3 = /tmp/M2-12078-0/0
│ │ │  
│ │ │  o3 : File
    
│ │ │              
    
│ │ │                
    i4 : moveFile(src,dst,Verbose=>true)
│ │ │ ---moving: /tmp/M2-11448-0/0 -> /tmp/M2-11448-0/1
    
│ │ │ +--moving: /tmp/M2-12078-0/0 -> /tmp/M2-12078-0/1
│ │ │              
    
│ │ │                
    i5 : get dst
│ │ │  
│ │ │  o5 = hi there
    
│ │ │              
    
│ │ │                
    i6 : bak = moveFile(dst,Verbose=>true)
│ │ │ ---backup file created: /tmp/M2-11448-0/1.bak
│ │ │ +--backup file created: /tmp/M2-12078-0/1.bak
│ │ │  
│ │ │ -o6 = /tmp/M2-11448-0/1.bak
    
│ │ │ +o6 = /tmp/M2-12078-0/1.bak
│ │ │              
    
│ │ │                
    i7 : removeFile bak
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : elapsedTime nanosleep 500000000
│ │ │ - -- .500147s elapsed
│ │ │ + -- .500914s 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
│ │ │ │ - -- .500147s elapsed
│ │ │ │ + -- .500914s elapsed
│ │ │ │  
│ │ │ │  o1 = 0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _s_l_e_e_p -- sleep for a while
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _n_a_n_o_s_l_e_e_p is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_options_lp__Function_rp.html
│ │ │ @@ -108,55 +108,55 @@
│ │ │  o3 : OptionTable
    
│ │ │              
    
│ │ │                
    i4 : methods codim
│ │ │  
│ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ -     {1 => (codim, Ideal)         }
│ │ │ -     {2 => (codim, PolynomialRing)}
│ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ -     {5 => (codim, Module)        }
│ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ -     {7 => (codim, BettiTally)    }
│ │ │ +o4 = {0 => (codim, BettiTally)    }
│ │ │ +     {1 => (codim, Variety)       }
│ │ │ +     {2 => (codim, Ideal)         }
│ │ │ +     {3 => (codim, PolynomialRing)}
│ │ │ +     {4 => (codim, QuotientRing)  }
│ │ │ +     {5 => (codim, MonomialIdeal) }
│ │ │ +     {6 => (codim, Module)        }
│ │ │ +     {7 => (codim, CoherentSheaf) }
│ │ │  
│ │ │  o4 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i5 : options oo
│ │ │  
│ │ │ -o5 = {0 => (OptionTable{Generic => false})}
│ │ │ +o5 = {0 => (OptionTable{})                }
│ │ │       {1 => (OptionTable{Generic => false})}
│ │ │       {2 => (OptionTable{Generic => false})}
│ │ │       {3 => (OptionTable{Generic => false})}
│ │ │       {4 => (OptionTable{Generic => false})}
│ │ │       {5 => (OptionTable{Generic => false})}
│ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ -     {7 => (OptionTable{})                }
│ │ │ +     {7 => (OptionTable{Generic => false})}
│ │ │  
│ │ │  o5 : NumberedVerticalList
    
│ │ │              
    
│ │ │                
    i6 : methods intersect
│ │ │  
│ │ │  o6 = {0 => (intersect, List)                          }
│ │ │       {1 => (intersect, Module)                        }
│ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ -     {6 => (intersect, Set)                           }
│ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ +     {8 => (intersect, Set)                           }
│ │ │       {9 => (intersect, Ideal)                         }
│ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │       {11 => (intersect, CCi)                          }
│ │ │       {12 => (intersect, RRi)                          }
│ │ │       {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │       {14 => (intersect, Cone, Cone)                   }
│ │ │       {15 => (intersect, Matrix)                       }
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,47 +34,47 @@
│ │ │ │  i3 : options(codim, Ideal)
│ │ │ │  
│ │ │ │  o3 = OptionTable{Generic => false}
│ │ │ │  
│ │ │ │  o3 : OptionTable
│ │ │ │  i4 : methods codim
│ │ │ │  
│ │ │ │ -o4 = {0 => (codim, Variety)       }
│ │ │ │ -     {1 => (codim, Ideal)         }
│ │ │ │ -     {2 => (codim, PolynomialRing)}
│ │ │ │ -     {3 => (codim, QuotientRing)  }
│ │ │ │ -     {4 => (codim, MonomialIdeal) }
│ │ │ │ -     {5 => (codim, Module)        }
│ │ │ │ -     {6 => (codim, CoherentSheaf) }
│ │ │ │ -     {7 => (codim, BettiTally)    }
│ │ │ │ +o4 = {0 => (codim, BettiTally)    }
│ │ │ │ +     {1 => (codim, Variety)       }
│ │ │ │ +     {2 => (codim, Ideal)         }
│ │ │ │ +     {3 => (codim, PolynomialRing)}
│ │ │ │ +     {4 => (codim, QuotientRing)  }
│ │ │ │ +     {5 => (codim, MonomialIdeal) }
│ │ │ │ +     {6 => (codim, Module)        }
│ │ │ │ +     {7 => (codim, CoherentSheaf) }
│ │ │ │  
│ │ │ │  o4 : NumberedVerticalList
│ │ │ │  i5 : options oo
│ │ │ │  
│ │ │ │ -o5 = {0 => (OptionTable{Generic => false})}
│ │ │ │ +o5 = {0 => (OptionTable{})                }
│ │ │ │       {1 => (OptionTable{Generic => false})}
│ │ │ │       {2 => (OptionTable{Generic => false})}
│ │ │ │       {3 => (OptionTable{Generic => false})}
│ │ │ │       {4 => (OptionTable{Generic => false})}
│ │ │ │       {5 => (OptionTable{Generic => false})}
│ │ │ │       {6 => (OptionTable{Generic => false})}
│ │ │ │ -     {7 => (OptionTable{})                }
│ │ │ │ +     {7 => (OptionTable{Generic => false})}
│ │ │ │  
│ │ │ │  o5 : NumberedVerticalList
│ │ │ │  i6 : methods intersect
│ │ │ │  
│ │ │ │  o6 = {0 => (intersect, List)                          }
│ │ │ │       {1 => (intersect, Module)                        }
│ │ │ │       {2 => (intersect, Matrix, Matrix, Matrix, Matrix)}
│ │ │ │       {3 => (intersect, RRi, RRi)                      }
│ │ │ │ -     {4 => (intersect, Matrix, Matrix)                }
│ │ │ │ -     {5 => (intersect, Set, Set)                      }
│ │ │ │ -     {6 => (intersect, Set)                           }
│ │ │ │ -     {7 => (intersect, Polyhedron, Cone)              }
│ │ │ │ -     {8 => (intersect, Cone, Polyhedron)              }
│ │ │ │ +     {4 => (intersect, Set, Set)                      }
│ │ │ │ +     {5 => (intersect, Matrix, Matrix)                }
│ │ │ │ +     {6 => (intersect, Polyhedron, Cone)              }
│ │ │ │ +     {7 => (intersect, Cone, Polyhedron)              }
│ │ │ │ +     {8 => (intersect, Set)                           }
│ │ │ │       {9 => (intersect, Ideal)                         }
│ │ │ │       {10 => (intersect, Ideal, Ideal)                 }
│ │ │ │       {11 => (intersect, CCi)                          }
│ │ │ │       {12 => (intersect, RRi)                          }
│ │ │ │       {13 => (intersect, Polyhedron, Polyhedron)       }
│ │ │ │       {14 => (intersect, Cone, Cone)                   }
│ │ │ │       {15 => (intersect, Matrix)                       }
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallel_spprogramming_spwith_spthreads_spand_sptasks.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │              
    		
│ │ │                
    i2 : L = shuffle toList (1..10000);
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ - -- .659735s elapsed
    
│ │ │ + -- .625351s elapsed
│ │ │              
    
│ │ │                
    i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ - -- .211134s elapsed
    
│ │ │ + -- .167328s elapsed
│ │ │              
    
│ │ │          
    
│ │ │            
    You will have to try it on your examples to see how much they speed up.
    
│ │ │            
    Warning: Threads computing in parallel can give wrong answers if their code is not "thread safe", meaning they make modifications to memory without ensuring the modifications get safely communicated to other threads. (Thread safety can slow computations some.) Currently, modifications to Macaulay2 variables and mutable hash tables are thread safe, but not changes inside mutable lists. Also, access to external libraries such as singular, etc., may not currently be thread safe.
    
│ │ │            
    The rest of this document describes how to control parallel tasks more directly.
    
│ │ │ @@ -105,15 +105,15 @@
│ │ │  o5 = 5
    
│ │ │              
    
│ │ │                
    i6 : allowableThreads = maxAllowableThreads
│ │ │  
│ │ │ -o6 = 7
    
│ │ │ +o6 = 17
│ │ │              
    
│ │ │          
    
│ │ │            
    To run a function in another thread use schedule, as in the following example.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -229,15 +229,15 @@
│ │ │                
    i17 : schedule t';
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i18 : t'
│ │ │  
│ │ │ -o18 = <<task, created>>
│ │ │ +o18 = <<task, running>>
│ │ │  
│ │ │  o18 : Task
    
│ │ │              
    
│ │ │                
    i19 : taskResult t'
│ │ │ ├── html2text {}
│ │ │ │ @@ -17,17 +17,17 @@
│ │ │ │  big computation. If the list is long, it will be split into chunks for each
│ │ │ │  core, reducing the overhead. But the speedup is still limited by the different
│ │ │ │  threads competing for memory, including cpu caches; it is like running
│ │ │ │  Macaulay2 on a computer that is running other big programs at the same time. We
│ │ │ │  can see this using _e_l_a_p_s_e_d_T_i_m_e.
│ │ │ │  i2 : L = shuffle toList (1..10000);
│ │ │ │  i3 : elapsedTime         apply(1..100, n -> sort L);
│ │ │ │ - -- .659735s elapsed
│ │ │ │ + -- .625351s elapsed
│ │ │ │  i4 : elapsedTime parallelApply(1..100, n -> sort L);
│ │ │ │ - -- .211134s elapsed
│ │ │ │ + -- .167328s 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)
│ │ │ │  
│ │ │ │ @@ -92,15 +92,15 @@
│ │ │ │  o16 = <>
│ │ │ │  
│ │ │ │  o16 : Task
│ │ │ │  Start it running with _s_c_h_e_d_u_l_e.
│ │ │ │  i17 : schedule t';
│ │ │ │  i18 : t'
│ │ │ │  
│ │ │ │ -o18 = <>
│ │ │ │ +o18 = <>
│ │ │ │  
│ │ │ │  o18 : Task
│ │ │ │  i19 : taskResult t'
│ │ │ │  
│ │ │ │  o19 = | 980z2-18y-201z+13 35yz-4y+2z-1 10y2-y-12z+1 5x2-4y+2z-1 |
│ │ │ │  
│ │ │ │                1      4
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_parallelism_spin_spengine_spcomputations.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalBetti I
│ │ │ - -- 1.15969s elapsed
│ │ │ + -- 1.49412s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o4 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -165,15 +165,15 @@
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime minimalBetti(I, ParallelizeByDegree => true)
│ │ │ - -- 1.14895s elapsed
│ │ │ + -- 1.38782s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o6 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -195,15 +195,15 @@
│ │ │  
│ │ │  o8 = 1
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime minimalBetti(I)
│ │ │ - -- 1.62269s elapsed
│ │ │ + -- 1.48296s elapsed
│ │ │  
│ │ │              0  1   2   3   4    5   6   7   8  9 10
│ │ │  o9 = total: 1 35 140 385 819 1080 819 385 140 35  1
│ │ │           0: 1  .   .   .   .    .   .   .   .  .  .
│ │ │           1: . 35 140 189  84    .   .   .   .  .  .
│ │ │           2: .  .   . 196 735 1080 735 196   .  .  .
│ │ │           3: .  .   .   .   .    .  84 189 140 35  .
│ │ │ @@ -236,15 +236,15 @@
│ │ │  
│ │ │  o12 : Ideal of S
    
│ │ │              
    
│ │ │                
    i13 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 2.37571s elapsed
│ │ │ + -- 1.76019s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o13 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o13 : Complex
    
│ │ │ @@ -263,15 +263,15 @@
│ │ │  
│ │ │  o15 : Ideal of S
│ │ │              
    
│ │ │                
    i16 : elapsedTime freeResolution(I, Strategy => Nonminimal)
│ │ │ - -- 1.51803s elapsed
│ │ │ + -- 1.78929s elapsed
│ │ │  
│ │ │         1      35      241      841      1781      2464      2294      1432      576      135      14
│ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S    <-- S    <-- S
│ │ │                                                                                                    
│ │ │        0      1       2        3        4         5         6         7         8        9        10
│ │ │  
│ │ │  o16 : Complex
    
│ │ │ @@ -304,15 +304,15 @@
│ │ │  
│ │ │  o19 : Ideal of S
│ │ │              
    
│ │ │                
    i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.09157s elapsed
│ │ │ + -- 3.04216s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o20 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -327,15 +327,15 @@
│ │ │  
│ │ │  o22 : Ideal of S
│ │ │              
    
│ │ │                
    i23 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 7.21927s elapsed
│ │ │ + -- 6.99351s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o23 : Matrix S  <-- S
    
│ │ │              
    
│ │ │ @@ -350,15 +350,15 @@
│ │ │  
│ │ │  o25 : Ideal of S
│ │ │              
    
│ │ │                
    i26 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ - -- 4.08357s elapsed
│ │ │ + -- 3.13269s elapsed
│ │ │  
│ │ │                1      108
│ │ │  o26 : Matrix S  <-- S
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -401,15 +401,15 @@
│ │ │  o30 : Ideal of ---<|a, b, c|>
│ │ │                 101
│ │ │              
    
│ │ │                
    i31 : elapsedTime NCGB(I, 22);
│ │ │ - -- .823549s elapsed
│ │ │ + -- .748632s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o31 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │                
    i34 : elapsedTime NCGB(I, 22);
│ │ │ - -- 1.45319s elapsed
│ │ │ + -- 1.2163s elapsed
│ │ │  
│ │ │                 ZZ            1       ZZ            148
│ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │                101                   101
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -93,30 +93,30 @@
│ │ │ │  0,5   1,5   2,5   3,5   4,5   0,6   1,6   2,6   3,6   4,6   5,6
│ │ │ │  i3 : S = ring I
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : elapsedTime minimalBetti I
│ │ │ │ - -- 1.15969s elapsed
│ │ │ │ + -- 1.49412s 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)
│ │ │ │ - -- 1.14895s elapsed
│ │ │ │ + -- 1.38782s 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)
│ │ │ │ - -- 1.62269s elapsed
│ │ │ │ + -- 1.48296s 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)
│ │ │ │ - -- 2.37571s elapsed
│ │ │ │ + -- 1.76019s 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)
│ │ │ │ - -- 1.51803s elapsed
│ │ │ │ + -- 1.78929s elapsed
│ │ │ │  
│ │ │ │         1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o16 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-
│ │ │ │  - S    <-- S    <-- S
│ │ │ │  
│ │ │ │  
│ │ │ │ @@ -195,37 +195,37 @@
│ │ │ │  o18 = S
│ │ │ │  
│ │ │ │  o18 : PolynomialRing
│ │ │ │  i19 : I = ideal random(S^1, S^{4:-5});
│ │ │ │  
│ │ │ │  o19 : Ideal of S
│ │ │ │  i20 : elapsedTime groebnerBasis(I, Strategy => "F4");
│ │ │ │ - -- 4.09157s elapsed
│ │ │ │ + -- 3.04216s 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");
│ │ │ │ - -- 7.21927s elapsed
│ │ │ │ + -- 6.99351s 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");
│ │ │ │ - -- 4.08357s elapsed
│ │ │ │ + -- 3.13269s 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);
│ │ │ │ - -- .823549s elapsed
│ │ │ │ + -- .748632s 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);
│ │ │ │ - -- 1.45319s elapsed
│ │ │ │ + -- 1.2163s elapsed
│ │ │ │  
│ │ │ │                 ZZ            1       ZZ            148
│ │ │ │  o34 : Matrix (---<|a, b, c|>)  <-- (---<|a, b, c|>)
│ │ │ │                101                   101
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_i_m_a_l_B_e_t_t_i -- minimal betti numbers of (the minimal free resolution of)
│ │ │ │        a homogeneous ideal or module
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_poincare.html
│ │ │ @@ -375,36 +375,36 @@
│ │ │  
│ │ │  o27 = 3
│ │ │              
    
│ │ │                
    i28 : time poincare I
│ │ │ - -- used 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00270475s (cpu); 9.833e-06s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o28 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i29 : time gens gb I;
│ │ │  
│ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ +   -- registering gb 16 at 0x7f9895d5ac40
│ │ │  
│ │ │     -- [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 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00931577s (cpu); 0.00985886s (thread); 0s (gc)
│ │ │  
│ │ │                1      11
│ │ │  o29 : Matrix R  <-- R
    
│ │ │              
    
│ │ │          
    
│ │ │ @@ -416,15 +416,15 @@
│ │ │                
    i30 : R = QQ[a..d];
    
│ │ │              
    
│ │ │                
    i31 : I = ideal random(R^1, R^{3:-3});
│ │ │  
│ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ +   -- registering gb 17 at 0x7f9895d5aa80
│ │ │  
│ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │     -- number of monomials                = 0
│ │ │     -- #reduction steps = 0
│ │ │     -- #spairs done = 0
│ │ │     -- ncalls = 0
│ │ │     -- nloop = 0
│ │ │ @@ -433,24 +433,24 @@
│ │ │  o31 : Ideal of R
    
│ │ │              
    
│ │ │                
    i32 : time p = poincare I
│ │ │  
│ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ +   -- registering gb 18 at 0x7f9895d5a8c0
│ │ │  
│ │ │     -- [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 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ +   --  -- used 0.00321219s (cpu); 0.00352901s (thread); 0s (gc)
│ │ │  
│ │ │              3     6    9
│ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │  
│ │ │  o32 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i37 : time gens gb J;
│ │ │  
│ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ +   -- registering gb 19 at 0x7f9895d5a700
│ │ │  
│ │ │     -- [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 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ +   --  -- used 0.0439623s (cpu); 0.0445398s (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 0.000690775s (cpu); 1.8044e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00270475s (cpu); 9.833e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o28 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o28 : ZZ[T]
│ │ │ │  i29 : time gens gb I;
│ │ │ │  
│ │ │ │ -   -- registering gb 16 at 0x7fad46272c40
│ │ │ │ +   -- registering gb 16 at 0x7f9895d5ac40
│ │ │ │  
│ │ │ │     -- [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 0.0192888s (cpu); 0.0198812s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00931577s (cpu); 0.00985886s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      11
│ │ │ │  o29 : Matrix R  <-- R
│ │ │ │  In this case, the savings is minimal, but often it can be dramatic. Another
│ │ │ │  important situation is to compute a Gröbner basis using a different monomial
│ │ │ │  order.
│ │ │ │  i30 : R = QQ[a..d];
│ │ │ │  i31 : I = ideal random(R^1, R^{3:-3});
│ │ │ │  
│ │ │ │ -   -- registering gb 17 at 0x7fad46272a80
│ │ │ │ +   -- registering gb 17 at 0x7f9895d5aa80
│ │ │ │  
│ │ │ │     -- [gb]number of (nonminimal) gb elements = 0
│ │ │ │     -- number of monomials                = 0
│ │ │ │     -- #reduction steps = 0
│ │ │ │     -- #spairs done = 0
│ │ │ │     -- ncalls = 0
│ │ │ │     -- nloop = 0
│ │ │ │     -- nsaved = 0
│ │ │ │     --
│ │ │ │  o31 : Ideal of R
│ │ │ │  i32 : time p = poincare I
│ │ │ │  
│ │ │ │ -   -- registering gb 18 at 0x7fad462728c0
│ │ │ │ +   -- registering gb 18 at 0x7f9895d5a8c0
│ │ │ │  
│ │ │ │     -- [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 0.0079971s (cpu); 0.00761358s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.00321219s (cpu); 0.00352901s (thread); 0s (gc)
│ │ │ │  
│ │ │ │              3     6    9
│ │ │ │  o32 = 1 - 3T  + 3T  - T
│ │ │ │  
│ │ │ │  o32 : ZZ[T]
│ │ │ │  i33 : S = QQ[a..d, MonomialOrder => Eliminate 2]
│ │ │ │  
│ │ │ │ @@ -281,30 +281,30 @@
│ │ │ │  
│ │ │ │  o35 : ZZ[T]
│ │ │ │  i36 : gbTrace = 3
│ │ │ │  
│ │ │ │  o36 = 3
│ │ │ │  i37 : time gens gb J;
│ │ │ │  
│ │ │ │ -   -- registering gb 19 at 0x7fad46272700
│ │ │ │ +   -- registering gb 19 at 0x7f9895d5a700
│ │ │ │  
│ │ │ │     -- [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 0.0839937s (cpu); 0.0833296s (thread); 0s (gc)
│ │ │ │ +   --  -- used 0.0439623s (cpu); 0.0445398s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1      39
│ │ │ │  o37 : Matrix S  <-- S
│ │ │ │  i38 : selectInSubring(1, gens gb J)
│ │ │ │  
│ │ │ │  o38 = | 188529931266160087758259645374082357642621166724936033369975727480205
│ │ │ │        -----------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_printing_spto_spa_spfile.html
│ │ │ @@ -102,22 +102,22 @@
│ │ │  o2 : File
│ │ │              
    
│ │ │                
    i3 : fn = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-11803-0/0
    
│ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │              
    
│ │ │                
    i4 : fn << "hi there" << endl << close
│ │ │  
│ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o4 : File
    
│ │ │              
    
│ │ │                
    i5 : get fn
│ │ │ @@ -156,15 +156,15 @@
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : fn << f << close
│ │ │  
│ │ │ -o9 = /tmp/M2-11803-0/0
│ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o9 : File
    
│ │ │              
    
│ │ │                
    i10 : get fn
│ │ │ @@ -174,15 +174,15 @@
│ │ │        + 1
    
│ │ │              
    
│ │ │                
    i11 : fn << toExternalString f << close
│ │ │  
│ │ │ -o11 = /tmp/M2-11803-0/0
│ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │  
│ │ │  o11 : File
    
│ │ │              
    
│ │ │                
    i12 : get fn
│ │ │ ├── html2text {}
│ │ │ │ @@ -36,18 +36,18 @@
│ │ │ │  -- ho there --
│ │ │ │  
│ │ │ │  o2 = stdio
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-11803-0/0
│ │ │ │ +o3 = /tmp/M2-12793-0/0
│ │ │ │  i4 : fn << "hi there" << endl << close
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-11803-0/0
│ │ │ │ +o4 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  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/M2-11803-0/0
│ │ │ │ +o9 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  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/M2-11803-0/0
│ │ │ │ +o11 = /tmp/M2-12793-0/0
│ │ │ │  
│ │ │ │  o11 : File
│ │ │ │  i12 : get fn
│ │ │ │  
│ │ │ │  o12 = x^10+10*x^9+45*x^8+120*x^7+210*x^6+252*x^5+210*x^4+120*x^3+45*x^2+10*x+
│ │ │ │        1
│ │ │ │  i13 : value get fn
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_process__I__D.html
│ │ │ @@ -69,15 +69,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : processID()
│ │ │  
│ │ │ -o1 = 11022
    
│ │ │ +o1 = 11242
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 = 11022
│ │ │ │ +o1 = 11242
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _g_r_o_u_p_I_D -- the process group identifier
│ │ │ │      * _s_e_t_G_r_o_u_p_I_D -- set the process group identifier
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _p_r_o_c_e_s_s_I_D is a _c_o_m_p_i_l_e_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_profile.html
│ │ │ @@ -96,35 +96,35 @@
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -136,15 +136,15 @@
│ │ │  
│ │ │  o9 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -178,15 +178,15 @@
│ │ │  o14 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i2 : profileSummary
│ │ │  
│ │ │  o2 = #run  %time   position                         
│ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72 
│ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22  
│ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61     
│ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74 
│ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52 
│ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ -     1     .0041s  elapsed total                    
    
│ │ │ +     1     93.86   ../../m2/matrix1.m2:273:4-276:58 
│ │ │ +     1     91.14   ../../m2/matrix1.m2:275:22-275:43
│ │ │ +     1     42.26   ../../m2/matrix1.m2:183:25-183:52
│ │ │ +     1     29.1    ../../m2/matrix1.m2:104:5-146:72 
│ │ │ +     1     27.95   ../../m2/matrix1.m2:130:10-145:16
│ │ │ +     1     23.28   ../../m2/matrix1.m2:171:4-171:42 
│ │ │ +     1     21.94   ../../m2/set.m2:129:5-129:61     
│ │ │ +     1     19.76   ../../m2/matrix1.m2:35:10-39:22  
│ │ │ +     1     3.55    ../../m2/matrix1.m2:102:5-102:29 
│ │ │ +     1     2.29    ../../m2/matrix1.m2:131:13-131:78
│ │ │ +     1     2.19    ../../m2/matrix1.m2:86:5-99:11   
│ │ │ +     1     1.63    ../../m2/matrix1.m2:275:7-275:16 
│ │ │ +     1     1.49    ../../m2/matrix1.m2:172:4-174:74 
│ │ │ +     1     1.36    ../../m2/matrix1.m2:270:4-271:73 
│ │ │ +     1     1.27    ../../m2/matrix1.m2:137:20-137:64
│ │ │ +     1     1.12    ../../m2/matrix1.m2:88:10-88:46  
│ │ │ +     1     1.12    ../../m2/matrix1.m2:101:5-101:91 
│ │ │ +     20    .92     ../../m2/matrix1.m2:181:14-182:67
│ │ │ +     20    .72     ../../m2/matrix1.m2:37:43-37:71  
│ │ │ +     20    .66     ../../m2/matrix1.m2:180:17-180:29
│ │ │ +     1     .0032s  elapsed total                    
│ │ │              
    
│ │ │                
    i3 : coverageSummary
│ │ │  
│ │ │  o3 = covered lines:
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,35 +25,35 @@
│ │ │ │                4       5
│ │ │ │  o1 : Matrix ZZ  <-- ZZ
│ │ │ │  Afterwards, running profileSummary and coverageSummary produces easy to read
│ │ │ │  tables summarizing the accumulated data so far in different ways.
│ │ │ │  i2 : profileSummary
│ │ │ │  
│ │ │ │  o2 = #run  %time   position
│ │ │ │ -     1     94.95   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ -     1     92.56   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ -     1     44.48   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ -     1     31.39   ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ -     1     30.23   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ -     1     22.58   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ -     1     21.8    ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ -     1     21.29   ../../m2/set.m2:129:5-129:61
│ │ │ │ -     1     3.45    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ -     1     2.36    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ -     1     2.15    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ -     1     1.35    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ -     1     1.34    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ -     1     1.21    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ -     1     1.11    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ -     1     1.1     ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ -     1     1.09    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ -     20    .52     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ -     1     .43     ../../m2/modules.m2:282:4-282:52
│ │ │ │ -     20    .36     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ -     1     .0041s  elapsed total
│ │ │ │ +     1     93.86   ../../m2/matrix1.m2:273:4-276:58
│ │ │ │ +     1     91.14   ../../m2/matrix1.m2:275:22-275:43
│ │ │ │ +     1     42.26   ../../m2/matrix1.m2:183:25-183:52
│ │ │ │ +     1     29.1    ../../m2/matrix1.m2:104:5-146:72
│ │ │ │ +     1     27.95   ../../m2/matrix1.m2:130:10-145:16
│ │ │ │ +     1     23.28   ../../m2/matrix1.m2:171:4-171:42
│ │ │ │ +     1     21.94   ../../m2/set.m2:129:5-129:61
│ │ │ │ +     1     19.76   ../../m2/matrix1.m2:35:10-39:22
│ │ │ │ +     1     3.55    ../../m2/matrix1.m2:102:5-102:29
│ │ │ │ +     1     2.29    ../../m2/matrix1.m2:131:13-131:78
│ │ │ │ +     1     2.19    ../../m2/matrix1.m2:86:5-99:11
│ │ │ │ +     1     1.63    ../../m2/matrix1.m2:275:7-275:16
│ │ │ │ +     1     1.49    ../../m2/matrix1.m2:172:4-174:74
│ │ │ │ +     1     1.36    ../../m2/matrix1.m2:270:4-271:73
│ │ │ │ +     1     1.27    ../../m2/matrix1.m2:137:20-137:64
│ │ │ │ +     1     1.12    ../../m2/matrix1.m2:88:10-88:46
│ │ │ │ +     1     1.12    ../../m2/matrix1.m2:101:5-101:91
│ │ │ │ +     20    .92     ../../m2/matrix1.m2:181:14-182:67
│ │ │ │ +     20    .72     ../../m2/matrix1.m2:37:43-37:71
│ │ │ │ +     20    .66     ../../m2/matrix1.m2:180:17-180:29
│ │ │ │ +     1     .0032s  elapsed total
│ │ │ │  i3 : coverageSummary
│ │ │ │  
│ │ │ │  o3 = covered lines:
│ │ │ │       ../../m2/lists.m2:148:24-148:32
│ │ │ │       ../../m2/lists.m2:148:34-148:58
│ │ │ │       ../../m2/matrix.m2:30:5-30:35
│ │ │ │       ../../m2/matrix.m2:31:5-31:46
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_random__K__Rational__Point.html
│ │ │ @@ -104,15 +104,15 @@
│ │ │  
│ │ │  o5 : Sequence
    
│ │ │              
    
│ │ │                
    i6 : time randomKRationalPoint(I)
│ │ │ - -- used 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ + -- used 0.448087s (cpu); 0.119215s (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 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ + -- used 0.311606s (cpu); 0.196472s (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 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ + -- used 4.7779s (cpu); 1.86455s (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 0.468269s (cpu); 0.20272s (thread); 0s (gc)
│ │ │ │ + -- used 0.448087s (cpu); 0.119215s (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 0.372139s (cpu); 0.315367s (thread); 0s (gc)
│ │ │ │ + -- used 0.311606s (cpu); 0.196472s (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 5.46061s (cpu); 2.7311s (thread); 0s (gc)
│ │ │ │ + -- used 4.7779s (cpu); 1.86455s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o15 = 58
│ │ │ │  ********** WWaayyss ttoo uussee rraannddoommKKRRaattiioonnaallPPooiinntt:: **********
│ │ │ │      * randomKRationalPoint(Ideal)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_a_n_d_o_m_K_R_a_t_i_o_n_a_l_P_o_i_n_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n.
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_read__Directory.html
│ │ │ @@ -73,38 +73,38 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12474-0/0
    
│ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-12474-0/0
    
│ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │              
    
│ │ │                
    i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │  
│ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ +o3 = /tmp/M2-14164-0/0/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/M2-12474-0/0
│ │ │ │ +o1 = /tmp/M2-14164-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12474-0/0
│ │ │ │ +o2 = /tmp/M2-14164-0/0
│ │ │ │  i3 : (fn = dir | "/" | "foo") << "hi there" << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12474-0/0/foo
│ │ │ │ +o3 = /tmp/M2-14164-0/0/foo
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : readDirectory dir
│ │ │ │  
│ │ │ │ -o4 = {., .., foo}
│ │ │ │ +o4 = {.., ., foo}
│ │ │ │  
│ │ │ │  o4 : List
│ │ │ │  i5 : removeFile fn
│ │ │ │  i6 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_m_o_v_e_D_i_r_e_c_t_o_r_y -- remove a directory
│ │ │ │      * _r_e_m_o_v_e_F_i_l_e -- remove a file
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_reading_spfiles.html
│ │ │ @@ -57,22 +57,22 @@
│ │ │        
    
│ │ │  Sometimes a file will contain a single expression whose value you wish to have access to.  For example, it might be a polynomial produced by another program.  The function get can be used to obtain the entire contents of a file as a single string.  We illustrate this here with a file whose name is expression.        
    
│ │ │  First we create the file by writing the desired text to it.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11978-0/0
    
│ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │              
    
│ │ │                
    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/M2-11978-0/0
│ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  o2 : File
    
│ │ │              
    
│ │ │  Now we get the contents of the file, as a single string.        
│ │ │            
│ │ │ @@ -121,15 +121,15 @@
│ │ │  Often a file will contain code written in the Macaulay2 language. Let's create such a file.        
    
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : fn << "sample = 2^100
│ │ │       print sample
│ │ │       " << close
│ │ │  
│ │ │ -o7 = /tmp/M2-11978-0/0
│ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │  
│ │ │  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/M2-11978-0/0
│ │ │ │ +o1 = /tmp/M2-13148-0/0
│ │ │ │  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/M2-11978-0/0
│ │ │ │ +o2 = /tmp/M2-13148-0/0
│ │ │ │  
│ │ │ │  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/M2-11978-0/0
│ │ │ │ +o7 = /tmp/M2-13148-0/0
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  Now verify that it contains the desired text with _g_e_t.
│ │ │ │  i8 : get fn
│ │ │ │  
│ │ │ │  o8 = sample = 2^100
│ │ │ │       print sample
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_readlink.html
│ │ │ @@ -73,15 +73,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,15 +11,15 @@
│ │ │ │            o fn, a _s_t_r_i_n_g, a filename or path to a file
│ │ │ │      * Outputs:
│ │ │ │            o a _s_t_r_i_n_g, the resolved path to a symbolic link, or null if the file
│ │ │ │              was not a symbolic link.
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : p = temporaryFileName ()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12734-0/0
│ │ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │ │  i2 : symlinkFile ("foo", p)
│ │ │ │  i3 : readlink p
│ │ │ │  
│ │ │ │  o3 = foo
│ │ │ │  i4 : removeFile p
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_l_p_a_t_h -- convert a filename to one passing through no symbolic links
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_realpath.html
│ │ │ @@ -73,57 +73,57 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
    
│ │ │                
    i1 : p = temporaryFileName ()
│ │ │  
│ │ │ -o1 = /tmp/M2-12734-0/0
    
│ │ │ +o1 = /tmp/M2-14684-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile ("foo", p)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -135,15 +135,15 @@
│ │ │          
    
│ │ │                
    i1 : realpath "."
│ │ │  
│ │ │ -o1 = /tmp/M2-11022-0/91-rundir/
    
│ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │              
    
│ │ │                
    i2 : p = temporaryFileName()
│ │ │  
│ │ │ -o2 = /tmp/M2-12753-0/0
    
│ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i3 : q = temporaryFileName()
│ │ │  
│ │ │ -o3 = /tmp/M2-12753-0/1
    
│ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │              
    
│ │ │                
    i4 : symlinkFile(p,q)
    
│ │ │              
    
│ │ │                
    i5 : p << close
│ │ │  
│ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │  
│ │ │  o5 : File
    
│ │ │              
    
│ │ │                
    i6 : readlink q
│ │ │  
│ │ │ -o6 = /tmp/M2-12753-0/0
    
│ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i7 : realpath q
│ │ │  
│ │ │ -o7 = /tmp/M2-12753-0/0
    
│ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │              
    
│ │ │                
    i8 : removeFile p
    
│ │ │              
    
│ │ │          
    The empty string is interpreted as a reference to the current directory.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : realpath ""
│ │ │  
│ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
    
│ │ │ +o10 = /tmp/M2-11242-0/91-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/M2-11022-0/91-rundir/
│ │ │ │ +o1 = /tmp/M2-11242-0/91-rundir/
│ │ │ │  i2 : p = temporaryFileName()
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12753-0/0
│ │ │ │ +o2 = /tmp/M2-14723-0/0
│ │ │ │  i3 : q = temporaryFileName()
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12753-0/1
│ │ │ │ +o3 = /tmp/M2-14723-0/1
│ │ │ │  i4 : symlinkFile(p,q)
│ │ │ │  i5 : p << close
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12753-0/0
│ │ │ │ +o5 = /tmp/M2-14723-0/0
│ │ │ │  
│ │ │ │  o5 : File
│ │ │ │  i6 : readlink q
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12753-0/0
│ │ │ │ +o6 = /tmp/M2-14723-0/0
│ │ │ │  i7 : realpath q
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12753-0/0
│ │ │ │ +o7 = /tmp/M2-14723-0/0
│ │ │ │  i8 : removeFile p
│ │ │ │  i9 : removeFile q
│ │ │ │  The empty string is interpreted as a reference to the current directory.
│ │ │ │  i10 : realpath ""
│ │ │ │  
│ │ │ │ -o10 = /tmp/M2-11022-0/91-rundir/
│ │ │ │ +o10 = /tmp/M2-11242-0/91-rundir/
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  Every component of the path must exist in the file system and be accessible to
│ │ │ │  the user. Terminal slashes will be dropped. Warning: under most operating
│ │ │ │  systems, the value returned is an absolute path (one starting at the root of
│ │ │ │  the file system), but under Solaris, this system call may, in certain
│ │ │ │  circumstances, return a relative path when given a relative path.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_register__Finalizer.html
│ │ │ @@ -81,24 +81,23 @@
│ │ │                  		
│ │ │                
    i1 : for i from 1 to 9 do (x := 0 .. 10000 ; registerFinalizer(x, "-- finalizing sequence #"|i|" --"))
    
│ │ │              
    
│ │ │                
    i2 : collectGarbage() 
│ │ │ ---finalization: (1)[1]: -- finalizing sequence #2 --
│ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ ---finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
    
│ │ │ +--finalization: (1)[8]: -- finalizing sequence #9 --
│ │ │ +--finalization: (2)[6]: -- finalizing sequence #7 --
│ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ +--finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ +--finalization: (7)[4]: -- finalizing sequence #5 --
│ │ │ +--finalization: (8)[7]: -- finalizing sequence #8 --
│ │ │ +--finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    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 {}
│ │ │ │ @@ -14,24 +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)[1]: -- finalizing sequence #2 --
│ │ │ │ ---finalization: (2)[5]: -- finalizing sequence #6 --
│ │ │ │ ---finalization: (3)[3]: -- finalizing sequence #4 --
│ │ │ │ ---finalization: (4)[8]: -- finalizing sequence #9 --
│ │ │ │ ---finalization: (5)[6]: -- finalizing sequence #7 --
│ │ │ │ ---finalization: (6)[7]: -- finalizing sequence #8 --
│ │ │ │ ---finalization: (7)[2]: -- finalizing sequence #3 --
│ │ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (1)[8]: -- finalizing sequence #9 --
│ │ │ │ +--finalization: (2)[6]: -- finalizing sequence #7 --
│ │ │ │ +--finalization: (3)[1]: -- finalizing sequence #2 --
│ │ │ │ +--finalization: (4)[5]: -- finalizing sequence #6 --
│ │ │ │ +--finalization: (5)[3]: -- finalizing sequence #4 --
│ │ │ │ +--finalization: (6)[2]: -- finalizing sequence #3 --
│ │ │ │ +--finalization: (7)[4]: -- finalizing sequence #5 --
│ │ │ │ +--finalization: (8)[7]: -- finalizing sequence #8 --
│ │ │ │  --finalization: (9)[0]: -- finalizing sequence #1 --
│ │ │ │ ---finalization: (8)[4]: -- finalizing sequence #5 --
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  This function should mainly be used for debugging. Having a large number of
│ │ │ │  finalizers might degrade the performance of the program. Moreover, registering
│ │ │ │  two or more objects that are members of a circular chain of pointers for
│ │ │ │  finalization will result in a memory leak, with none of the objects in the
│ │ │ │  chain being freed, even if nothing else points to any of them.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_remove__Directory.html
│ │ │ @@ -76,29 +76,29 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : dir = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11650-0/0
    
│ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │              
    
│ │ │                
    i2 : makeDirectory dir
│ │ │  
│ │ │ -o2 = /tmp/M2-11650-0/0
    
│ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │              
    
│ │ │                
    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/M2-11650-0/0
│ │ │ │ +o1 = /tmp/M2-12480-0/0
│ │ │ │  i2 : makeDirectory dir
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11650-0/0
│ │ │ │ +o2 = /tmp/M2-12480-0/0
│ │ │ │  i3 : readDirectory dir
│ │ │ │  
│ │ │ │ -o3 = {., ..}
│ │ │ │ +o3 = {.., .}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : removeDirectory dir
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_e_a_d_D_i_r_e_c_t_o_r_y -- read the contents of a directory
│ │ │ │      * _m_a_k_e_D_i_r_e_c_t_o_r_y -- make a directory
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__Path.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
    Description
    
│ │ │          
    This string may be concatenated with an absolute path to get one understandable by external programs.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-11116-0/0
    
│ │ │ +o1 = /tmp/M2-11406-0/0
│ │ │              
    
│ │ │                
    i2 : rootPath | fn
│ │ │  
│ │ │ -o2 = /tmp/M2-11116-0/0
    
│ │ │ +o2 = /tmp/M2-11406-0/0
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │          
    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/M2-11116-0/0
│ │ │ │ +o1 = /tmp/M2-11406-0/0
│ │ │ │  i2 : rootPath | fn
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-11116-0/0
│ │ │ │ +o2 = /tmp/M2-11406-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_U_R_I
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_P_a_t_h is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2025:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_root__U__R__I.html
│ │ │ @@ -70,22 +70,22 @@
│ │ │          
      Description
      
│ │ │          
      This string may be concatenated with an absolute path to get one understandable by an external browser.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12417-0/0
      
│ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │              
      
│ │ │                
      i2 : rootURI | fn
│ │ │  
│ │ │ -o2 = file:///tmp/M2-12417-0/0
      
│ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │              
      
│ │ │        
    
│ │ │        
    
│ │ │          
    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/M2-12417-0/0
│ │ │ │ +o1 = /tmp/M2-14047-0/0
│ │ │ │  i2 : rootURI | fn
│ │ │ │  
│ │ │ │ -o2 = file:///tmp/M2-12417-0/0
│ │ │ │ +o2 = file:///tmp/M2-14047-0/0
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _r_o_o_t_P_a_t_h
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _r_o_o_t_U_R_I is a _s_t_r_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ │ │  1.26.06+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_system.m2:2041:0.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_saving_sppolynomials_spand_spmatrices_spin_spfiles.html
│ │ │ @@ -95,22 +95,22 @@
│ │ │  o4 : R-module, submodule of R
│ │ │              
    
│ │ │                
    i5 : f = temporaryFileName()
│ │ │  
│ │ │ -o5 = /tmp/M2-12265-0/0
    
│ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │              
    
│ │ │                
    i6 : f << toString (p,m,M) << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │  
│ │ │  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/M2-12265-0/0
│ │ │ │ +o5 = /tmp/M2-13735-0/0
│ │ │ │  i6 : f << toString (p,m,M) << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12265-0/0
│ │ │ │ +o6 = /tmp/M2-13735-0/0
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -73,22 +73,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : serialNumber asdf
│ │ │  
│ │ │ -o1 = 1367325
    
│ │ │ +o1 = 1367343
│ │ │              
    
│ │ │                
    i2 : serialNumber foo
│ │ │  
│ │ │ -o2 = 1367327
    
│ │ │ +o2 = 1367345
│ │ │              
    
│ │ │                
    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 = 1367325
│ │ │ │ +o1 = 1367343
│ │ │ │  i2 : serialNumber foo
│ │ │ │  
│ │ │ │ -o2 = 1367327
│ │ │ │ +o2 = 1367345
│ │ │ │  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
│ │ │ @@ -371,21 +371,21 @@
│ │ │                  		
│ │ │                
    i28 : B = mutableMatrix(CC_53, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i30 : time X = solve(A,B);
│ │ │ - -- used 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
    
│ │ │ + -- used 0.00020554s (cpu); 0.000199105s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
    
│ │ │ + -- used 8.6518e-05s (cpu); 8.6233e-05s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i32 : norm(A*X-B)
│ │ │  
│ │ │  o32 = 5.111850690840453e-15
│ │ │ @@ -416,21 +416,21 @@
│ │ │              
    		
│ │ │                
    i36 : B = mutableMatrix(CC_100, N, 2); fillMatrix B;
    
│ │ │              
    
│ │ │                
    i38 : time X = solve(A,B);
│ │ │ - -- used 0.232067s (cpu); 0.232067s (thread); 0s (gc)
    
│ │ │ + -- used 0.117585s (cpu); 0.117586s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
    
│ │ │ + -- used 0.118674s (cpu); 0.118681s (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 0.000183394s (cpu); 0.000176741s (thread); 0s (gc)
│ │ │ │ + -- used 0.00020554s (cpu); 0.000199105s (thread); 0s (gc)
│ │ │ │  i31 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.000144531s (cpu); 0.000145172s (thread); 0s (gc)
│ │ │ │ + -- used 8.6518e-05s (cpu); 8.6233e-05s (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 0.232067s (cpu); 0.232067s (thread); 0s (gc)
│ │ │ │ + -- used 0.117585s (cpu); 0.117586s (thread); 0s (gc)
│ │ │ │  i39 : time X = solve(A,B, MaximalRank=>true);
│ │ │ │ - -- used 0.215747s (cpu); 0.215685s (thread); 0s (gc)
│ │ │ │ + -- used 0.118674s (cpu); 0.118681s (thread); 0s (gc)
│ │ │ │  i40 : norm(A*X-B)
│ │ │ │  
│ │ │ │  o40 = 1.491578274689709814082355885932e-28
│ │ │ │  
│ │ │ │  o40 : RR (of precision 100)
│ │ │ │  Giving the option ClosestFit=>true, in the case when the field is RR or CC,
│ │ │ │  uses a least squares algorithm to find a best fit solution.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__Directory_lp__String_cm__String_rp.html
│ │ │ @@ -85,93 +85,93 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : src = temporaryFileName() | "/"
│ │ │  
│ │ │ -o1 = /tmp/M2-12037-0/0/
    
│ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │              
    
│ │ │                
    i2 : dst = temporaryFileName() | "/"
│ │ │  
│ │ │ -o2 = /tmp/M2-12037-0/1/
    
│ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │              
    
│ │ │                
    i3 : makeDirectory (src|"a/")
│ │ │  
│ │ │ -o3 = /tmp/M2-12037-0/0/a/
    
│ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │              
    
│ │ │                
    i4 : makeDirectory (src|"b/")
│ │ │  
│ │ │ -o4 = /tmp/M2-12037-0/0/b/
    
│ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │              
    
│ │ │                
    i5 : makeDirectory (src|"b/c/")
│ │ │  
│ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
    
│ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │              
    
│ │ │                
    i6 : src|"a/f" << "hi there" << close
│ │ │  
│ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │  
│ │ │  o6 : File
    
│ │ │              
    
│ │ │                
    i7 : src|"a/g" << "hi there" << close
│ │ │  
│ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │  
│ │ │  o7 : File
    
│ │ │              
    
│ │ │                
    i8 : src|"b/c/g" << "ho there" << close
│ │ │  
│ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │  
│ │ │  o8 : File
    
│ │ │              
    
│ │ │                
    i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
    
│ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/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/M2-12037-0/1/b/c/g
│ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
    
│ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │              
    
│ │ │  Now we remove the files and directories we created.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -112,15 +112,15 @@
│ │ │  
│ │ │  o5 : Ideal of R
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    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/M2-12037-0/0/
│ │ │ │ +o1 = /tmp/M2-13267-0/0/
│ │ │ │  i2 : dst = temporaryFileName() | "/"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-12037-0/1/
│ │ │ │ +o2 = /tmp/M2-13267-0/1/
│ │ │ │  i3 : makeDirectory (src|"a/")
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-12037-0/0/a/
│ │ │ │ +o3 = /tmp/M2-13267-0/0/a/
│ │ │ │  i4 : makeDirectory (src|"b/")
│ │ │ │  
│ │ │ │ -o4 = /tmp/M2-12037-0/0/b/
│ │ │ │ +o4 = /tmp/M2-13267-0/0/b/
│ │ │ │  i5 : makeDirectory (src|"b/c/")
│ │ │ │  
│ │ │ │ -o5 = /tmp/M2-12037-0/0/b/c/
│ │ │ │ +o5 = /tmp/M2-13267-0/0/b/c/
│ │ │ │  i6 : src|"a/f" << "hi there" << close
│ │ │ │  
│ │ │ │ -o6 = /tmp/M2-12037-0/0/a/f
│ │ │ │ +o6 = /tmp/M2-13267-0/0/a/f
│ │ │ │  
│ │ │ │  o6 : File
│ │ │ │  i7 : src|"a/g" << "hi there" << close
│ │ │ │  
│ │ │ │ -o7 = /tmp/M2-12037-0/0/a/g
│ │ │ │ +o7 = /tmp/M2-13267-0/0/a/g
│ │ │ │  
│ │ │ │  o7 : File
│ │ │ │  i8 : src|"b/c/g" << "ho there" << close
│ │ │ │  
│ │ │ │ -o8 = /tmp/M2-12037-0/0/b/c/g
│ │ │ │ +o8 = /tmp/M2-13267-0/0/b/c/g
│ │ │ │  
│ │ │ │  o8 : File
│ │ │ │  i9 : symlinkDirectory(src,dst,Verbose=>true)
│ │ │ │ ---symlinking: ../../../0/b/c/g -> /tmp/M2-12037-0/1/b/c/g
│ │ │ │ ---symlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ │ ---symlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ │ +--symlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ │ +--symlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ │ +--symlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/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/M2-12037-0/1/b/c/g
│ │ │ │ ---unsymlinking: ../../0/a/g -> /tmp/M2-12037-0/1/a/g
│ │ │ │ ---unsymlinking: ../../0/a/f -> /tmp/M2-12037-0/1/a/f
│ │ │ │ +--unsymlinking: ../../0/a/g -> /tmp/M2-13267-0/1/a/g
│ │ │ │ +--unsymlinking: ../../0/a/f -> /tmp/M2-13267-0/1/a/f
│ │ │ │ +--unsymlinking: ../../../0/b/c/g -> /tmp/M2-13267-0/1/b/c/g
│ │ │ │  Now we remove the files and directories we created.
│ │ │ │  i12 : rm = d -> if isDirectory d then removeDirectory d else removeFile d
│ │ │ │  
│ │ │ │  o12 = rm
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : scan(reverse findFiles src, rm)
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_symlink__File.html
│ │ │ @@ -77,15 +77,15 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │            o dst, a _s_t_r_i_n_g
│ │ │ │      * Consequences:
│ │ │ │            o a symbolic link at the location in the directory tree specified by
│ │ │ │              dst is created, pointing to src
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  i1 : fn = temporaryFileName()
│ │ │ │  
│ │ │ │ -o1 = /tmp/M2-12094-0/0
│ │ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │ │  i2 : symlinkFile("qwert", fn)
│ │ │ │  i3 : fileExists fn
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : readlink fn
│ │ │ │  
│ │ │ │  o4 = qwert
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_temporary__File__Name.html
│ │ │ @@ -69,22 +69,22 @@
│ │ │        
    
│ │ │          
    Description
    
│ │ │  The file name is so unique that even with various suffixes appended, no collision with existing files will occur.  The files will be removed when the program terminates, unless it terminates as the result of an error.        
    
│ │ │                
    i1 : fn = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-12094-0/0
    
│ │ │ +o1 = /tmp/M2-13384-0/0
│ │ │              
    
│ │ │                
    i2 : symlinkFile("qwert", fn)
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : temporaryFileName () | ".tex"
│ │ │  
│ │ │ -o1 = /tmp/M2-13097-0/0.tex
    
│ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │              
    
│ │ │                
    i2 : temporaryFileName () | ".html"
│ │ │  
│ │ │ -o2 = /tmp/M2-13097-0/1.html
    
│ │ │ +o2 = /tmp/M2-15437-0/1.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/M2-13097-0/0.tex
│ │ │ │ +o1 = /tmp/M2-15437-0/0.tex
│ │ │ │  i2 : temporaryFileName () | ".html"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-13097-0/1.html
│ │ │ │ +o2 = /tmp/M2-15437-0/1.html
│ │ │ │  This function will work under Unix, and also under Windows if you have a
│ │ │ │  directory on the same drive called /tmp.
│ │ │ │  If the name of the temporary file will be given to an external program, it may
│ │ │ │  be necessary to concatenate it with _r_o_o_t_P_a_t_h or _r_o_o_t_U_R_I to enable the external
│ │ │ │  program to find the file.
│ │ │ │  The temporary file name is derived from the value of the environment variable
│ │ │ │  TMPDIR, if it has one.
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_time.html
│ │ │ @@ -64,15 +64,15 @@
│ │ │        
│ │ │        
    
│ │ │          
    Description
    
│ │ │  time e evaluates e, prints the amount of cpu time used, and returns the value of e.  The time used by the the current thread and garbage collection during the evaluation of e is also shown.        
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : time 3^30
│ │ │ - -- used 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ + -- used 1.6199e-05s (cpu); 5.786e-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 1.1622e-05s (cpu); 6.291e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.6199e-05s (cpu); 5.786e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o1 = 205891132094649
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_i_m_i_n_g -- time a computation
│ │ │ │      * _c_p_u_T_i_m_e -- seconds of cpu time used since Macaulay2 began
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_i_n_g -- time a computation using time elapsed
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_timing.html
│ │ │ @@ -59,24 +59,24 @@
│ │ │  timing e evaluates e and returns a list of type Time of the form {t,v}, where t is the number of seconds of cpu timing used, and v is the value of the expression.        
    
│ │ │  The default method for printing such timing results is to display the timing separately in a comment below the computed value.        
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : timing 3^30
│ │ │  
│ │ │  o1 = 205891132094649
│ │ │ -     -- .000013355 seconds
│ │ │ +     -- .000013321 seconds
│ │ │  
│ │ │  o1 : Time
    
│ │ │              
    
│ │ │                
    i2 : peek oo
│ │ │  
│ │ │ -o2 = Time{.000013355, 205891132094649}
    
│ │ │ +o2 = Time{.000013321, 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
│ │ │ │ -     -- .000013355 seconds
│ │ │ │ +     -- .000013321 seconds
│ │ │ │  
│ │ │ │  o1 : Time
│ │ │ │  i2 : peek oo
│ │ │ │  
│ │ │ │ -o2 = Time{.000013355, 205891132094649}
│ │ │ │ +o2 = Time{.000013321, 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/_using_spa_sp__Hilbert_sphint_spfor_spa_sp__Groebner_spbasis_spcalculation.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │  
│ │ │  o3 : ZZ[T]
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime Mgb = gb M
│ │ │ - -- 1.4495s elapsed
│ │ │ + -- 1.73571s elapsed
│ │ │  
│ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o4 : GroebnerBasis
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .415018s elapsed
│ │ │ + -- .463688s elapsed
│ │ │  
│ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o6 : GroebnerBasis
    
│ │ │              
    
│ │ │ @@ -162,15 +162,15 @@
│ │ │  
│ │ │  o10 : Ideal of R
    
│ │ │              
    
│ │ │                
    i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ - -- .517879s elapsed
│ │ │ + -- .548971s elapsed
│ │ │  
│ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │  
│ │ │  o11 : GroebnerBasis
    
│ │ │              
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,24 +29,24 @@
│ │ │ │  i3 : hf = poincare ideal (a^4,b^5,c^5,d^6)
│ │ │ │  
│ │ │ │            4     5    6     9     10     11    14     15    16    20
│ │ │ │  o3 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │ │  
│ │ │ │  o3 : ZZ[T]
│ │ │ │  i4 : elapsedTime Mgb = gb M
│ │ │ │ - -- 1.4495s elapsed
│ │ │ │ + -- 1.73571s elapsed
│ │ │ │  
│ │ │ │  o4 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o4 : GroebnerBasis
│ │ │ │  i5 : M = ideal M_*;
│ │ │ │  
│ │ │ │  o5 : Ideal of R
│ │ │ │  i6 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ │ - -- .415018s elapsed
│ │ │ │ + -- .463688s elapsed
│ │ │ │  
│ │ │ │  o6 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o6 : GroebnerBasis
│ │ │ │  However, obtaining the Hilbert function is not always easy to provide in this
│ │ │ │  way. In this case, one must work with the _d_e_g_r_e_e_s_R_i_n_g of the ring in question.
│ │ │ │  i7 : S = degreesRing R
│ │ │ │ @@ -65,15 +65,15 @@
│ │ │ │  o9 = 1 - T  - 2T  - T  + 2T  + 2T   + 2T   - T   - 2T   - T   + T
│ │ │ │  
│ │ │ │  o9 : S
│ │ │ │  i10 : M = ideal M_*;
│ │ │ │  
│ │ │ │  o10 : Ideal of R
│ │ │ │  i11 : elapsedTime Mgb = gb(M, Hilbert => hf)
│ │ │ │ - -- .517879s elapsed
│ │ │ │ + -- .548971s elapsed
│ │ │ │  
│ │ │ │  o11 = GroebnerBasis[status: done; S-pairs encountered up to degree 17]
│ │ │ │  
│ │ │ │  o11 : GroebnerBasis
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _c_a_n_U_s_e_H_i_l_b_e_r_t_H_i_n_t -- whether certain Groebner computations can make use
│ │ │ │        of the Hilbert function
│ │ ├── ./usr/share/doc/Macaulay2/Macaulay2Doc/html/_version.html
│ │ │ @@ -110,15 +110,15 @@
│ │ │                 "memtailor version" => 1.4
│ │ │                 "mpfi version" => 1.5.5
│ │ │                 "mpfr version" => 4.2.2
│ │ │                 "mpsolve version" => 3.2.2
│ │ │                 "mysql version" => not present
│ │ │                 "normaliz version" => 3.11.1
│ │ │                 "ntl version" => 11.5.1
│ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ +               "operating system release" => 6.12.100+deb13-cloud-amd64
│ │ │                 "operating system" => Linux
│ │ │                 "packages" => Style FirstPackage Macaulay2Doc Parsing Classic Browse Benchmark Text SimpleDoc PackageTemplate Saturation PrimaryDecomposition FourierMotzkin Dmodules WeylAlgebras HolonomicSystems BernsteinSato ConnectionMatrices Depth Elimination GenericInitialIdeal IntegralClosure HyperplaneArrangements LexIdeals Markov NoetherNormalization Points ReesAlgebra Regularity SchurRings SymmetricPolynomials SchurFunctors SimplicialComplexes LLLBases TangentCone ChainComplexExtras Varieties Schubert2 PushForward LocalRings BoijSoederberg BGG Bruns InvolutiveBases ConwayPolynomials EdgeIdeals FourTiTwo StatePolytope Polyhedra Truncations Polymake gfanInterface PieriMaps Normaliz Posets XML OpenMath SCSCP RationalPoints MapleInterface ConvexInterface SRdeformations NumericalAlgebraicGeometry BeginningMacaulay2 FormalGroupLaws Graphics WeylGroups HodgeIntegrals Cyclotomic Binomials Kronecker Nauty ToricVectorBundles ModuleDeformations PHCpack SimplicialDecomposability BooleanGB AdjointIdeal Parametrization Serialization NAGtypes NormalToricVarieties DGAlgebras Graphs GraphicalModels BIBasis KustinMiller Units NautyGraphs VersalDeformations CharacteristicClasses RandomIdeals RandomObjects RandomCurves TensorComplexes MonomialAlgebras QthPower EliminationMatrices EllipticIntegrals Triplets CompleteIntersectionResolutions EagonResolution MCMApproximations MultiplierIdeals InvariantRing QuillenSuslin EnumerationCurves Book3264Examples WeilDivisors EllipticCurves HighestWeights MinimalPrimes Bertini TorAlgebra Permanents BinomialEdgeIdeals TateOnProducts LatticePolytopes FiniteFittingIdeals HigherCIOperators LieAlgebraRepresentations ConformalBlocks M0nbar AnalyzeSheafOnP1 MultiplierIdealsDim2 RunExternalM2 NumericalSchubertCalculus ToricTopology Cremona Resultants VectorFields SLPexpressions Miura ResidualIntersections Visualize EquivariantGB ExampleSystems RationalMaps FastMinors RandomPoints SwitchingFields SpectralSequences SectionRing K3Carpets ChainComplexOperations NumericalCertification PhylogeneticTrees MonodromySolver ReactionNetworks PackageCitations NumericSolutions GradedLieAlgebras InverseSystems Pullback EngineTests SVDComplexes RandomComplexes CohomCalg Topcom Triangulations ReflexivePolytopesDB AbstractToricVarieties TestIdeals FrobeniusThresholds NonPrincipalTestIdeals Seminormalization AlgebraicSplines TriangularSets Chordal Tropical SymbolicPowers Complexes OldChainComplexes GroebnerWalk RandomMonomialIdeals Matroids NumericalImplicitization CoincidentRootLoci RelativeCanonicalResolution RandomCurvesOverVerySmallFiniteFields StronglyStableIdeals SLnEquivariantMatrices CorrespondenceScrolls NCAlgebra SpaceCurves ExteriorIdeals ToricInvariants SegreClasses SemidefiniteProgramming SumsOfSquares MultiGradedRationalMap AssociativeAlgebras VirtualResolutions Quasidegrees DiffAlg DeterminantalRepresentations FGLM SpechtModule SchurComplexes SimplicialPosets SlackIdeals PositivityToricBundles SparseResultants DecomposableSparseSystems MixedMultiplicity PencilsOfQuadrics ThreadedGB AdjunctionForSurfaces VectorGraphics GKMVarieties MonomialIntegerPrograms NoetherianOperators Hadamard StatGraphs GraphicalModelsMLE EigenSolver MultiplicitySequence ResolutionsOfStanleyReisnerRings NumericalLinearAlgebra ResLengthThree MonomialOrbits MultiprojectiveVarieties SpecialFanoFourfolds RationalPoints2 SuperLinearAlgebra SubalgebraBases AInfinity LinearTruncations ThinSincereQuivers Python BettiCharacters Jets FunctionFieldDesingularization HomotopyLieAlgebra TSpreadIdeals RealRoots ExteriorModules K3Surfaces GroebnerStrata QuaternaryQuartics CotangentSchubert OnlineLookup MergeTeX Probability Isomorphism DirectSummands CodingTheory WhitneyStratifications JSON ForeignFunctions GeometricDecomposability PseudomonomialPrimaryDecomposition PolyominoIdeals MatchingFields CellularResolutions SagbiGbDetection A1BrouwerDegrees QuadraticIdealExamplesByRoos TerraciniLoci MatrixSchubert RInterface OIGroebnerBases PlaneCurveLinearSeries Valuations SchurVeronese VNumber TropicalToric MultigradedBGG AbstractSimplicialComplexes MultigradedImplicitization Msolve Permutations SCMAlgebras NumericalSemigroups ExteriorExtensions Oscillators IncidenceCorrespondenceCohomology ToricHigherDirectImages Brackets IntegerProgramming GameTheory AllMarkovBases Tableaux CpMackeyFunctors JSONRPC SimplicialModules MatrixFactorizations PathSignatures MacaulayPosets MRDI EliminationTemplates WittVectors Padic WeierstrassSemigroups ResultantComplexes EuclideanDistanceDegree NeuralIdeals TestAudit LanguageServer
│ │ │                 "pointer size" => 8
│ │ │                 "python version" => 3.14.9
│ │ │                 "readline version" => 8.3
│ │ │                 "scscp version" => not present
│ │ │                 "tbb version" => 2023.1
│ │ │ ├── html2text {}
│ │ │ │ @@ -67,15 +67,15 @@
│ │ │ │                 "memtailor version" => 1.4
│ │ │ │                 "mpfi version" => 1.5.5
│ │ │ │                 "mpfr version" => 4.2.2
│ │ │ │                 "mpsolve version" => 3.2.2
│ │ │ │                 "mysql version" => not present
│ │ │ │                 "normaliz version" => 3.11.1
│ │ │ │                 "ntl version" => 11.5.1
│ │ │ │ -               "operating system release" => 6.12.113+deb13-amd64
│ │ │ │ +               "operating system release" => 6.12.100+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/MacaulayPosets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  UG9zZXRNYXA=
│ │ │  #:len=820
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNsYXNzIG9mIGFsbCBtb3JwaGlz
│ │ │  bXMgYmV0d2VlbiBwb3NldHMiLCAibGluZW51bSIgPT4gMTYyNCwgU2VlQWxzbyA9PiBESVZ7SEVB
│ │ ├── ./usr/share/doc/Macaulay2/MapleInterface/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  c3RvcmU=
│ │ │  #:len=651
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RvcmUgcmVzdWx0IG9mIGEgTWFwbGUg
│ │ │  Y29tcHV0YXRpb24gaW4gYSBmaWxlLiIsIERlc2NyaXB0aW9uID0+IDE6KERJVntQQVJBe1RFWHsi
│ │ ├── ./usr/share/doc/Macaulay2/Markov/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  Z2F1c3NJZGVhbChSaW5nLExpc3Qp
│ │ │  #:len=239
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjE4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnYXVzc0lkZWFsLFJpbmcsTGlzdCksImdhdXNzSWRl
│ │ ├── ./usr/share/doc/Macaulay2/Markov/example-output/___Markov.out
│ │ │ @@ -70,15 +70,15 @@
│ │ │       |   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2|   1,2,2,2 2,2,2,1    1,2,2,1 2,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │       |- p       p        + p       p       |- p       p        + p       p       |
│ │ │       |   1,1,2,1 1,2,1,1    1,1,1,1 1,2,2,1|   1,1,2,2 1,2,1,2    1,1,1,2 1,2,2,2|
│ │ │       +-------------------------------------+-------------------------------------+
│ │ │  
│ │ │  i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ + -- used 1.62869s (cpu); 1.15174s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ ├── ./usr/share/doc/Macaulay2/Markov/html/index.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │          
    
│ │ │            
    This ideal has 5 primary components.  The first is the one that has statistical significance. The significance of the other components is still poorly understood.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i8 : time netList primaryDecomposition J
│ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ + -- used 1.62869s (cpu); 1.15174s (thread); 0s (gc)
│ │ │  
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │  o8 = |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,2,1,2   1,2,1,1   1,1,2,2 2,1,2,1    1,1,2,1 2,1,2,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │       +---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
│ │ │       |ideal (p       , p       , p       , p       , p       p        - p       p       , p       p        - p       p       )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
│ │ │       |        1,2,2,2   1,2,2,1   1,1,2,2   1,1,2,1   1,2,1,2 2,2,1,1    1,2,1,1 2,2,1,2   1,1,1,2 2,1,1,1    1,1,1,1 2,1,1,2                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
│ │ │ ├── html2text {}
│ │ │ │ @@ -102,15 +102,15 @@
│ │ │ │  1,2,2,2|
│ │ │ │       +-------------------------------------+-----------------------------------
│ │ │ │  --+
│ │ │ │  This ideal has 5 primary components. The first is the one that has statistical
│ │ │ │  significance. The significance of the other components is still poorly
│ │ │ │  understood.
│ │ │ │  i8 : time netList primaryDecomposition J
│ │ │ │ - -- used 1.78876s (cpu); 1.24178s (thread); 0s (gc)
│ │ │ │ + -- used 1.62869s (cpu); 1.15174s (thread); 0s (gc)
│ │ │ │  
│ │ │ │       +-------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ │ │  -------------------------------------------------------------------------------
│ │ ├── ./usr/share/doc/Macaulay2/MatchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  YWxnZWJyYWljTWF0cm9pZENpcmN1aXRz
│ │ │  #:len=1461
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIGJhc2VzIG9mIHRoZSBhbGdlYnJh
│ │ │  aWMgbWF0cm9pZCIsICJsaW5lbnVtIiA9PiAxOTI3LCBJbnB1dHMgPT4ge1NQQU57VFR7IkwifSwi
│ │ ├── ./usr/share/doc/Macaulay2/MatrixFactorizations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=41
│ │ │  SG9tKFpaZEZhY3Rvcml6YXRpb25NYXAsWlpkRmFjdG9yaXphdGlvbik=
│ │ │  #:len=413
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMzMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoSG9tLFpaZEZhY3Rvcml6YXRpb25NYXAsWlpkRmFj
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aXNBU01VbmlvbihMaXN0KQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNBU01VbmlvbixMaXN0KSwiaXNBU01VbmlvbihM
│ │ ├── ./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 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ + -- used 0.0843846s (cpu); 0.0289058s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
│ │ │  
│ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ + -- used 0.0178294s (cpu); 0.0178368s (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 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ + -- used 0.000257429s (cpu); 0.000252396s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
│ │ │  
│ │ │  i17 : time regularity comodule I
│ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ + -- used 0.0211422s (cpu); 0.0211476s (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 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ + -- used 0.00456881s (cpu); 0.0045671s (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 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ + -- used 0.00346967s (cpu); 0.00347142s (thread); 0s (gc)
│ │ │  
│ │ │        2        2      2               2
│ │ │  o3 = x x x  - x x  - x x  - x x x  + x  + x x  + x x
│ │ │        1 2 3    1 2    1 3    1 2 3    1    1 2    1 3
│ │ │  
│ │ │  o3 : QQ[x ..x ]
│ │ │           1   4
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_sp__A__S__M_spvarieties.html
│ │ │ @@ -388,23 +388,23 @@
│ │ │          
    
│ │ │            
    Additionally, this package facilitates investigating homological invariants of ASM ideals such as regularity (schubertRegularity) and codimension (schubertCodim). efficiently by computing the associated invariants for their antidiagonal initial ideals, which are known to be squarefree by [Wei17]. Therefore the extremal Betti numbers (which encode regularity, depth, and projective dimension) of ASM ideals coincide with those of their antidiagonal initial ideals by [CV20].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time schubertRegularity B
│ │ │ - -- used 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ + -- used 0.0843846s (cpu); 0.0289058s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = 1
    
│ │ │              
    
│ │ │                
    i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ + -- used 0.0178294s (cpu); 0.0178368s (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 0.118965s (cpu); 0.0492731s (thread); 0s (gc)
│ │ │ │ + -- used 0.0843846s (cpu); 0.0289058s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = 1
│ │ │ │  i24 : time regularity comodule schubertDeterminantalIdeal B
│ │ │ │ - -- used 0.0269913s (cpu); 0.026995s (thread); 0s (gc)
│ │ │ │ + -- used 0.0178294s (cpu); 0.0178368s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o24 = 1
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg AASSMM vvaarriieettiieess **********
│ │ │ │      * _i_s_P_a_r_t_i_a_l_A_S_M_(_M_a_t_r_i_x_) -- whether a matrix is a partial alternating sign
│ │ │ │        matrix
│ │ │ │      * _p_a_r_t_i_a_l_A_S_M_T_o_A_S_M_(_M_a_t_r_i_x_) -- extend a partial alternating sign matrix to an
│ │ │ │        alternating sign matrix
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/___Investigating_spmatrix_sp__Schubert_spvarieties.html
│ │ │ @@ -320,23 +320,23 @@
│ │ │          
    
│ │ │            
    Finally, this package contains functions for investigating homological invariants of matrix Schubert varieties efficiently through combinatorial algorithms produced in [PSW24] via schubertRegularityschubertCodim.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i16 : time schubertRegularity p
│ │ │ - -- used 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ + -- used 0.000257429s (cpu); 0.000252396s (thread); 0s (gc)
│ │ │  
│ │ │  o16 = 5
    
│ │ │              
    
│ │ │                
    i17 : time regularity comodule I
│ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ + -- used 0.0211422s (cpu); 0.0211476s (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 0.000305362s (cpu); 0.000300744s (thread); 0s (gc)
│ │ │ │ + -- used 0.000257429s (cpu); 0.000252396s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o16 = 5
│ │ │ │  i17 : time regularity comodule I
│ │ │ │ - -- used 0.0193429s (cpu); 0.0193488s (thread); 0s (gc)
│ │ │ │ + -- used 0.0211422s (cpu); 0.0211476s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 = 5
│ │ │ │  ********** FFuunnccttiioonnss ffoorr iinnvveessttiiggaattiinngg mmaattrriixx SScchhuubbeerrtt vvaarriieettiieess **********
│ │ │ │      * _a_n_t_i_D_i_a_g_I_n_i_t_(_L_i_s_t_) -- compute the (unique) antidiagonal initial ideal of
│ │ │ │        an ASM ideal
│ │ │ │      * _r_a_n_k_T_a_b_l_e_(_L_i_s_t_) -- compute a table of rank conditions that determines the
│ │ │ │        corresponding ASM or matrix Schubert variety
│ │ ├── ./usr/share/doc/Macaulay2/MatrixSchubert/html/_grothendieck__Polynomial.html
│ │ │ @@ -85,28 +85,28 @@
│ │ │  
│ │ │  o1 : List
    
│ │ │              
    
│ │ │                
    i2 : time grothendieckPolynomial w
│ │ │ - -- used 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ + -- used 0.00456881s (cpu); 0.0045671s (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 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ + -- used 0.00346967s (cpu); 0.00347142s (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 0.0048739s (cpu); 0.00487285s (thread); 0s (gc)
│ │ │ │ + -- used 0.00456881s (cpu); 0.0045671s (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 0.00246725s (cpu); 0.00245375s (thread); 0s (gc)
│ │ │ │ + -- used 0.00346967s (cpu); 0.00347142s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  ZmxhdHMoTWF0cm9pZCxaWixTdHJpbmcp
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEzOSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxhdHMsTWF0cm9pZCxaWixTdHJpbmcpLCJmbGF0
│ │ ├── ./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 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ + -- used 0.0531863s (cpu); 0.0531874s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
│ │ │  
│ │ │  i11 : time fVector R10
│ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ + -- used 0.0392037s (cpu); 0.0390752s (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 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ + -- used 0.000190236s (cpu); 0.000189168s (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);
│ │ │ - -- .0437144s elapsed
│ │ │ + -- .0406338s 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 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
│ │ │ + -- used 0.038272s (cpu); 0.0382701s (thread); 0s (gc)
│ │ │  
│ │ │  i8 : #autF7
│ │ │  
│ │ │  o8 = 168
│ │ │  
│ │ │  i9 :
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/example-output/_has__Minor.out
│ │ │ @@ -9,12 +9,12 @@
│ │ │  o1 : Sequence
│ │ │  
│ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │  
│ │ │  o2 = false
│ │ │  
│ │ │  i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ + -- used 1.73835s (cpu); 1.15934s (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 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ + -- used 0.013819s (cpu); 0.0138171s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -56,15 +56,15 @@
│ │ │  i7 : N = relabel M6
│ │ │  
│ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │  
│ │ │  o7 : Matroid
│ │ │  
│ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ + -- used 2.59063s (cpu); 2.24639s (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 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ + -- used 0.000570851s (cpu); 0.000566223s (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
│ │ │ - -- .0127952s elapsed
│ │ │ + -- .0108991s elapsed
│ │ │  
│ │ │  o6 = HashTable{0 => 1 }
│ │ │                 1 => 6
│ │ │                 2 => 15
│ │ │                 3 => 20
│ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/___Matroid.html
│ │ │ @@ -153,23 +153,23 @@
│ │ │  
│ │ │  o9 : List
│ │ │              
    
│ │ │                
    i10 : time isWellDefined R10
│ │ │ - -- used 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ + -- used 0.0531863s (cpu); 0.0531874s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    
│ │ │              
    
│ │ │                
    i11 : time fVector R10
│ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ + -- used 0.0392037s (cpu); 0.0390752s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = HashTable{0 => 1 }
│ │ │                  1 => 10
│ │ │                  2 => 45
│ │ │                  3 => 75
│ │ │                  4 => 30
│ │ │                  5 => 1
│ │ │ @@ -187,15 +187,15 @@
│ │ │  
│ │ │  o12 : List
    
│ │ │              
    
│ │ │                
    i13 : time fVector R10
│ │ │ - -- used 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ + -- used 0.000190236s (cpu); 0.000189168s (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 0.0514784s (cpu); 0.0514772s (thread); 0s (gc)
│ │ │ │ + -- used 0.0531863s (cpu); 0.0531874s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : time fVector R10
│ │ │ │ - -- used 0.0421595s (cpu); 0.0421627s (thread); 0s (gc)
│ │ │ │ + -- used 0.0392037s (cpu); 0.0390752s (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 0.00021336s (cpu); 0.000213109s (thread); 0s (gc)
│ │ │ │ + -- used 0.000190236s (cpu); 0.000189168s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 = HashTable{0 => 1 }
│ │ │ │                  1 => 10
│ │ │ │                  2 => 45
│ │ │ │                  3 => 75
│ │ │ │                  4 => 30
│ │ │ │                  5 => 1
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_all__Minors.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : Matroid
    
│ │ │              
    
│ │ │                
    i3 : elapsedTime L = allMinors(V, U25);
│ │ │ - -- .0437144s elapsed
    
│ │ │ + -- .0406338s 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);
│ │ │ │ - -- .0437144s elapsed
│ │ │ │ + -- .0406338s elapsed
│ │ │ │  i4 : #L
│ │ │ │  
│ │ │ │  o4 = 64
│ │ │ │  i5 : netList L_{0..4}
│ │ │ │  
│ │ │ │       +----------+-------+
│ │ │ │  o5 = |set {5, 3}|set {2}|
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_get__Isos.html
│ │ │ @@ -140,15 +140,15 @@
│ │ │  
│ │ │  o6 : Matroid
│ │ │              
    
│ │ │                
    i7 : time autF7 = getIsos(F7, F7);
│ │ │ - -- used 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
    
│ │ │ + -- used 0.038272s (cpu); 0.0382701s (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 0.0390737s (cpu); 0.0390572s (thread); 0s (gc)
│ │ │ │ + -- used 0.038272s (cpu); 0.0382701s (thread); 0s (gc)
│ │ │ │  i8 : #autF7
│ │ │ │  
│ │ │ │  o8 = 168
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ │ │        isomorphic matroids
│ │ │ │      * _q_u_i_c_k_I_s_o_m_o_r_p_h_i_s_m_T_e_s_t -- quick checks for isomorphism between matroids
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_has__Minor.html
│ │ │ @@ -101,15 +101,15 @@
│ │ │  
│ │ │  o2 = false
│ │ │              
    
│ │ │                
    i3 : time hasMinor(M6, M5)
│ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ + -- used 1.73835s (cpu); 1.15934s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │       elements, a "matroid" of rank 5 on 15 elements)
│ │ │ │  
│ │ │ │  o1 : Sequence
│ │ │ │  i2 : hasMinor(M4, uniformMatroid(2,4))
│ │ │ │  
│ │ │ │  o2 = false
│ │ │ │  i3 : time hasMinor(M6, M5)
│ │ │ │ - -- used 2.4626s (cpu); 1.80647s (thread); 0s (gc)
│ │ │ │ + -- used 1.73835s (cpu); 1.15934s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _m_i_n_o_r -- minor of matroid
│ │ │ │      * _i_s_B_i_n_a_r_y -- whether a matroid is representable over F_2
│ │ │ │  ********** WWaayyss ttoo uussee hhaassMMiinnoorr:: **********
│ │ │ │      * hasMinor(Matroid,Matroid)
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_isomorphism_lp__Matroid_cm__Matroid_rp.html
│ │ │ @@ -123,15 +123,15 @@
│ │ │  
│ │ │  o4 : Matroid
    
│ │ │              
    
│ │ │                
    i5 : time isomorphism(M5, minorM6)
│ │ │ - -- used 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ + -- used 0.013819s (cpu); 0.0138171s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = HashTable{0 => 1}
│ │ │                 1 => 0
│ │ │                 2 => 3
│ │ │                 3 => 2
│ │ │                 4 => 6
│ │ │                 5 => 5
│ │ │ @@ -169,15 +169,15 @@
│ │ │  
│ │ │  o7 : Matroid
    
│ │ │              
    
│ │ │                
    i8 : time phi = isomorphism(N,M6)
│ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ + -- used 2.59063s (cpu); 2.24639s (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 0.0170407s (cpu); 0.017038s (thread); 0s (gc)
│ │ │ │ + -- used 0.013819s (cpu); 0.0138171s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{0 => 1}
│ │ │ │                 1 => 0
│ │ │ │                 2 => 3
│ │ │ │                 3 => 2
│ │ │ │                 4 => 6
│ │ │ │                 5 => 5
│ │ │ │ @@ -74,15 +74,15 @@
│ │ │ │  o6 : HashTable
│ │ │ │  i7 : N = relabel M6
│ │ │ │  
│ │ │ │  o7 = a "matroid" of rank 5 on 15 elements
│ │ │ │  
│ │ │ │  o7 : Matroid
│ │ │ │  i8 : time phi = isomorphism(N,M6)
│ │ │ │ - -- used 3.84691s (cpu); 3.25885s (thread); 0s (gc)
│ │ │ │ + -- used 2.59063s (cpu); 2.24639s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = HashTable{0 => 11 }
│ │ │ │                 1 => 0
│ │ │ │                 2 => 1
│ │ │ │                 3 => 6
│ │ │ │                 4 => 9
│ │ │ │                 5 => 8
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_quick__Isomorphism__Test.html
│ │ │ @@ -142,15 +142,15 @@
│ │ │  
│ │ │  o9 = true
    
│ │ │              
    
│ │ │                
    i10 : time quickIsomorphismTest(M0, M1)
│ │ │ - -- used 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ + -- used 0.000570851s (cpu); 0.000566223s (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 0.000721082s (cpu); 0.000717165s (thread); 0s (gc)
│ │ │ │ + -- used 0.000570851s (cpu); 0.000566223s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = false
│ │ │ │  i11 : value oo === false
│ │ │ │  
│ │ │ │  o11 = true
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_o_m_o_r_p_h_i_s_m_(_M_a_t_r_o_i_d_,_M_a_t_r_o_i_d_) -- computes an isomorphism between
│ │ ├── ./usr/share/doc/Macaulay2/Matroids/html/_set__Representation.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime fVector M
│ │ │ - -- .0127952s elapsed
│ │ │ + -- .0108991s 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
│ │ │ │ - -- .0127952s elapsed
│ │ │ │ + -- .0108991s elapsed
│ │ │ │  
│ │ │ │  o6 = HashTable{0 => 1 }
│ │ │ │                 1 => 6
│ │ │ │                 2 => 15
│ │ │ │                 3 => 20
│ │ │ │                 4 => 1
│ │ ├── ./usr/share/doc/Macaulay2/MergeTeX/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  bWVyZ2VUZVgoLi4uLFBhdGg9Pi4uLik=
│ │ │  #:len=236
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1ttZXJnZVRlWCxQYXRoXSwibWVyZ2VUZVgoLi4uLFBh
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  ZGVjb21wb3NlKElkZWFsLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1soZGVjb21wb3NlLElkZWFsKSxTdHJhdGVneV0sImRl
│ │ ├── ./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 .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00112391)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0309643)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000340306)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000019748)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000630212
│ │ │ - -- .0384912s elapsed
│ │ │ + --  Time taken : .000713235
│ │ │ + -- .0270881s 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)
│ │ │ - -- .000442257s elapsed
│ │ │ + -- .000510632s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
│ │ │  
│ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0121996s elapsed
│ │ │ + -- .0185482s 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)
│ │ │ - -- .151496s elapsed
│ │ │ + -- .0801s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00157389s elapsed
│ │ │ + -- .00173948s elapsed
│ │ │  
│ │ │  o8 = true
│ │ │  
│ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00123922s elapsed
│ │ │ + -- .0013597s elapsed
│ │ │  
│ │ │  o9 = false
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/___Hybrid.html
│ │ │ @@ -77,21 +77,21 @@
│ │ │  
│ │ │  o3 : Ideal of R
    
│ │ │              
    
│ │ │                
    i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid{Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim = 0
│ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim = 0
│ │ │ +  Strategy: Linear            (time .00112391)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Birational        (time .0309643)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: Factorization     (time .000340306)  #primes = 0 #prunedViaCodim = 0
│ │ │ +  Strategy: DecomposeMonomials(time .000019748)  #primes = 1 #prunedViaCodim = 0
│ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ - --  Time taken : .000630212
│ │ │ - -- .0384912s elapsed
    
│ │ │ + --  Time taken : .000713235
│ │ │ + -- .0270881s elapsed
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │          
    See also
    
│ │ │          
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,23 +11,23 @@
│ │ │ │  i1 : debug MinimalPrimes
│ │ │ │  i2 : R = ZZ/101[w..z];
│ │ │ │  i3 : I = ideal(w*x^2-42*y*z, x^6+12*w*y+x^3*z, w^2-47*x^4*z-47*x*z^2);
│ │ │ │  
│ │ │ │  o3 : Ideal of R
│ │ │ │  i4 : elapsedTime minimalPrimes(ideal I_*, Strategy => Hybrid
│ │ │ │  {Linear,Birational,Factorization,DecomposeMonomials}, Verbosity => 2);
│ │ │ │ -  Strategy: Linear            (time .00103587)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Birational        (time .0131375)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ -  Strategy: Factorization     (time .000340419)  #primes = 0 #prunedViaCodim =
│ │ │ │ +  Strategy: Linear            (time .00112391)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Birational        (time .0309643)  #primes = 0 #prunedViaCodim = 0
│ │ │ │ +  Strategy: Factorization     (time .000340306)  #primes = 0 #prunedViaCodim =
│ │ │ │  0
│ │ │ │ -  Strategy: DecomposeMonomials(time .000024566)  #primes = 1 #prunedViaCodim =
│ │ │ │ +  Strategy: DecomposeMonomials(time .000019748)  #primes = 1 #prunedViaCodim =
│ │ │ │  0
│ │ │ │   -- Converting annotated ideals to ideals and selecting minimal primes...
│ │ │ │ - --  Time taken : .000630212
│ │ │ │ - -- .0384912s elapsed
│ │ │ │ + --  Time taken : .000713235
│ │ │ │ + -- .0270881s elapsed
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _p_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n_(_._._._,_S_t_r_a_t_e_g_y_=_>_._._._)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ │ │  The object _H_y_b_r_i_d is a _s_e_l_f_ _i_n_i_t_i_a_l_i_z_i_n_g_ _t_y_p_e, with ancestor classes _L_i_s_t <
│ │ │ │  _V_i_s_i_b_l_e_L_i_s_t < _B_a_s_i_c_L_i_s_t < _T_h_i_n_g.
│ │ │ │  ===============================================================================
│ │ │ │  The source of this document is in /build/reproducible-path/macaulay2-
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical.html
│ │ │ @@ -136,25 +136,25 @@
│ │ │  
│ │ │  o5 : Ideal of R
    
│ │ │              
    
│ │ │                
    i6 : elapsedTime radical(ideal I_*, Strategy => Monomial)
│ │ │ - -- .000442257s elapsed
│ │ │ + -- .000510632s elapsed
│ │ │  
│ │ │  o6 = ideal (a, b, c)
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ - -- .0121996s elapsed
│ │ │ + -- .0185482s 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)
│ │ │ │ - -- .000442257s elapsed
│ │ │ │ + -- .000510632s elapsed
│ │ │ │  
│ │ │ │  o6 = ideal (a, b, c)
│ │ │ │  
│ │ │ │  o6 : Ideal of R
│ │ │ │  i7 : elapsedTime radical(ideal I_*, Unmixed => true)
│ │ │ │ - -- .0121996s elapsed
│ │ │ │ + -- .0185482s elapsed
│ │ │ │  
│ │ │ │  o7 = ideal (c, b, a)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  For another example, see _P_r_i_m_a_r_y_D_e_c_o_m_p_o_s_i_t_i_o_n.
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ │ │  Eisenbud, Huneke, Vasconcelos, Invent. Math. 110 207-235 (1992).
│ │ ├── ./usr/share/doc/Macaulay2/MinimalPrimes/html/_radical__Containment.html
│ │ │ @@ -130,31 +130,31 @@
│ │ │  
│ │ │  o6 = true
    
│ │ │              
    
│ │ │                
    i7 : elapsedTime radicalContainment(x_0, I)
│ │ │ - -- .151496s elapsed
│ │ │ + -- .0801s elapsed
│ │ │  
│ │ │  o7 = true
    
│ │ │              
    
│ │ │                
    i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ - -- .00157389s elapsed
│ │ │ + -- .00173948s elapsed
│ │ │  
│ │ │  o8 = true
    
│ │ │              
    
│ │ │                
    i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ - -- .00123922s elapsed
│ │ │ + -- .0013597s 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)
│ │ │ │ - -- .151496s elapsed
│ │ │ │ + -- .0801s elapsed
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : elapsedTime radicalContainment(x_0, I, Strategy => "Kollar")
│ │ │ │ - -- .00157389s elapsed
│ │ │ │ + -- .00173948s elapsed
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : elapsedTime radicalContainment(x_n, I, Strategy => "Kollar")
│ │ │ │ - -- .00123922s elapsed
│ │ │ │ + -- .0013597s 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/Miura/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  c2NhbGFyTXVsdGlwbGljYXRpb24=
│ │ │  #:len=1318
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│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bXVsdGlSZWVzSWRlYWwoSWRlYWwp
│ │ │  #:len=280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjA3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./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 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ + -- used 0.387396s (cpu); 0.106838s (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 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ + -- used 0.0729774s (cpu); 0.0174987s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o11 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ ├── ./usr/share/doc/Macaulay2/MixedMultiplicity/html/_multi__Rees__Ideal.html
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o9 : Ideal of U
│ │ │              
    
│ │ │                
    i10 : time multiReesIdeal J
│ │ │ - -- used 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ + -- used 0.387396s (cpu); 0.106838s (thread); 0s (gc)
│ │ │  
│ │ │                                                                               
│ │ │  o10 = ideal (c*X  - b*X , b*X  - a*X , a*X  - c*X , c*X  - a*X , b*X  - c*X ,
│ │ │                  1      2     1      2     1      2     0      2     0      2 
│ │ │        -----------------------------------------------------------------------
│ │ │                      2                 2   2
│ │ │        a*X  - b*X , X  - X X , X X  - X , X  - X X )
│ │ │ @@ -195,15 +195,15 @@
│ │ │  o10 : Ideal of U[X ..X ]
│ │ │                    0   2
    
│ │ │              
    
│ │ │                
    i11 : time multiReesIdeal (J, a)
│ │ │ - -- used 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ + -- used 0.0729774s (cpu); 0.0174987s (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 0.165102s (cpu); 0.107224s (thread); 0s (gc)
│ │ │ │ + -- used 0.387396s (cpu); 0.106838s (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 0.0216638s (cpu); 0.0211096s (thread); 0s (gc)
│ │ │ │ + -- used 0.0729774s (cpu); 0.0174987s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  ZGVmb3JtTUNNTW9kdWxl
│ │ │  #:len=2237
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmVyc2FsIGRlZm9ybWF0aW9uIG9mIE1D
│ │ │  TS1tb2R1bGUgb24gaHlwZXJzdXJmYWNlIiwgRGVzY3JpcHRpb24gPT4gKCJUaGlzIGlzIHRoZSBt
│ │ ├── ./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 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ + -- used 0.634883s (cpu); 0.338118s (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 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ + -- used 0.642754s (cpu); 0.457132s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = (S', cokernel | x2-xxi_4^3-xxi_2+xi_2xi_4^3-3xi_3xi_4^2+xi_2^2-xi_1
│ │ │                      | xxi_4-y+xi_3                                       
│ │ │        -----------------------------------------------------------------------
│ │ │        x2xi_4^2+xyxi_4+2xxi_3xi_4+y2+yxi_3+xi_3^2 |)
│ │ │        -x2-xxi_2-xi_1                             |
│ │ ├── ./usr/share/doc/Macaulay2/ModuleDeformations/html/_deform__M__C__M__Module_lp__Module_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │                               1
│ │ │  o7 : R-module, submodule of R
    
│ │ │              
    
│ │ │                
    i8 : (S,N) = time deformMCMModule N0 
│ │ │ - -- used 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ + -- used 0.634883s (cpu); 0.338118s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = (S, cokernel {6} | x2-xxi_2-xi_1+xi_2^2-yxi_4^2-2xi_3xi_4^2+xi_2xi_4^3
│ │ │                    {8} | xxi_4-y+xi_3                                       
│ │ │       ------------------------------------------------------------------------
│ │ │       xyxi_4+2xxi_3xi_4-xxi_2xi_4^2+y2+yxi_3+xi_3^2-xi_1xi_4^2 |)
│ │ │       -x2-xxi_2-xi_1                                           |
│ │ │  
│ │ │ @@ -191,15 +191,15 @@
│ │ │                               2
│ │ │  o10 : R-module, quotient of R
    
│ │ │              
    
│ │ │                
    i11 : (S',N') = time deformMCMModule N0'
│ │ │ - -- used 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ + -- used 0.642754s (cpu); 0.457132s (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 0.684142s (cpu); 0.379036s (thread); 0s (gc)
│ │ │ │ + -- used 0.634883s (cpu); 0.338118s (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 0.682333s (cpu); 0.475269s (thread); 0s (gc)
│ │ │ │ + -- used 0.642754s (cpu); 0.457132s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c3BhcnNlTW9ub2Ryb215U29sdmUoLi4uLE51bWJlck9mUmVwZWF0cz0+Li4uKQ==
│ │ │  #:len=345
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzcGFyc2VNb25vZHJvbXlTb2x2ZSxOdW1iZXJPZlJl
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_dynamic__Flower__Solve.out
│ │ │ @@ -3,41 +3,41 @@
│ │ │  i1 : R = CC[a,b,c,d][x,y];
│ │ │  
│ │ │  i2 : polys = polySystem {a*x+b*y^2,c*x*y+d};
│ │ │  
│ │ │  i3 : (p0, x0) = createSeedPair polys;
│ │ │  
│ │ │  i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ - -- .00616338s elapsed
│ │ │ +--backup directory created: /tmp/M2-33141-0/2
│ │ │ + -- .00322685s elapsed
│ │ │    H01: 1
│ │ │ - -- .00900117s elapsed
│ │ │ + -- .00334371s elapsed
│ │ │    H10: 1
│ │ │ - -- .000649635s elapsed
│ │ │ + -- .000568627s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00324889s elapsed
│ │ │ + -- .00311715s elapsed
│ │ │    H01: 1
│ │ │ - -- .00435076s elapsed
│ │ │ + -- .00353501s elapsed
│ │ │    H10: 1
│ │ │ - -- .000544598s elapsed
│ │ │ + -- .000564146s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00428387s elapsed
│ │ │ + -- .0029798s elapsed
│ │ │    H01: 1
│ │ │ - -- .00298132s elapsed
│ │ │ + -- .00329803s elapsed
│ │ │    H10: 1
│ │ │ - -- .000466532s elapsed
│ │ │ + -- .000517887s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .0044506s elapsed
│ │ │ + -- .00311868s elapsed
│ │ │    H01: 1
│ │ │ - -- .00309306s elapsed
│ │ │ + -- .00297752s elapsed
│ │ │    H10: 1
│ │ │ - -- .000457295s elapsed
│ │ │ + -- .000481962s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/example-output/_get__Track__Time.out
│ │ │ @@ -6,16 +6,16 @@
│ │ │  i2 : declareVariable \ {A,B,C,D,X,Y};
│ │ │  
│ │ │  i3 : F = gateSystem(matrix{{A,B,C,D}},matrix{{X,Y}},matrix{{A*(X-1)^2-B}, {C*(Y+2)^2+D}});
│ │ │  
│ │ │  i4 : (p0,x0) = createSeedPair F;
│ │ │  
│ │ │  i5 : time (V,npaths) = monodromySolve(F,p0,{x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
│ │ │ + -- used 0.0555346s (cpu); 0.0555319s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : getTrackTime V.Graph
│ │ │  
│ │ │ -o6 = .006001457999999999
│ │ │ +o6 = .009915771999999998
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/MonodromySolver/html/_dynamic__Flower__Solve.html
│ │ │ @@ -101,41 +101,41 @@
│ │ │              
    		
│ │ │                
    i3 : (p0, x0) = createSeedPair polys;
    
│ │ │              
    
│ │ │                
    i4 : (L, npaths) = dynamicFlowerSolve(polys.PolyMap,p0,{x0})
│ │ │ ---backup directory created: /tmp/M2-25512-0/2
│ │ │ - -- .00616338s elapsed
│ │ │ +--backup directory created: /tmp/M2-33141-0/2
│ │ │ + -- .00322685s elapsed
│ │ │    H01: 1
│ │ │ - -- .00900117s elapsed
│ │ │ + -- .00334371s elapsed
│ │ │    H10: 1
│ │ │ - -- .000649635s elapsed
│ │ │ + -- .000568627s elapsed
│ │ │  number of paths tracked: 2
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00324889s elapsed
│ │ │ + -- .00311715s elapsed
│ │ │    H01: 1
│ │ │ - -- .00435076s elapsed
│ │ │ + -- .00353501s elapsed
│ │ │    H10: 1
│ │ │ - -- .000544598s elapsed
│ │ │ + -- .000564146s elapsed
│ │ │  number of paths tracked: 4
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .00428387s elapsed
│ │ │ + -- .0029798s elapsed
│ │ │    H01: 1
│ │ │ - -- .00298132s elapsed
│ │ │ + -- .00329803s elapsed
│ │ │    H10: 1
│ │ │ - -- .000466532s elapsed
│ │ │ + -- .000517887s elapsed
│ │ │  number of paths tracked: 6
│ │ │  found 1 points in the fiber so far
│ │ │ - -- .0044506s elapsed
│ │ │ + -- .00311868s elapsed
│ │ │    H01: 1
│ │ │ - -- .00309306s elapsed
│ │ │ + -- .00297752s elapsed
│ │ │    H10: 1
│ │ │ - -- .000457295s elapsed
│ │ │ + -- .000481962s elapsed
│ │ │  number of paths tracked: 8
│ │ │  found 1 points in the fiber so far
│ │ │  
│ │ │  o4 = ({{.892712+.673395*ii, .29398+.632944*ii}}, 8)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │                  		
│ │ │                
    i4 : (p0,x0) = createSeedPair F;
    
│ │ │              
    
│ │ │                
    i5 : time (V,npaths) = monodromySolve(F,p0,{x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
    
│ │ │ + -- used 0.0555346s (cpu); 0.0555319s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i6 : getTrackTime V.Graph
│ │ │  
│ │ │ -o6 = .006001457999999999
│ │ │ +o6 = .009915771999999998
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │          
    
│ │ │            
    time measures the elapsed time for the whole call, including graph construction, and other overhead. getTrackTime reports only the cumulative time spent tracking paths along edges of the homotopy graph.
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -16,18 +16,18 @@
│ │ │ │   -- setting random seed to 0
│ │ │ │  i2 : declareVariable \ {A,B,C,D,X,Y};
│ │ │ │  i3 : F = gateSystem(matrix{{A,B,C,D}},matrix{{X,Y}},matrix{{A*(X-1)^2-B}, {C*
│ │ │ │  (Y+2)^2+D}});
│ │ │ │  i4 : (p0,x0) = createSeedPair F;
│ │ │ │  i5 : time (V,npaths) = monodromySolve(F,p0,
│ │ │ │  {x0},NumberOfEdges=>3,NumberOfNodes=>5,TargetSolutionCount=>4,Verbose=>false);
│ │ │ │ - -- used 0.0535401s (cpu); 0.0535372s (thread); 0s (gc)
│ │ │ │ + -- used 0.0555346s (cpu); 0.0555319s (thread); 0s (gc)
│ │ │ │  i6 : getTrackTime V.Graph
│ │ │ │  
│ │ │ │ -o6 = .006001457999999999
│ │ │ │ +o6 = .009915771999999998
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  time measures the elapsed time for the whole call, including graph
│ │ │ │  construction, and other overhead. getTrackTime reports only the cumulative time
│ │ │ │  spent tracking paths along edges of the homotopy graph.
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _e_l_a_p_s_e_d_T_i_m_e -- time a computation including time elapsed
│ │ ├── ./usr/share/doc/Macaulay2/MonomialAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  ZGVjb21wb3NlSG9tb2dlbmVvdXNNQShMaXN0KQ==
│ │ │  #:len=308
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│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZGVjb21wb3NlSG9tb2dlbmVvdXNNQSxMaXN0KSwi
│ │ ├── ./usr/share/doc/Macaulay2/MonomialIntegerPrograms/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  R3JhZGVkQmV0dGlz
│ │ │  #:len=431
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtEZXNjcmlwdGlvbiA9PiB7fSwgImxpbmVudW0iID0+IDExNDYs
│ │ │  IHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7IkdyYWRlZEJldHRp
│ │ ├── ./usr/share/doc/Macaulay2/MonomialOrbits/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  aGlsYmVydFJlcHJlc2VudGF0aXZlcw==
│ │ │  #:len=3397
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZCByZXByZXNlbnRhdGl2ZXMgb2Yg
│ │ │  bW9ub21pYWwgaWRlYWxzIHVuZGVyIHBlcm11dGF0aW9ucyBvZiB0aGUgdmFyaWFibGVzIiwgImxp
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  bXNvbHZlUlVSKElkZWFsKQ==
│ │ │  #:len=227
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjQ0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtc29sdmVSVVIsSWRlYWwpLCJtc29sdmVSVVIoSWRl
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/example-output/___Msolve.out
│ │ │ @@ -9,16 +9,16 @@
│ │ │  i2 : I = ideal(x, y, z)
│ │ │  
│ │ │  o2 = ideal (x, y, z)
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/M2-38527-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1792814436
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54074-0/0-in.ms -o /tmp/M2-54074-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1792836101
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -29,15 +29,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1099986659
│ │ │ +Initial prime = 1118389211
│ │ │  
│ │ │  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)
│ │ │ @@ -54,18 +54,18 @@
│ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │  overall(elapsed)        0.00 sec
│ │ │  overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.00 sec  96.6%
│ │ │ -convert                 0.00 sec   0.6%
│ │ │ -linear algebra          0.00 sec   0.2%
│ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ +update                  0.00 sec  69.7%
│ │ │ +convert                 0.00 sec   3.0%
│ │ │ +linear algebra          0.00 sec   1.3%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -79,18 +79,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1120193111
│ │ │ +New prime = 1116865681
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ +multi-mod overall(elapsed)      0.01 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -106,15 +106,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.08 sec (elapsed) /  0.27 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.05 sec (cpu)
│ │ │  ------------------------------------------------------------------------------------
│ │ │  
│ │ │  o3 = | z y x |
│ │ │  
│ │ │               1      3
│ │ │  o3 : Matrix R  <-- R
│ │ ├── ./usr/share/doc/Macaulay2/Msolve/html/index.html
│ │ │ @@ -83,16 +83,16 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │
    │ │ │ 
    i3 : msolveGB(I, Verbosity => 2, Threads => 6) 
│ │ │ - -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-38527-0/0-in.ms -o /tmp/M2-38527-0/0-out.ms
│ │ │ -Initial seed for pseudo-random number generator is 1792814436
│ │ │ + -- running: /usr/bin/msolve -g 2 -t 6 -v 2 -f /tmp/M2-54074-0/0-in.ms -o /tmp/M2-54074-0/0-out.ms
│ │ │ +Initial seed for pseudo-random number generator is 1792836101
│ │ │  
│ │ │  --------------- INPUT DATA ---------------
│ │ │  #variables                       3
│ │ │  #equations                       3
│ │ │  #invalid equations               0
│ │ │  field characteristic             0
│ │ │  homogeneous input?               1
│ │ │ @@ -103,15 +103,15 @@
│ │ │  initial hash table size     131072 (2^17)
│ │ │  max pair selection             ALL
│ │ │  reduce gb                        1
│ │ │  #threads                         6
│ │ │  info level                       2
│ │ │  generate pbm files               0
│ │ │  ------------------------------------------
│ │ │ -Initial prime = 1099986659
│ │ │ +Initial prime = 1118389211
│ │ │  
│ │ │  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)
│ │ │ @@ -128,18 +128,18 @@
│ │ │  reduce final basis        3 x 3          33.33%        3 new       0 zero         0.00 | 0.00         
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │  overall(elapsed)        0.00 sec
│ │ │  overall(cpu)            0.00 sec
│ │ │  select                  0.00 sec   0.0%
│ │ │ -symbolic prep.          0.00 sec   0.0%
│ │ │ -update                  0.00 sec  96.6%
│ │ │ -convert                 0.00 sec   0.6%
│ │ │ -linear algebra          0.00 sec   0.2%
│ │ │ +symbolic prep.          0.00 sec   0.5%
│ │ │ +update                  0.00 sec  69.7%
│ │ │ +convert                 0.00 sec   3.0%
│ │ │ +linear algebra          0.00 sec   1.3%
│ │ │  reduce gb               0.00 sec   0.0%
│ │ │  -----------------------------------------
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  size of basis                     3
│ │ │  #terms in basis                   3
│ │ │  #pairs reduced                    0
│ │ │ @@ -153,18 +153,18 @@
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ---------- COMPUTATIONAL DATA -----------
│ │ │  [3]
│ │ │  #polynomials to lift              3
│ │ │  -----------------------------------------
│ │ │ -New prime = 1120193111
│ │ │ +New prime = 1116865681
│ │ │  
│ │ │  ---------------- TIMINGS ----------------
│ │ │ -multi-mod overall(elapsed)      0.02 sec
│ │ │ +multi-mod overall(elapsed)      0.01 sec
│ │ │  learning phase                  0.00 Gops/sec
│ │ │  application phase               0.00 Gops/sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  multi-modular steps
│ │ │  ------------------------------------------------------------------------------------------------------
│ │ │  {1}{2}<100.00%> 
│ │ │ @@ -180,15 +180,15 @@
│ │ │  ---------------- TIMINGS ----------------
│ │ │  CRT     (elapsed)               0.00 sec
│ │ │  ratrecon(elapsed)               0.00 sec
│ │ │  -----------------------------------------
│ │ │  
│ │ │  
│ │ │  ------------------------------------------------------------------------------------
│ │ │ -msolve overall time           0.08 sec (elapsed) /  0.27 sec (cpu)
│ │ │ +msolve overall time           0.01 sec (elapsed) /  0.05 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
│ │ │ - -- .00225028s elapsed
│ │ │ + -- .00218476s elapsed
│ │ │  computing total degree: 2
│ │ │  number of monomials = 21
│ │ │  number of distinct multidegrees = 18
│ │ │ - -- .00849938s elapsed
│ │ │ + -- .00877331s 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
│ │ │ │ - -- .00225028s elapsed
│ │ │ │ + -- .00218476s elapsed
│ │ │ │  computing total degree: 2
│ │ │ │  number of monomials = 21
│ │ │ │  number of distinct multidegrees = 18
│ │ │ │ - -- .00849938s elapsed
│ │ │ │ + -- .00877331s 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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=11
│ │ │  Z3JHcihJZGVhbCk=
│ │ │  #:len=249
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzc4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnckdyLElkZWFsKSwiZ3JHcihJZGVhbCkiLCJNdWx0
│ │ ├── ./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
│ │ │ - -- .0242277s elapsed
│ │ │ + -- .0225676s elapsed
│ │ │  
│ │ │  o3 = 2
│ │ │  
│ │ │  i4 : elapsedTime monjMult I
│ │ │ - -- .159695s elapsed
│ │ │ + -- .0664976s elapsed
│ │ │  
│ │ │  o4 = 2
│ │ │  
│ │ │  i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .270321s elapsed
│ │ │ + -- .120257s 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
│ │ │ - -- .0952293s elapsed
│ │ │ + -- .0768035s 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
│ │ │ - -- .189553s elapsed
│ │ │ + -- .0786743s elapsed
│ │ │  
│ │ │  o3 = 192
│ │ │  
│ │ │  i4 : elapsedTime jMult I
│ │ │ - -- 1.87284s elapsed
│ │ │ + -- 1.22602s elapsed
│ │ │  
│ │ │  o4 = 192
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_j__Mult.html
│ │ │ @@ -93,31 +93,31 @@
│ │ │  
│ │ │  o2 : Ideal of R
    │ │ │
    │ │ │ 
    i3 : elapsedTime jMult I
│ │ │ - -- .0242277s elapsed
│ │ │ + -- .0225676s elapsed
│ │ │  
│ │ │  o3 = 2
    │ │ │
    │ │ │ 
    i4 : elapsedTime monjMult I
│ │ │ - -- .159695s elapsed
│ │ │ + -- .0664976s elapsed
│ │ │  
│ │ │  o4 = 2
    │ │ │
    │ │ │ 
    i5 : elapsedTime multiplicitySequence I
│ │ │ - -- .270321s elapsed
│ │ │ + -- .120257s elapsed
│ │ │  
│ │ │  o5 = HashTable{2 => 3}
│ │ │                 3 => 2
│ │ │  
│ │ │  o5 : HashTable
    │ │ │
    │ │ │ 
    i3 : elapsedTime monAnalyticSpread I
│ │ │ - -- .0952293s elapsed
│ │ │ + -- .0768035s 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 │ │ │ │ - -- .0952293s elapsed │ │ │ │ + -- .0768035s elapsed │ │ │ │ │ │ │ │ o3 = 2 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_P -- the Newton polyhedron of a monomial ideal │ │ │ │ ********** WWaayyss ttoo uussee mmoonnAAnnaallyyttiiccSSpprreeaadd:: ********** │ │ │ │ * monAnalyticSpread(Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ ├── ./usr/share/doc/Macaulay2/MultiplicitySequence/html/_monj__Mult.html │ │ │ @@ -97,23 +97,23 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime monjMult I
│ │ │ - -- .189553s elapsed
│ │ │ + -- .0786743s elapsed
│ │ │  
│ │ │  o3 = 192
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime jMult I
│ │ │ - -- 1.87284s elapsed
│ │ │ + -- 1.22602s 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 │ │ │ │ - -- .189553s elapsed │ │ │ │ + -- .0786743s elapsed │ │ │ │ │ │ │ │ o3 = 192 │ │ │ │ i4 : elapsedTime jMult I │ │ │ │ - -- 1.87284s elapsed │ │ │ │ + -- 1.22602s 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/MultiplierIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=41 │ │ │ bG9nQ2Fub25pY2FsVGhyZXNob2xkKENlbnRyYWxBcnJhbmdlbWVudCk= │ │ │ #:len=361 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA3MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobG9nQ2Fub25pY2FsVGhyZXNob2xkLENlbnRyYWxB │ │ ├── ./usr/share/doc/Macaulay2/MultiplierIdealsDim2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ TXVsdElkZWFs │ │ │ #:len=1363 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIG11bHRpcGxpZXIg │ │ │ aWRlYWwgb2YgYSBnaXZlbiBudW1iZXIuIiwgImxpbmVudW0iID0+IDY0OCwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ dGV4TWF0aChSQVQp │ │ │ #:len=209 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDE5OCwgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4TWF0 │ │ ├── ./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 3.4873s (cpu); 1.91387s (thread); 0s (gc) │ │ │ + -- used 3.1092s (cpu); 1.71027s (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 3.66337s (cpu); 2.10606s (thread); 0s (gc) │ │ │ + -- used 3.31702s (cpu); 1.83992s (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 8.41326s (cpu); 5.16056s (thread); 0s (gc) │ │ │ + -- used 5.55357s (cpu); 3.94331s (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 5.22987s (cpu); 3.71826s (thread); 0s (gc) │ │ │ + -- used 3.71941s (cpu); 3.12601s (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 0.20103s (cpu); 0.125952s (thread); 0s (gc) │ │ │ + -- used 0.187309s (cpu); 0.116136s (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 2.02267s (cpu); 1.47252s (thread); 0s (gc) │ │ │ + -- used 1.80277s (cpu); 1.17268s (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 4.01388s (cpu); 2.8819s (thread); 0s (gc) │ │ │ + -- used 3.72776s (cpu); 2.15538s (thread); 0s (gc) │ │ │ │ │ │ o4 = 2 │ │ │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc) │ │ │ + -- used 0.386582s (cpu); 0.268978s (thread); 0s (gc) │ │ │ │ │ │ o5 = 2 │ │ │ │ │ │ i6 : time degree Phi │ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc) │ │ │ + -- used 0.200953s (cpu); 0.201473s (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 0.43902s (cpu); 0.366735s (thread); 0s (gc) │ │ │ + -- used 0.389629s (cpu); 0.315382s (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 0.00332575s (cpu); 0.000166912s (thread); 0s (gc) │ │ │ + -- used 0.000765344s (cpu); 0.000154512s (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 0.00316389s (cpu); 0.000237795s (thread); 0s (gc) │ │ │ + -- used 0.00199745s (cpu); 0.000246182s (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 1.3359s (cpu); 1.00639s (thread); 0s (gc) │ │ │ + -- used 1.06788s (cpu); 0.882959s (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 0.000157475s (cpu); 0.000414858s (thread); 0s (gc) │ │ │ + -- used 0.000112437s (cpu); 0.0003882s (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 0.0281486s (cpu); 0.0267017s (thread); 0s (gc) │ │ │ + -- used 0.0345314s (cpu); 0.0213747s (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 0.134338s (cpu); 0.060805s (thread); 0s (gc) │ │ │ + -- used 0.132959s (cpu); 0.0543993s (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 0.124474s (cpu); 0.123543s (thread); 0s (gc) │ │ │ + -- used 0.123534s (cpu); 0.113657s (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 0.103109s (cpu); 0.102376s (thread); 0s (gc) │ │ │ + -- used 0.121507s (cpu); 0.109754s (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 5.47152s (cpu); 2.82366s (thread); 0s (gc) │ │ │ + -- used 8.4436s (cpu); 2.31525s (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 0.418413s (cpu); 0.331336s (thread); 0s (gc) │ │ │ + -- used 0.352399s (cpu); 0.280054s (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 1.45295s (cpu); 1.03172s (thread); 0s (gc) │ │ │ + -- used 1.13481s (cpu); 0.928269s (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 0.157948s (cpu); 0.0767566s (thread); 0s (gc) │ │ │ + -- used 0.189869s (cpu); 0.0770155s (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 0.186263s (cpu); 0.107471s (thread); 0s (gc) │ │ │ + -- used 0.22146s (cpu); 0.0935344s (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 0.48562s (cpu); 0.313057s (thread); 0s (gc) │ │ │ + -- used 0.537769s (cpu); 0.281111s (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 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc) │ │ │ + -- used 0.00237262s (cpu); 6.863e-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 0.396692s (cpu); 0.213364s (thread); 0s (gc) │ │ │ + -- used 0.551135s (cpu); 0.232243s (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 1.74335s (cpu); 0.941708s (thread); 0s (gc) │ │ │ + -- used 1.56542s (cpu); 0.778042s (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 0.386427s (cpu); 0.269828s (thread); 0s (gc) │ │ │ + -- used 0.336211s (cpu); 0.210827s (thread); 0s (gc) │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : time Psi = first graph Phi; │ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc) │ │ │ + -- used 0.149498s (cpu); 0.080731s (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 4.42649s (cpu); 3.16319s (thread); 0s (gc) │ │ │ + -- used 3.2133s (cpu); 2.67866s (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 0.00800583s (cpu); 0.00919368s (thread); 0s (gc) │ │ │ + -- used 0.00802891s (cpu); 0.00956977s (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 0.709324s (cpu); 0.523301s (thread); 0s (gc) │ │ │ + -- used 0.437391s (cpu); 0.367622s (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 0.453684s (cpu); 0.376266s (thread); 0s (gc) │ │ │ + -- used 0.41182s (cpu); 0.305461s (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 0.00102199s (cpu); 0.00133739s (thread); 0s (gc) │ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc) │ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc) │ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc) │ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc) │ │ │ + -- used 0.00162598s (cpu); 0.00115661s (thread); 0s (gc) │ │ │ + -- used 0.181937s (cpu); 0.124109s (thread); 0s (gc) │ │ │ + -- used 0.290668s (cpu); 0.166895s (thread); 0s (gc) │ │ │ + -- used 0.2007s (cpu); 0.141229s (thread); 0s (gc) │ │ │ + -- used 0.176345s (cpu); 0.11901s (thread); 0s (gc) │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ o2 : List │ │ │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc) │ │ │ + -- used 0.175518s (cpu); 0.0748323s (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 0.0200872s (cpu); 0.0193746s (thread); 0s (gc) │ │ │ + -- used 0.0730555s (cpu); 0.0232712s (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 2.20856s (cpu); 1.14282s (thread); 0s (gc) │ │ │ + -- used 1.41965s (cpu); 0.896369s (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 0.774927s (cpu); 0.592841s (thread); 0s (gc) │ │ │ + -- used 1.49653s (cpu); 0.58933s (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 0.410659s (cpu); 0.270055s (thread); 0s (gc) │ │ │ + -- used 0.279033s (cpu); 0.15779s (thread); 0s (gc) │ │ │ │ │ │ o2 = multi-rational map consisting of 3 rational maps │ │ │ source variety: threefold in PP^3 x PP^2 cut out by 2 hypersurfaces of multi-degree (1,1) │ │ │ target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of multi-degrees (0,1,1)^3 (1,0,1)^2 (1,1,0)^2 │ │ │ base locus: empty subscheme of PP^3 x PP^2 │ │ │ dominance: true │ │ │ multidegree: {10, 14, 19, 25} │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Embedded__Projective__Variety_sp_eq_eq_eq_gt_sp__Embedded__Projective__Variety.html │ │ │ @@ -108,15 +108,15 @@ │ │ │ │ │ │ o5 = curve in PP^8 cut out by 17 hypersurfaces of degrees 1^2 2^15 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time f = X ===> Y;
│ │ │ - -- used 3.4873s (cpu); 1.91387s (thread); 0s (gc)
│ │ │ + -- used 3.1092s (cpu); 1.71027s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : MultirationalMap (automorphism of PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : f X
│ │ │ @@ -148,15 +148,15 @@
│ │ │  
│ │ │  o10 : ProjectiveVariety, 6-dimensional subvariety of PP^8
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time g = V ===> W;
│ │ │ - -- used 3.66337s (cpu); 2.10606s (thread); 0s (gc)
│ │ │ + -- used 3.31702s (cpu); 1.83992s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : MultirationalMap (automorphism of PP^8)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : g||W
│ │ │ @@ -257,15 +257,15 @@
│ │ │  
│ │ │  o16 = 6-dimensional subvariety of PP^9 cut out by 5 hypersurfaces of degree 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : time h = Z ===> GG_K(1,4)
│ │ │ - -- used 8.41326s (cpu); 5.16056s (thread); 0s (gc)
│ │ │ + -- used 5.55357s (cpu); 3.94331s (thread); 0s (gc)
│ │ │  
│ │ │  o17 = h
│ │ │  
│ │ │  o17 : MultirationalMap (isomorphism from PP^9 to PP^9)
    │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 3.4873s (cpu); 1.91387s (thread); 0s (gc) │ │ │ │ + -- used 3.1092s (cpu); 1.71027s (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 3.66337s (cpu); 2.10606s (thread); 0s (gc) │ │ │ │ + -- used 3.31702s (cpu); 1.83992s (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 8.41326s (cpu); 5.16056s (thread); 0s (gc) │ │ │ │ + -- used 5.55357s (cpu); 3.94331s (thread); 0s (gc) │ │ │ │ │ │ │ │ o17 = h │ │ │ │ │ │ │ │ o17 : MultirationalMap (isomorphism from PP^9 to PP^9) │ │ │ │ i18 : h || GG_K(1,4) │ │ │ │ │ │ │ │ o18 = multi-rational map consisting of one single rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp^_st_st_sp__Multiprojective__Variety.html │ │ │ @@ -94,15 +94,15 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^2 x PP^4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time X = Phi^* Y;
│ │ │ - -- used 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ + -- used 3.71941s (cpu); 3.12601s (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 5.22987s (cpu); 3.71826s (thread); 0s (gc)
│ │ │ │ + -- used 3.71941s (cpu); 3.12601s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : ProjectiveVariety, curve in PP^3 x PP^2 x PP^4 (subvariety of codimension
│ │ │ │  2 in threefold in PP^3 x PP^2 x PP^4 cut out by 12 hypersurfaces of multi-
│ │ │ │  degrees (0,0,2)^1 (0,1,1)^2 (1,0,1)^7 (1,1,0)^2 )
│ │ │ │  i5 : dim X, degree X, degrees X
│ │ │ │  
│ │ │ │  o5 = (1, 31, {({0, 0, 2}, 1), ({0, 0, 3}, 4), ({0, 1, 1}, 4), ({0, 4, 1}, 1),
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/___Multirational__Map_sp__Multiprojective__Variety.html
│ │ │ @@ -100,15 +100,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time Phi Z;
│ │ │ - -- used 0.20103s (cpu); 0.125952s (thread); 0s (gc)
│ │ │ + -- used 0.187309s (cpu); 0.116136s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : dim oo, degree oo, degrees oo
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o6 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time Phi (point Z + point Z + point Z)
│ │ │ - -- used 2.02267s (cpu); 1.47252s (thread); 0s (gc)
│ │ │ + -- used 1.80277s (cpu); 1.17268s (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
    │ │ │ │ │ │ │ │ │ │ │ │ ├── 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 0.20103s (cpu); 0.125952s (thread); 0s (gc) │ │ │ │ + -- used 0.187309s (cpu); 0.116136s (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 2.02267s (cpu); 1.47252s (thread); 0s (gc) │ │ │ │ + -- used 1.80277s (cpu); 1.17268s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of │ │ │ │ multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3 │ │ │ │ │ │ │ │ o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7 │ │ │ │ i8 : dim oo, degree oo, degrees oo │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_cm__Option_rp.html │ │ │ @@ -98,31 +98,31 @@ │ │ │ ------------------------------------------------------------------------ │ │ │ multi-degrees (0,2)^1 (1,1)^3 (2,1)^8 (4,0)^1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time degree(Phi,Strategy=>"random point")
│ │ │ - -- used 4.01388s (cpu); 2.8819s (thread); 0s (gc)
│ │ │ + -- used 3.72776s (cpu); 2.15538s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time degree(Phi,Strategy=>"0-th projective degree")
│ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc)
│ │ │ + -- used 0.386582s (cpu); 0.268978s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time degree Phi
│ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc)
│ │ │ + -- used 0.200953s (cpu); 0.201473s (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 4.01388s (cpu); 2.8819s (thread); 0s (gc) │ │ │ │ + -- used 3.72776s (cpu); 2.15538s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = 2 │ │ │ │ i5 : time degree(Phi,Strategy=>"0-th projective degree") │ │ │ │ - -- used 0.504268s (cpu); 0.357315s (thread); 0s (gc) │ │ │ │ + -- used 0.386582s (cpu); 0.268978s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 2 │ │ │ │ i6 : time degree Phi │ │ │ │ - -- used 0.330552s (cpu); 0.2545s (thread); 0s (gc) │ │ │ │ + -- used 0.200953s (cpu); 0.201473s (thread); 0s (gc) │ │ │ │ │ │ │ │ o6 = 2 │ │ │ │ Note, as in the example above, that calculation times may vary depending on the │ │ │ │ strategy used. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- degree of a multi-rational map │ │ │ │ * _d_e_g_r_e_e_M_a_p_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_degree_lp__Multirational__Map_rp.html │ │ │ @@ -86,15 +86,15 @@ │ │ │ │ │ │ o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time degree Phi
│ │ │ - -- used 0.43902s (cpu); 0.366735s (thread); 0s (gc)
│ │ │ + -- used 0.389629s (cpu); 0.315382s (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 0.43902s (cpu); 0.366735s (thread); 0s (gc) │ │ │ │ + -- used 0.389629s (cpu); 0.315382s (thread); 0s (gc) │ │ │ │ │ │ │ │ o3 = 1 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_,_O_p_t_i_o_n_) -- degree of a multi-rational map using a │ │ │ │ probabilistic approach │ │ │ │ * _d_e_g_r_e_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- degree of a rational map │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_describe_lp__Multirational__Map_rp.html │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ │ │ o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^4 x PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time ? Phi
│ │ │ - -- used 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ + -- used 0.000765344s (cpu); 0.000154512s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │       target variety: PP^4 x PP^5
│ │ │       ------------------------------------------------------------------------
│ │ │       hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
    │ │ │ │ │ │ @@ -101,27 +101,27 @@ │ │ │ │ │ │ o3 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time ? Phi
│ │ │ - -- used 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ + -- used 0.00199745s (cpu); 0.000246182s (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 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ + -- used 1.06788s (cpu); 0.882959s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = multi-rational map consisting of 2 rational maps
│ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │       target variety: PP^4 x PP^5
│ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │       dominance: false
│ │ │       image: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of multi-degrees (0,2)^1 (1,1)^8 
│ │ │ @@ -131,15 +131,15 @@
│ │ │       degree sequence (map 2/2): [(0,1), (2,0)]
│ │ │       coefficient ring: ZZ/65521
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time ? Phi
│ │ │ - -- used 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ + -- used 0.000112437s (cpu); 0.0003882s (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 0.00332575s (cpu); 0.000166912s (thread); 0s (gc)
│ │ │ │ + -- used 0.000765344s (cpu); 0.000154512s (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 0.00316389s (cpu); 0.000237795s (thread); 0s (gc)
│ │ │ │ + -- used 0.00199745s (cpu); 0.000246182s (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 1.3359s (cpu); 1.00639s (thread); 0s (gc)
│ │ │ │ + -- used 1.06788s (cpu); 0.882959s (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 0.000157475s (cpu); 0.000414858s (thread); 0s (gc)
│ │ │ │ + -- used 0.000112437s (cpu); 0.0003882s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = multi-rational map consisting of 2 rational maps
│ │ │ │       source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9
│ │ │ │  hypersurfaces of multi-degrees (0,2)^1 (1,1)^8
│ │ │ │       target variety: PP^4 x PP^5
│ │ │ │       base locus: empty subscheme of PP^4 x PP^5
│ │ │ │       dominance: false
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_graph_lp__Multirational__Map_rp.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (Phi1,Phi2) = graph Phi
│ │ │ - -- used 0.0281486s (cpu); 0.0267017s (thread); 0s (gc)
│ │ │ + -- used 0.0345314s (cpu); 0.0213747s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (Phi1, Phi2)
│ │ │  
│ │ │  o2 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -112,15 +112,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (Phi21,Phi22) = graph Phi2
│ │ │ - -- used 0.134338s (cpu); 0.060805s (thread); 0s (gc)
│ │ │ + -- used 0.132959s (cpu); 0.0543993s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = (Phi21, Phi22)
│ │ │  
│ │ │  o5 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -136,15 +136,15 @@ │ │ │ │ │ │ o7 : MultirationalMap (dominant rational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to hypersurface in PP^5) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time (Phi211,Phi212) = graph Phi21
│ │ │ - -- used 0.124474s (cpu); 0.123543s (thread); 0s (gc)
│ │ │ + -- used 0.123534s (cpu); 0.113657s (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 0.0281486s (cpu); 0.0267017s (thread); 0s (gc) │ │ │ │ + -- used 0.0345314s (cpu); 0.0213747s (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 0.134338s (cpu); 0.060805s (thread); 0s (gc) │ │ │ │ + -- used 0.132959s (cpu); 0.0543993s (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 0.124474s (cpu); 0.123543s (thread); 0s (gc) │ │ │ │ + -- used 0.123534s (cpu); 0.113657s (thread); 0s (gc) │ │ │ │ │ │ │ │ o8 = (Phi211, Phi212) │ │ │ │ │ │ │ │ o8 : Sequence │ │ │ │ i9 : Phi211; │ │ │ │ │ │ │ │ o9 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_image_lp__Multirational__Map_rp.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ o4 : MultirationalMap (rational map from PP^4 to PP^7 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time Z = image Phi;
│ │ │ - -- used 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ + -- used 0.121507s (cpu); 0.109754s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : dim Z, degree Z, degrees Z
│ │ │ @@ -120,15 +120,15 @@
│ │ │            
│ │ │          
│ │ │          
    Alternatively, the calculation can be performed using the Segre embedding as follows:
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── 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 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ │ + -- used 0.41182s (cpu); 0.305461s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = {66, 46, 31, 20}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : (degree source Phi,degree image Phi)
│ │ │ │  
│ │ │ │  o4 = (66, 20)
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Z__Z_cm__Multirational__Map_rp.html
│ │ │ @@ -82,29 +82,29 @@
│ │ │  
│ │ │  o1 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^5 to PP^5)
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i7 : time Z' = projectiveVariety (map segre target Phi) image(segre Phi,"F4");
│ │ │ - -- used 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ + -- used 8.4436s (cpu); 2.31525s (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 0.103109s (cpu); 0.102376s (thread); 0s (gc)
│ │ │ │ + -- used 0.121507s (cpu); 0.109754s (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 5.47152s (cpu); 2.82366s (thread); 0s (gc)
│ │ │ │ + -- used 8.4436s (cpu); 2.31525s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^4
│ │ │ │  i8 : assert(Z == Z')
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _M_u_l_t_i_p_r_o_j_e_c_t_i_v_e_V_a_r_i_e_t_y -- direct image via a multi-
│ │ │ │        rational map
│ │ │ │      * _i_m_a_g_e_(_R_a_t_i_o_n_a_l_M_a_p_) -- closure of the image of a rational map
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse2.html
│ │ │ @@ -88,15 +88,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from PP^6 to GG(2,4))
│ │ │              
    
│ │ │                
    i3 : time Psi = inverse2 Phi;
│ │ │ - -- used 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ + -- used 0.352399s (cpu); 0.280054s (thread); 0s (gc)
│ │ │  
│ │ │  o3 : MultirationalMap (birational map from GG(2,4) to PP^6)
    
│ │ │              
    
│ │ │                
    i4 : assert(Phi * Psi == 1)
    
│ │ │ @@ -108,15 +108,15 @@
│ │ │  
│ │ │  o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))
│ │ │              
    
│ │ │                
    i6 : time Psi' = inverse2 Phi';
│ │ │ - -- used 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ + -- used 1.13481s (cpu); 0.928269s (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 0.418413s (cpu); 0.331336s (thread); 0s (gc)
│ │ │ │ + -- used 0.352399s (cpu); 0.280054s (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 1.45295s (cpu); 1.03172s (thread); 0s (gc)
│ │ │ │ + -- used 1.13481s (cpu); 0.928269s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)
│ │ │ │  i7 : assert(Phi' * Psi' == 1)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_ _<_=_=_>_ _M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p -- equality of multi-rational maps
│ │ │ │        with checks on internal data
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_inverse_lp__Multirational__Map_rp.html
│ │ │ @@ -93,45 +93,45 @@
│ │ │  
│ │ │  o2 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5)
│ │ │              
    
│ │ │                
    i3 : time inverse Phi;
│ │ │ - -- used 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ + -- used 0.189869s (cpu); 0.0770155s (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 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ + -- used 0.22146s (cpu); 0.0935344s (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 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ + -- used 0.537769s (cpu); 0.281111s (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 0.157948s (cpu); 0.0767566s (thread); 0s (gc)
│ │ │ │ + -- used 0.189869s (cpu); 0.0770155s (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 0.186263s (cpu); 0.107471s (thread); 0s (gc)
│ │ │ │ + -- used 0.22146s (cpu); 0.0935344s (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 0.48562s (cpu); 0.313057s (thread); 0s (gc)
│ │ │ │ + -- used 0.537769s (cpu); 0.281111s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x
│ │ │ │  PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5)
│ │ │ │  i8 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1)
│ │ │ │  i9 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1)
│ │ │ │  i10 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1)
│ │ │ │  ********** RReeffeerreenncceess **********
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Isomorphism_lp__Multirational__Map_rp.html
│ │ │ @@ -88,45 +88,45 @@
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
│ │ │              
    
│ │ │                
    i4 : time isIsomorphism Phi
│ │ │ - -- used 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00237262s (cpu); 6.863e-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 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ + -- used 0.551135s (cpu); 0.232243s (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 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ + -- used 1.56542s (cpu); 0.778042s (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 0.000119785s (cpu); 1.2293e-05s (thread); 0s (gc)
│ │ │ │ + -- used 0.00237262s (cpu); 6.863e-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 0.396692s (cpu); 0.213364s (thread); 0s (gc)
│ │ │ │ + -- used 0.551135s (cpu); 0.232243s (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 1.74335s (cpu); 0.941708s (thread); 0s (gc)
│ │ │ │ + -- used 1.56542s (cpu); 0.778042s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : assert(o8 and (not o6) and (not o4))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_n_v_e_r_s_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- inverse of a birational map
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a multi-rational map is a
│ │ │ │        morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_is__Morphism_lp__Multirational__Map_rp.html
│ │ │ @@ -85,31 +85,31 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^4 x PP^2)
│ │ │              
    
│ │ │                
    i3 : time isMorphism Phi
│ │ │ - -- used 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ + -- used 0.336211s (cpu); 0.210827s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = false
    
│ │ │              
    
│ │ │                
    i4 : time Psi = first graph Phi;
│ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ + -- used 0.149498s (cpu); 0.080731s (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 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ + -- used 3.2133s (cpu); 2.67866s (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 0.386427s (cpu); 0.269828s (thread); 0s (gc)
│ │ │ │ + -- used 0.336211s (cpu); 0.210827s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : time Psi = first graph Phi;
│ │ │ │ - -- used 0.168774s (cpu); 0.0862052s (thread); 0s (gc)
│ │ │ │ + -- used 0.149498s (cpu); 0.080731s (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 4.42649s (cpu); 3.16319s (thread); 0s (gc)
│ │ │ │ + -- used 3.2133s (cpu); 2.67866s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : assert((not o3) and o5)
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _i_s_I_s_o_m_o_r_p_h_i_s_m_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- whether a birational map is an
│ │ │ │        isomorphism
│ │ │ │      * _i_s_M_o_r_p_h_i_s_m_(_R_a_t_i_o_n_a_l_M_a_p_) -- whether a rational map is a morphism
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_linearly__Normal__Embedding.html
│ │ │ @@ -84,30 +84,30 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, curve in PP^7
│ │ │              
    
│ │ │                
    i3 : time f = linearlyNormalEmbedding X;
│ │ │ - -- used 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ + -- used 0.00802891s (cpu); 0.00956977s (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 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ + -- used 0.437391s (cpu); 0.367622s (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 0.00800583s (cpu); 0.00919368s (thread); 0s (gc)
│ │ │ │ + -- used 0.00802891s (cpu); 0.00956977s (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 0.709324s (cpu); 0.523301s (thread); 0s (gc)
│ │ │ │ + -- used 0.437391s (cpu); 0.367622s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : MultirationalMap (birational map from Y to curve in PP^7)
│ │ │ │  i6 : assert(isIsomorphism g)
│ │ │ │  i7 : describe g
│ │ │ │  
│ │ │ │  o7 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: curve in PP^3 cut out by 6 hypersurfaces of degree 4
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_multidegree_lp__Multirational__Map_rp.html
│ │ │ @@ -86,15 +86,15 @@
│ │ │  
│ │ │  o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^4 to PP^2 x PP^4)
│ │ │              
    
│ │ │                
    i3 : time multidegree Phi
│ │ │ - -- used 0.453684s (cpu); 0.376266s (thread); 0s (gc)
│ │ │ + -- used 0.41182s (cpu); 0.305461s (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 0.00102199s (cpu); 0.00133739s (thread); 0s (gc)
│ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc)
│ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc)
│ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc)
│ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc)
│ │ │ + -- used 0.00162598s (cpu); 0.00115661s (thread); 0s (gc)
│ │ │ + -- used 0.181937s (cpu); 0.124109s (thread); 0s (gc)
│ │ │ + -- used 0.290668s (cpu); 0.166895s (thread); 0s (gc)
│ │ │ + -- used 0.2007s (cpu); 0.141229s (thread); 0s (gc)
│ │ │ + -- used 0.176345s (cpu); 0.11901s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = {51, 28, 14, 6, 2}
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time assert(oo == multidegree Phi)
│ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc)
    
│ │ │ + -- used 0.175518s (cpu); 0.0748323s (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 0.00102199s (cpu); 0.00133739s (thread); 0s (gc) │ │ │ │ - -- used 0.222006s (cpu); 0.14506s (thread); 0s (gc) │ │ │ │ - -- used 0.340129s (cpu); 0.192965s (thread); 0s (gc) │ │ │ │ - -- used 0.227149s (cpu); 0.156054s (thread); 0s (gc) │ │ │ │ - -- used 0.198029s (cpu); 0.122886s (thread); 0s (gc) │ │ │ │ + -- used 0.00162598s (cpu); 0.00115661s (thread); 0s (gc) │ │ │ │ + -- used 0.181937s (cpu); 0.124109s (thread); 0s (gc) │ │ │ │ + -- used 0.290668s (cpu); 0.166895s (thread); 0s (gc) │ │ │ │ + -- used 0.2007s (cpu); 0.141229s (thread); 0s (gc) │ │ │ │ + -- used 0.176345s (cpu); 0.11901s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = {51, 28, 14, 6, 2} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : time assert(oo == multidegree Phi) │ │ │ │ - -- used 0.164641s (cpu); 0.0837544s (thread); 0s (gc) │ │ │ │ + -- used 0.175518s (cpu); 0.0748323s (thread); 0s (gc) │ │ │ │ ********** RReeffeerreenncceess ********** │ │ │ │ ArXiv preprint: _C_o_m_p_u_t_a_t_i_o_n_s_ _w_i_t_h_ _r_a_t_i_o_n_a_l_ _m_a_p_s_ _b_e_t_w_e_e_n_ _m_u_l_t_i_-_p_r_o_j_e_c_t_i_v_e │ │ │ │ _v_a_r_i_e_t_i_e_s. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_u_l_t_i_d_e_g_r_e_e_(_M_u_l_t_i_r_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a multi-rational │ │ │ │ map │ │ │ │ * _p_r_o_j_e_c_t_i_v_e_D_e_g_r_e_e_s_(_R_a_t_i_o_n_a_l_M_a_p_) -- projective degrees of a rational map │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_point_lp__Multiprojective__Variety_rp.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ o2 : ProjectiveVariety, 4-dimensional subvariety of PP^3 x PP^2 x PP^9     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time p := point X
│ │ │ - -- used 0.0200872s (cpu); 0.0193746s (thread); 0s (gc)
│ │ │ + -- used 0.0730555s (cpu); 0.0232712s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = point of coordinates ([421369, 39917, -212481, 1],[-128795, -176966, 1],[3870, -390108, -496127, -308581, 46649, 164926, -446111, 48038, 415309, 1])
│ │ │  
│ │ │  o3 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9
    │ │ │ │ │ │ │ │ │ │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o4 : ProjectiveVariety, hypersurface in PP^3 x PP^2 x PP^9 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time q = point Y
│ │ │ - -- used 2.20856s (cpu); 1.14282s (thread); 0s (gc)
│ │ │ + -- used 1.41965s (cpu); 0.896369s (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 0.0200872s (cpu); 0.0193746s (thread); 0s (gc) │ │ │ │ + -- used 0.0730555s (cpu); 0.0232712s (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 2.20856s (cpu); 1.14282s (thread); 0s (gc) │ │ │ │ + -- used 1.41965s (cpu); 0.896369s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = q │ │ │ │ │ │ │ │ o5 : ProjectiveVariety, a point in PP^3 x PP^2 x PP^9 │ │ │ │ i6 : assert(isSubset(p,X) and isSubset(q,Y)) │ │ │ │ The list of homogeneous coordinates can be obtained with the operator |-. │ │ │ │ i7 : |- p │ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_segre_lp__Multirational__Map_rp.html │ │ │ @@ -106,15 +106,15 @@ │ │ │ │ │ │ o5 : MultirationalMap (rational map from PP^4 to hypersurface in PP^5 x PP^4 x PP^4) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time segre Phi;
│ │ │ - -- used 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ + -- used 1.49653s (cpu); 0.58933s (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 0.774927s (cpu); 0.592841s (thread); 0s (gc)
│ │ │ │ + -- used 1.49653s (cpu); 0.58933s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : RationalMap (rational map from PP^4 to PP^149)
│ │ │ │  i7 : describe segre Phi
│ │ │ │  
│ │ │ │  o7 = rational map defined by forms of degree 6
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: PP^149
│ │ ├── ./usr/share/doc/Macaulay2/MultiprojectiveVarieties/html/_show_lp__Multirational__Map_rp.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │  
│ │ │  o1 : MultirationalMap (birational map from threefold in PP^3 x PP^2 to threefold in PP^3 x PP^2 x PP^2)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time describe Phi
│ │ │ - -- used 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ + -- used 0.279033s (cpu); 0.15779s (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 0.410659s (cpu); 0.270055s (thread); 0s (gc)
│ │ │ │ + -- used 0.279033s (cpu); 0.15779s (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/NAGtypes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  UG9seVNwYWNl
│ │ │  #:len=827
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwb2x5bm9taWFsIHZlY3RvciBzdWJz
│ │ │  cGFjZSIsICJsaW5lbnVtIiA9PiA4NTcsIFNlZUFsc28gPT4gRElWe0hFQURFUjJ7IlNlZSBhbHNv
│ │ ├── ./usr/share/doc/Macaulay2/NCAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  UVEgJSBOQ0dyb2VibmVyQmFzaXM=
│ │ │  #:len=317
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTY3MCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sICUsUVEsTkNHcm9lYm5lckJhc2lzKSwi
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  Z2VuZXJhdGVSYW5kb21HcmFwaHMoWlosWlosWlop
│ │ │  #:len=275
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTExMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2VuZXJhdGVSYW5kb21HcmFwaHMsWlosWlosWlop
│ │ ├── ./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 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ +o9 = (67, 82, 87, 92, 94, 96, 98, 95, 98, 96, 98, 96, 99, 95, 99, 97, 99, 99,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ +     99, 98, 98, 99, 97, 100, 94, 96, 99, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
│ │ │  
│ │ │  i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ +o10 = (19, 5, 3, 4, 2, 2, 4, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      0, 1, 0, 2, 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 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ +o2 = {DEg, D{C, DIK, Dh{, DRW}
│ │ │  
│ │ │  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, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, 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 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
│ │ │ + -- used 0.000502386s (cpu); 0.000491481s (thread); 0s (gc)
│ │ │  
│ │ │  i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
│ │ │ + -- used 0.0576543s (cpu); 0.0555317s (thread); 0s (gc)
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html
│ │ │ @@ -122,28 +122,28 @@
│ │ │                
    i8 : prob = n -> log(n)/n;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96,
│ │ │ +o9 = (67, 82, 87, 92, 94, 96, 98, 95, 98, 96, 98, 96, 99, 95, 99, 97, 99, 99,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99)
│ │ │ +     99, 98, 98, 99, 97, 100, 94, 96, 99, 98, 99)
│ │ │  
│ │ │  o9 : Sequence
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0,
│ │ │ +o10 = (19, 5, 3, 4, 2, 2, 4, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      1, 0, 0, 0, 0, 0)
│ │ │ +      0, 1, 0, 2, 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 = (63, 89, 84, 84, 91, 94, 100, 96, 95, 95, 95, 98, 98, 96, 100, 99, 96, │ │ │ │ +o9 = (67, 82, 87, 92, 94, 96, 98, 95, 98, 96, 98, 96, 99, 95, 99, 97, 99, 99, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 100, 97, 99, 98, 97, 100, 99, 94, 95, 98, 99, 99) │ │ │ │ + 99, 98, 98, 99, 97, 100, 94, 96, 99, 98, 99) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (12, 12, 6, 4, 3, 5, 2, 2, 1, 0, 1, 1, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 0, │ │ │ │ +o10 = (19, 5, 3, 4, 2, 2, 4, 1, 0, 1, 0, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 1, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 1, 0, 0, 0, 0, 0) │ │ │ │ + 0, 1, 0, 2, 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 │ │ │ @@ -105,15 +105,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ +o2 = {DEg, D{C, DIK, Dh{, DRW}
│ │ │  
│ │ │  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 = {D~{, D`{, DFC, D^S, D{G}
│ │ │ │ +o2 = {DEg, D{C, DIK, Dh{, DRW}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/Nauty/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -92,23 +92,23 @@
│ │ │                
    i1 : R = QQ[a..e];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : generateRandomRegularGraphs(R, 3, 2)
│ │ │  
│ │ │ -o2 = {Graph{"edges" => {{a, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │              "ring" => R                                          
│ │ │              "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ +     Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │             "ring" => R                                          
│ │ │             "vertices" => {a, b, c, d, e}                        
│ │ │       ------------------------------------------------------------------------
│ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, 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, c}, {b, c}, {b, d}, {a, e}, {d, e}}},
│ │ │ │ +o2 = {Graph{"edges" => {{a, b}, {b, d}, {c, d}, {a, e}, {c, e}}},
│ │ │ │              "ring" => R
│ │ │ │              "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, e}, {d, e}}},
│ │ │ │ +     Graph{"edges" => {{a, c}, {b, d}, {c, d}, {a, e}, {b, e}}},
│ │ │ │             "ring" => R
│ │ │ │             "vertices" => {a, b, c, d, e}
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     Graph{"edges" => {{a, b}, {a, c}, {b, d}, {c, e}, {d, e}}}}
│ │ │ │ +     Graph{"edges" => {{a, b}, {a, c}, {c, d}, {b, 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
│ │ │ @@ -121,21 +121,21 @@
│ │ │              
│ │ │                
    i4 : G = generateBipartiteGraphs 7;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time graphComplement G;
│ │ │ - -- used 0.000443261s (cpu); 0.000438352s (thread); 0s (gc)
    
│ │ │ + -- used 0.000502386s (cpu); 0.000491481s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time (graphComplement \ G);
│ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc)
    │ │ │ + -- used 0.0576543s (cpu); 0.0555317s (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 0.000443261s (cpu); 0.000438352s (thread); 0s (gc) │ │ │ │ + -- used 0.000502386s (cpu); 0.000491481s (thread); 0s (gc) │ │ │ │ i6 : time (graphComplement \ G); │ │ │ │ - -- used 0.0520587s (cpu); 0.0501016s (thread); 0s (gc) │ │ │ │ + -- used 0.0576543s (cpu); 0.0555317s (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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ YWRkRWRnZXMoLi4uLE5vTmV3NUN5Y2xlcz0+Li4uKQ== │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1thZGRFZGdlcyxOb05ldzVDeWNsZXNdLCJhZGRFZGdl │ │ ├── ./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 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96, │ │ │ +o9 = (62, 87, 90, 91, 91, 96, 97, 94, 98, 100, 98, 98, 97, 97, 99, 99, 98, │ │ │ ------------------------------------------------------------------------ │ │ │ - 97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98) │ │ │ + 97, 99, 98, 96, 99, 100, 98, 95, 97, 99, 99, 98) │ │ │ │ │ │ o9 : Sequence │ │ │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected)) │ │ │ │ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, │ │ │ +o10 = (15, 10, 7, 8, 4, 3, 1, 2, 4, 1, 0, 2, 3, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, │ │ │ ----------------------------------------------------------------------- │ │ │ - 0, 1, 0, 0, 0, 1) │ │ │ + 1, 0, 1, 0, 0, 1) │ │ │ │ │ │ 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 = {DJc, DRK, DhS, D_s, DkS} │ │ │ +o2 = {D|{, DiG, DP?, DMo, Djw} │ │ │ │ │ │ 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 = {DkK, DUW, DLo} │ │ │ +o1 = {Dhc, DYc, DdW} │ │ │ │ │ │ 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 0.000506259s (cpu); 0.000493143s (thread); 0s (gc) │ │ │ + -- used 0.000588575s (cpu); 0.000581323s (thread); 0s (gc) │ │ │ │ │ │ i5 : time (graphComplement \ G); │ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc) │ │ │ + -- used 0.0569302s (cpu); 0.0546241s (thread); 0s (gc) │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/___Example_co_sp__Generating_spand_spfiltering_spgraphs.html │ │ │ @@ -122,28 +122,28 @@ │ │ │ 
      i8 : prob = n -> log(n)/n;
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i9 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, 2*(prob n)), connected))
│ │ │  
│ │ │ -o9 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96,
│ │ │ +o9 = (62, 87, 90, 91, 91, 96, 97, 94, 98, 100, 98, 98, 97, 97, 99, 99, 98,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98)
│ │ │ +     97, 99, 98, 96, 99, 100, 98, 95, 97, 99, 99, 98)
│ │ │  
│ │ │  o9 : Sequence
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), connected))
│ │ │  
│ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0,
│ │ │ +o10 = (15, 10, 7, 8, 4, 3, 1, 2, 4, 1, 0, 2, 3, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      0, 1, 0, 0, 0, 1)
│ │ │ +      1, 0, 1, 0, 0, 1)
│ │ │  
│ │ │  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 = (71, 75, 84, 96, 95, 97, 96, 98, 96, 94, 93, 98, 98, 99, 96, 95, 96, 96, │ │ │ │ +o9 = (62, 87, 90, 91, 91, 96, 97, 94, 98, 100, 98, 98, 97, 97, 99, 99, 98, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 97, 98, 99, 98, 97, 97, 98, 94, 100, 98, 98) │ │ │ │ + 97, 99, 98, 96, 99, 100, 98, 95, 97, 99, 99, 98) │ │ │ │ │ │ │ │ o9 : Sequence │ │ │ │ i10 : apply(2..30, n -> #filterGraphs(generateRandomGraphs(n, 100, (prob n)/2), │ │ │ │ connected)) │ │ │ │ │ │ │ │ -o10 = (11, 14, 6, 2, 2, 2, 2, 1, 1, 2, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, │ │ │ │ +o10 = (15, 10, 7, 8, 4, 3, 1, 2, 4, 1, 0, 2, 3, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - 0, 1, 0, 0, 0, 1) │ │ │ │ + 1, 0, 1, 0, 0, 1) │ │ │ │ │ │ │ │ o10 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _b_u_i_l_d_G_r_a_p_h_F_i_l_t_e_r -- creates the appropriate filter string for use with │ │ │ │ filterGraphs and countGraphs │ │ │ │ * _f_i_l_t_e_r_G_r_a_p_h_s -- filters (i.e., selects) graphs in a list for given │ │ │ │ properties │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Graphs.html │ │ │ @@ -98,15 +98,15 @@ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : generateRandomGraphs(5, 5)
│ │ │  
│ │ │ -o2 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ +o2 = {D|{, DiG, DP?, DMo, Djw}
│ │ │  
│ │ │  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 = {DJc, DRK, DhS, D_s, DkS}
│ │ │ │ +o2 = {D|{, DiG, DP?, DMo, Djw}
│ │ │ │  
│ │ │ │  o2 : List
│ │ │ │  i3 : generateRandomGraphs(5, 5, RandomSeed => 314159)
│ │ │ │  
│ │ │ │  o3 = {DDO, Dx_, Dlw, Dx{, D_K}
│ │ │ │  
│ │ │ │  o3 : List
│ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_generate__Random__Regular__Graphs.html
│ │ │ @@ -82,15 +82,15 @@
│ │ │            
    This method generates a specified number of random graphs on a given number of vertices with a given regularity. Note that some graphs may be isomorphic.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : generateRandomRegularGraphs(5, 3, 2)
│ │ │  
│ │ │ -o1 = {DkK, DUW, DLo}
│ │ │ +o1 = {Dhc, DYc, DdW}
│ │ │  
│ │ │  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 = {DkK, DUW, DLo} │ │ │ │ +o1 = {Dhc, DYc, DdW} │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** CCaavveeaatt ********** │ │ │ │ The number of vertices $n$ must be positive as nauty cannot handle graphs with │ │ │ │ zero vertices. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _g_e_n_e_r_a_t_e_R_a_n_d_o_m_G_r_a_p_h_s -- generates random graphs on a given number of │ │ ├── ./usr/share/doc/Macaulay2/NautyGraphs/html/_graph__Complement.html │ │ │ @@ -115,21 +115,21 @@ │ │ │ │ │ │ 
    i3 : G = generateBipartiteGraphs 7;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time graphComplement G;
│ │ │ - -- used 0.000506259s (cpu); 0.000493143s (thread); 0s (gc)
    │ │ │ + -- used 0.000588575s (cpu); 0.000581323s (thread); 0s (gc)     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time (graphComplement \ G);
│ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc)
    │ │ │ + -- used 0.0569302s (cpu); 0.0546241s (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 0.000506259s (cpu); 0.000493143s (thread); 0s (gc) │ │ │ │ + -- used 0.000588575s (cpu); 0.000581323s (thread); 0s (gc) │ │ │ │ i5 : time (graphComplement \ G); │ │ │ │ - -- used 0.0534439s (cpu); 0.0512922s (thread); 0s (gc) │ │ │ │ + -- used 0.0569302s (cpu); 0.0546241s (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/NeuralIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ aXNDYW5vbmljYWwoLi4uLFBvbGFyaXplZD0+Li4uKQ== │ │ │ #:len=270 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tpc0Nhbm9uaWNhbCxQb2xhcml6ZWRdLCJpc0Nhbm9u │ │ ├── ./usr/share/doc/Macaulay2/NoetherNormalization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=38 │ │ │ bm9ldGhlck5vcm1hbGl6YXRpb24oLi4uLFZlcmJvc2U9Pi4uLik= │ │ │ #:len=326 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tub2V0aGVyTm9ybWFsaXphdGlvbixWZXJib3NlXSwi │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ bm9ldGhlcmlhbk9wZXJhdG9ycyhJZGVhbCxJZGVhbCk= │ │ │ #:len=2583 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiTm9ldGhlcmlhbiBvcGVyYXRvcnMgb2Yg │ │ │ YSBwcmltYXJ5IGNvbXBvbmVudCIsICJsaW5lbnVtIiA9PiAyNzQ5LCBJbnB1dHMgPT4ge1NQQU57 │ │ ├── ./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") │ │ │ - -- .167714s elapsed │ │ │ + -- .0877222s elapsed │ │ │ │ │ │ o6 = {| 1 |, | dx_1 |, | dx_2 |, | dx_1^2 |, | dx_1dx_2 |, | dx_2^2 |, | │ │ │ ------------------------------------------------------------------------ │ │ │ 2x_1x_3dx_1^3+3x_2x_3dx_1^2dx_2-3x_3x_4dx_1dx_2^2-2x_1x_4dx_2^3 |} │ │ │ │ │ │ o6 : List │ │ ├── ./usr/share/doc/Macaulay2/NoetherianOperators/html/___Strategy_sp_eq_gt_sp_dq__Punctual__Quot_dq.html │ │ │ @@ -125,15 +125,15 @@ │ │ │ │ │ │ o5 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime noetherianOperators(Q, Strategy => "PunctualQuot")
│ │ │ - -- .167714s elapsed
│ │ │ + -- .0877222s 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") │ │ │ │ - -- .167714s elapsed │ │ │ │ + -- .0877222s 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/NonPrincipalTestIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=24 │ │ │ aXNJbnZlcnRpYmxlSWRlYWwoSWRlYWwp │ │ │ #:len=308 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIwNiwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNJbnZlcnRpYmxlSWRlYWwsSWRlYWwpLCJpc0lu │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ VG9yaWNEaXZpc29yID09IFpa │ │ │ #:len=349 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQ0MSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc3ltYm9sID09LFRvcmljRGl2aXNvcixaWiksIlRv │ │ ├── ./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 0.212369s (cpu); 0.212369s (thread); 0s (gc) │ │ │ + -- used 0.217815s (cpu); 0.217815s (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 0.00146338s (cpu); 0.00146077s (thread); 0s (gc) │ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc) │ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc) │ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc) │ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc) │ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc) │ │ │ + -- used 0.00126334s (cpu); 0.00125987s (thread); 0s (gc) │ │ │ + -- used 0.000104675s (cpu); 0.000104566s (thread); 0s (gc) │ │ │ + -- used 7.9235e-05s (cpu); 7.945e-05s (thread); 0s (gc) │ │ │ + -- used 8.3906e-05s (cpu); 8.3979e-05s (thread); 0s (gc) │ │ │ + -- used 8.7471e-05s (cpu); 8.7626e-05s (thread); 0s (gc) │ │ │ + -- used 0.000106411s (cpu); 0.000106791s (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 1792812560 │ │ │ + -- setting random seed to 1792834952 │ │ │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ -o3 = {2, 4, 6} │ │ │ +o3 = {1, 4, 6} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9)); │ │ │ │ │ │ i7 : q │ │ │ │ │ │ -o7 = {1, 5, 6, 9, 9} │ │ │ +o7 = {2, 2, 3, 3, 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 │ │ │ - -- .033906s elapsed │ │ │ + -- .0413957s 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)) │ │ │ - -- .0131253s elapsed │ │ │ + -- .00202324s 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 │ │ │ - -- .134092s elapsed │ │ │ + -- .0807927s 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)) │ │ │ - -- .00103874s elapsed │ │ │ + -- .0010946s 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 │ │ │ - -- 46.8661s elapsed │ │ │ + -- 25.5108s elapsed │ │ │ │ │ │ o11 = 7909 │ │ │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3)) │ │ │ - -- .183473s elapsed │ │ │ + -- .0248134s 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 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc) │ │ │ + -- used 2.8503e-05s (cpu); 2.2234e-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 0.0871239s (cpu); 0.08713s (thread); 0s (gc) │ │ │ + -- used 0.0493004s (cpu); 0.0493079s (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 0.023116s (cpu); 0.0231153s (thread); 0s (gc) │ │ │ + -- used 0.0223976s (cpu); 0.0224148s (thread); 0s (gc) │ │ │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc) │ │ │ + -- used 0.00237006s (cpu); 0.0023743s (thread); 0s (gc) │ │ │ │ │ │ i21 : assert (set rays X2 === set rays X1 and max X1 === max X2) │ │ │ │ │ │ i22 : │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/___Chow_spring.html │ │ │ @@ -212,15 +212,15 @@ │ │ │
    │ │ │

    We end with a slightly larger example.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -259,20 +259,20 @@ │ │ │ │ │ │ o18 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i14 : Y = time smoothFanoToricVariety(5,100);
│ │ │ - -- used 0.212369s (cpu); 0.212369s (thread); 0s (gc)
    │ │ │ + -- used 0.217815s (cpu); 0.217815s (thread); 0s (gc) │ │ │
    │ │ │ 
    i15 : A2 = intersectionRing Y;
    │ │ │
    │ │ │ 
    i19 : for i to dim Y list time hilbertFunction (i, A2)
│ │ │ - -- used 0.00146338s (cpu); 0.00146077s (thread); 0s (gc)
│ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc)
│ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc)
│ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc)
│ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc)
│ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc)
│ │ │ + -- used 0.00126334s (cpu); 0.00125987s (thread); 0s (gc)
│ │ │ + -- used 0.000104675s (cpu); 0.000104566s (thread); 0s (gc)
│ │ │ + -- used 7.9235e-05s (cpu); 7.945e-05s (thread); 0s (gc)
│ │ │ + -- used 8.3906e-05s (cpu); 8.3979e-05s (thread); 0s (gc)
│ │ │ + -- used 8.7471e-05s (cpu); 8.7626e-05s (thread); 0s (gc)
│ │ │ + -- used 0.000106411s (cpu); 0.000106791s (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 0.212369s (cpu); 0.212369s (thread); 0s (gc) │ │ │ │ + -- used 0.217815s (cpu); 0.217815s (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 0.00146338s (cpu); 0.00146077s (thread); 0s (gc) │ │ │ │ - -- used 8.8586e-05s (cpu); 8.8917e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.0521e-05s (cpu); 8.0932e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.6783e-05s (cpu); 8.7574e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.3096e-05s (cpu); 8.3456e-05s (thread); 0s (gc) │ │ │ │ - -- used 8.5891e-05s (cpu); 8.6282e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.00126334s (cpu); 0.00125987s (thread); 0s (gc) │ │ │ │ + -- used 0.000104675s (cpu); 0.000104566s (thread); 0s (gc) │ │ │ │ + -- used 7.9235e-05s (cpu); 7.945e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.3906e-05s (cpu); 8.3979e-05s (thread); 0s (gc) │ │ │ │ + -- used 8.7471e-05s (cpu); 8.7626e-05s (thread); 0s (gc) │ │ │ │ + -- used 0.000106411s (cpu); 0.000106791s (thread); 0s (gc) │ │ │ │ │ │ │ │ o19 = {1, 6, 13, 13, 6, 1} │ │ │ │ │ │ │ │ o19 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _s_h_e_a_v_e_s -- information about coherent sheaves and total │ │ │ │ coordinate rings (a.k.a. Cox rings) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_is__Well__Defined_lp__Normal__Toric__Variety_rp.html │ │ │ @@ -98,22 +98,22 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i2 : setRandomSeed (currentTime ());
│ │ │ - -- setting random seed to 1792812560
    │ │ │ + -- setting random seed to 1792834952 │ │ │
    │ │ │ 
    i3 : a = sort apply (3, i -> random (7))
│ │ │  
│ │ │ -o3 = {2, 4, 6}
│ │ │ +o3 = {1, 4, 6}
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : assert isWellDefined kleinschmidt (4,a)
    │ │ │ @@ -131,15 +131,15 @@ │ │ │ 
    i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j -> random (1,9));
    │ │ │
    │ │ │ 
    i7 : q
│ │ │  
│ │ │ -o7 = {1, 5, 6, 9, 9}
│ │ │ +o7 = {2, 2, 3, 3, 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 1792812560 │ │ │ │ + -- setting random seed to 1792834952 │ │ │ │ i3 : a = sort apply (3, i -> random (7)) │ │ │ │ │ │ │ │ -o3 = {2, 4, 6} │ │ │ │ +o3 = {1, 4, 6} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : assert isWellDefined kleinschmidt (4,a) │ │ │ │ i5 : q = sort apply (5, j -> random (1,9)); │ │ │ │ i6 : while not all (subsets (q,#q-1), s -> gcd s === 1) do q = sort apply (5, j │ │ │ │ -> random (1,9)); │ │ │ │ i7 : q │ │ │ │ │ │ │ │ -o7 = {1, 5, 6, 9, 9} │ │ │ │ +o7 = {2, 2, 3, 3, 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 │ │ │ @@ -101,15 +101,15 @@ │ │ │ │ │ │ o2 : ToricDivisor on PP2 │ │ │
    │ │ │ 
    i3 : M1 = elapsedTime monomials D1
│ │ │ - -- .033906s elapsed
│ │ │ + -- .0413957s elapsed
│ │ │  
│ │ │         5     4     4   2 3       3   2 3   3 2     2 2   2   2   3 2   4   
│ │ │  o3 = {x , x x , x x , x x , x x x , x x , x x , x x x , x x x , x x , x x ,
│ │ │         2   1 2   0 2   1 2   0 1 2   0 2   1 2   0 1 2   0 1 2   0 2   1 2 
│ │ │       ------------------------------------------------------------------------
│ │ │          3     2 2     3       4     5     4   2 3   3 2   4     5
│ │ │       x x x , x x x , x x x , x x , x , x x , x x , x x , x x , x }
│ │ │ @@ -117,15 +117,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │
    │ │ │ 
    i4 : elapsedTime assert (set M1 === set first entries basis(degree D1, ring variety D1))
│ │ │ - -- .0131253s elapsed
    │ │ │ + -- .00202324s elapsed │ │ │
    │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ @@ -143,27 +143,27 @@ │ │ │ │ │ │ o6 : ToricDivisor on FF2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -176,23 +176,23 @@ │ │ │ │ │ │ o10 : ToricDivisor on X │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i7 : M2 = elapsedTime monomials D2
│ │ │ - -- .134092s elapsed
│ │ │ + -- .0807927s 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))
│ │ │ - -- .00103874s elapsed
    │ │ │ + -- .0010946s elapsed │ │ │
    │ │ │ 
    i9 : X = kleinschmidt (5, {1,2,3});
    │ │ │
    │ │ │ 
    i11 : m3 = elapsedTime # monomials D3
│ │ │ - -- 46.8661s elapsed
│ │ │ + -- 25.5108s elapsed
│ │ │  
│ │ │  o11 = 7909
    │ │ │
    │ │ │ 
    i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety D3))
│ │ │ - -- .183473s elapsed
    │ │ │ + -- .0248134s 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 │ │ │ │ - -- .033906s elapsed │ │ │ │ + -- .0413957s 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)) │ │ │ │ - -- .0131253s elapsed │ │ │ │ + -- .00202324s 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 │ │ │ │ - -- .134092s elapsed │ │ │ │ + -- .0807927s 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)) │ │ │ │ - -- .00103874s elapsed │ │ │ │ + -- .0010946s 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 │ │ │ │ - -- 46.8661s elapsed │ │ │ │ + -- 25.5108s elapsed │ │ │ │ │ │ │ │ o11 = 7909 │ │ │ │ i12 : elapsedTime assert (m3 === #first entries basis (degree D3, ring variety │ │ │ │ D3)) │ │ │ │ - -- .183473s elapsed │ │ │ │ + -- .0248134s elapsed │ │ │ │ By exploiting _l_a_t_t_i_c_e_P_o_i_n_t_s, this method function avoids using the _b_a_s_i_s │ │ │ │ function. │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _w_o_r_k_i_n_g_ _w_i_t_h_ _d_i_v_i_s_o_r_s -- information about toric divisors and their │ │ │ │ related groups │ │ │ │ * _r_i_n_g_(_N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_) -- make the total coordinate ring (a.k.a. Cox │ │ │ │ ring) │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Fan_rp.html │ │ │ @@ -130,25 +130,25 @@ │ │ │
    │ │ │

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

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -48,22 +48,22 @@ │ │ │ │ i5 : assert (transpose matrix rays X == rays F and max X == sort maxCones F) │ │ │ │ The recommended method for creating a _N_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y from a fan is │ │ │ │ _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_L_i_s_t_,_L_i_s_t_). In fact, this package avoids using objects from │ │ │ │ the _P_o_l_y_h_e_d_r_a package whenever possible. Here is a trivial example, namely │ │ │ │ projective 2-space, illustrating the substantial increase in time resulting │ │ │ │ from the use of a _P_o_l_y_h_e_d_r_a fan. │ │ │ │ i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}}) │ │ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc) │ │ │ │ + -- used 2.8503e-05s (cpu); 2.2234e-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 0.0871239s (cpu); 0.08713s (thread); 0s (gc) │ │ │ │ + -- used 0.0493004s (cpu); 0.0493079s (thread); 0s (gc) │ │ │ │ i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _m_a_k_i_n_g_ _n_o_r_m_a_l_ _t_o_r_i_c_ _v_a_r_i_e_t_i_e_s -- information about the basic constructors │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y -- make a normal toric variety │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _n_o_r_m_a_l_T_o_r_i_c_V_a_r_i_e_t_y_(_F_a_n_) -- make a normal toric variety from a 'Polyhedra' │ │ │ │ fan │ │ ├── ./usr/share/doc/Macaulay2/NormalToricVarieties/html/_normal__Toric__Variety_lp__Polyhedron_rp.html │ │ │ @@ -238,21 +238,21 @@ │ │ │ 2 3 │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -102,17 +102,17 @@ │ │ │ │ │ │ │ │ o18 = | 0 1 0 | │ │ │ │ | 0 0 1 | │ │ │ │ │ │ │ │ 2 3 │ │ │ │ o18 : Matrix ZZ <-- ZZ │ │ │ │ i19 : X1 = time normalToricVariety convexHull (vertMatrix); │ │ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc) │ │ │ │ + -- used 0.0223976s (cpu); 0.0224148s (thread); 0s (gc) │ │ │ │ i20 : X2 = time normalToricVariety vertMatrix; │ │ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc) │ │ │ │ + -- used 0.00237006s (cpu); 0.0023743s (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/Normaliz/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ aW50Y2xNb25JZGVhbChJZGVhbCxSaW5nRWxlbWVudCk= │ │ │ #:len=2753 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXphdGlvbiBvZiBSZWVzIGFs │ │ │ Z2VicmEiLCAibGluZW51bSIgPT4gMTg3NiwgSW5wdXRzID0+IHtTUEFOe1NQQU57ImFuICIsVE8y │ │ ├── ./usr/share/doc/Macaulay2/NumericSolutions/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ am9yZGFuRm9ybShNYXRyaXgp │ │ │ #:len=263 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODE0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhqb3JkYW5Gb3JtLE1hdHJpeCksImpvcmRhbkZvcm0o │ │ ├── ./usr/share/doc/Macaulay2/NumericalAlgebraicGeometry/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ aXNTdWJzZXQoTnVtZXJpY2FsVmFyaWV0eSxOdW1lcmljYWxWYXJpZXR5KQ== │ │ │ #:len=1399 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgY29udGFpbm1lbnQiLCAibGlu │ │ │ ZW51bSIgPT4gNzg5LCBJbnB1dHMgPT4ge1NQQU57VFR7IlYifSwiLCAiLFNQQU57ImEgIixUTzJ7 │ │ ├── ./usr/share/doc/Macaulay2/NumericalCertification/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=23 │ │ │ aW50ZXJ2YWxDQ2koUlJpLE51bWJlcik= │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDQsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGludGVydmFs │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=42 │ │ │ bnVtZXJpY2FsSGlsYmVydEZ1bmN0aW9uKFJpbmdNYXAsSWRlYWwsWlop │ │ │ #:len=367 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDUwLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhudW1lcmljYWxIaWxiZXJ0RnVuY3Rpb24sUmluZ01h │ │ ├── ./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 .00691559 seconds │ │ │ + -- used .00708199 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .005626 seconds │ │ │ + -- used .00544165 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00273274 seconds │ │ │ + -- used .00249389 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000418905 seconds │ │ │ + -- used .000356338 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 .00350208 seconds │ │ │ + -- used .00382913 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .00258173 seconds │ │ │ + -- used .00289964 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00100088 seconds │ │ │ + -- used .00106947 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000278743 seconds │ │ │ + -- used .000261173 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 .0119364 seconds │ │ │ + -- used .0118196 seconds │ │ │ Creating interpolation matrix ... │ │ │ - -- used .0107729 seconds │ │ │ + -- used .0112521 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00683732 seconds │ │ │ + -- used .0066749 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000713648 seconds │ │ │ + -- used .000661149 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 .00320623 seconds │ │ │ + -- used .0029113 seconds │ │ │ Performing normalization preconditioning ... │ │ │ - -- used .00838546 seconds │ │ │ + -- used .00818651 seconds │ │ │ Computing numerical kernel ... │ │ │ - -- used .000876183 seconds │ │ │ + -- used .000822101 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 0.117425s (cpu); 0.11742s (thread); 0s (gc) │ │ │ + -- used 0.0723312s (cpu); 0.0723259s (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) │ │ │ - -- .642071s elapsed │ │ │ + -- .407004s elapsed │ │ │ │ │ │ o7 = p │ │ │ │ │ │ o7 : Point │ │ │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/___Convert__To__Cone.html │ │ │ @@ -105,21 +105,21 @@ │ │ │ o4 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -33,21 +33,21 @@ │ │ │ │ o3 : Ideal of R │ │ │ │ i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue │ │ │ │ == 0 │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00691559 seconds │ │ │ │ + -- used .00708199 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .005626 seconds │ │ │ │ + -- used .00544165 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00273274 seconds │ │ │ │ + -- used .00249389 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000418905 seconds │ │ │ │ + -- used .000356338 seconds │ │ │ │ │ │ │ │ o5 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 3 forms in the ideal of the image has dimension 3 │ │ │ │ │ │ │ │ o5 : NumericalInterpolationTable │ │ │ │ i6 : extractImageEquations(T, AttemptZZ => true) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_extract__Image__Equations.html │ │ │ @@ -107,21 +107,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,21 +38,21 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : extractImageEquations(F, ideal 0_R, 2, AttemptZZ => true) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .00350208 seconds │ │ │ │ + -- used .00382913 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .00258173 seconds │ │ │ │ + -- used .00289964 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00100088 seconds │ │ │ │ + -- used .00106947 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000278743 seconds │ │ │ │ + -- used .000261173 seconds │ │ │ │ │ │ │ │ o3 = | y_1^2-y_0y_2 y_1y_2-y_0y_3 y_2^2-y_1y_3 | │ │ │ │ │ │ │ │ 1 3 │ │ │ │ o3 : Matrix (CC [y ..y ]) <-- (CC [y ..y ]) │ │ │ │ 53 0 3 53 0 3 │ │ │ │ Here is how to do the same computation symbolically. │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Hilbert__Function.html │ │ │ @@ -112,21 +112,21 @@ │ │ │ o2 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -152,19 +152,19 @@ │ │ │ 
    i6 : S = numericalImageSample(F, ideal 0_R, 60);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -57,39 +57,39 @@ │ │ │ │ │ │ │ │ o2 = | s3 s2t st2 t3 | │ │ │ │ │ │ │ │ 1 4 │ │ │ │ o2 : Matrix R <-- R │ │ │ │ i3 : numericalHilbertFunction(F, ideal 0_R, 4) │ │ │ │ Sampling image points ... │ │ │ │ - -- used .0119364 seconds │ │ │ │ + -- used .0118196 seconds │ │ │ │ Creating interpolation matrix ... │ │ │ │ - -- used .0107729 seconds │ │ │ │ + -- used .0112521 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00683732 seconds │ │ │ │ + -- used .0066749 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000713648 seconds │ │ │ │ + -- used .000661149 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 .00320623 seconds │ │ │ │ + -- used .0029113 seconds │ │ │ │ Performing normalization preconditioning ... │ │ │ │ - -- used .00838546 seconds │ │ │ │ + -- used .00818651 seconds │ │ │ │ Computing numerical kernel ... │ │ │ │ - -- used .000876183 seconds │ │ │ │ + -- used .000822101 seconds │ │ │ │ │ │ │ │ o7 = a "numerical interpolation table", indicating │ │ │ │ the space of degree 2 forms in the ideal of the image has dimension 1 │ │ │ │ │ │ │ │ o7 : NumericalInterpolationTable │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _N_u_m_e_r_i_c_a_l_I_n_t_e_r_p_o_l_a_t_i_o_n_T_a_b_l_e -- the class of all │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_numerical__Image__Dim.html │ │ │ @@ -145,15 +145,15 @@ │ │ │ 1 70 │ │ │ o8 : Matrix R <-- R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : X1 = time normalToricVariety ({{-1,-1},{1,0},{0,1}}, {{0,1},{1,2},{0,2}})
│ │ │ - -- used 5.1586e-05s (cpu); 4.5906e-05s (thread); 0s (gc)
│ │ │ + -- used 2.8503e-05s (cpu); 2.2234e-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 0.0871239s (cpu); 0.08713s (thread); 0s (gc)
    │ │ │ + -- used 0.0493004s (cpu); 0.0493079s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert (sort rays X1 == sort rays X2 and max X1 == max X2)
    │ │ │
    │ │ │ 
    i19 : X1 = time normalToricVariety convexHull (vertMatrix);
│ │ │ - -- used 0.023116s (cpu); 0.0231153s (thread); 0s (gc)
    │ │ │ + -- used 0.0223976s (cpu); 0.0224148s (thread); 0s (gc) │ │ │
    │ │ │ 
    i20 : X2 = time normalToricVariety vertMatrix;
│ │ │ - -- used 0.00248959s (cpu); 0.00249056s (thread); 0s (gc)
    │ │ │ + -- used 0.00237006s (cpu); 0.0023743s (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 .00691559 seconds
│ │ │ +     -- used .00708199 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .005626 seconds
│ │ │ +     -- used .00544165 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00273274 seconds
│ │ │ +     -- used .00249389 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000418905 seconds
│ │ │ +     -- used .000356338 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 .00350208 seconds
│ │ │ +     -- used .00382913 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .00258173 seconds
│ │ │ +     -- used .00289964 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00100088 seconds
│ │ │ +     -- used .00106947 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000278743 seconds
│ │ │ +     -- used .000261173 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 .0119364 seconds
│ │ │ +     -- used .0118196 seconds
│ │ │  Creating interpolation matrix ...
│ │ │ -     -- used .0107729 seconds
│ │ │ +     -- used .0112521 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00683732 seconds
│ │ │ +     -- used .0066749 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000713648 seconds
│ │ │ +     -- used .000661149 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 .00320623 seconds
│ │ │ +     -- used .0029113 seconds
│ │ │  Performing normalization preconditioning ...
│ │ │ -     -- used .00838546 seconds
│ │ │ +     -- used .00818651 seconds
│ │ │  Computing numerical kernel ...
│ │ │ -     -- used .000876183 seconds
│ │ │ +     -- used .000822101 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 0.117425s (cpu); 0.11742s (thread); 0s (gc)
│ │ │ + -- used 0.0723312s (cpu); 0.0723259s (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 0.117425s (cpu); 0.11742s (thread); 0s (gc) │ │ │ │ + -- used 0.0723312s (cpu); 0.0723259s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 = 69 │ │ │ │ ********** WWaayyss ttoo uussee nnuummeerriiccaallIImmaaggeeDDiimm:: ********** │ │ │ │ * numericalImageDim(List,Ideal) │ │ │ │ * numericalImageDim(List,Ideal,Point) │ │ │ │ * numericalImageDim(Matrix,Ideal) │ │ │ │ * numericalImageDim(Matrix,Ideal,Point) │ │ ├── ./usr/share/doc/Macaulay2/NumericalImplicitization/html/_real__Point.html │ │ │ @@ -137,15 +137,15 @@ │ │ │ │ │ │ o6 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime p = realPoint(I, Iterations => 100)
│ │ │ - -- .642071s elapsed
│ │ │ + -- .407004s 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) │ │ │ │ - -- .642071s elapsed │ │ │ │ + -- .407004s elapsed │ │ │ │ │ │ │ │ o7 = p │ │ │ │ │ │ │ │ o7 : Point │ │ │ │ i8 : matrix pack(5, p#Coordinates) │ │ │ │ │ │ │ │ o8 = | .722359 .289465 -.295808 .591752 -.454678 | │ │ ├── ./usr/share/doc/Macaulay2/NumericalLinearAlgebra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ bnVtZXJpY2FsS2VybmVsKC4uLixUb2xlcmFuY2U9Pi4uLik= │ │ │ #:len=352 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjI0LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tudW1lcmljYWxLZXJuZWwsVG9sZXJhbmNlXSwibnVt │ │ ├── ./usr/share/doc/Macaulay2/NumericalSchubertCalculus/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ UGllcmlSb290Q291bnQoLi4uLFZlcmJvc2U9Pi4uLik= │ │ │ #:len=334 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmVxdWVzdCB2ZXJib3NlIGZlZWRiYWNr │ │ │ IiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAzMzMsIHN5bWJvbCBEb2N1bWVudFRh │ │ ├── ./usr/share/doc/Macaulay2/NumericalSemigroups/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ c2VtaWdyb3VwRnJvbU11KExpc3Qp │ │ │ #:len=289 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzAwNCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2VtaWdyb3VwRnJvbU11LExpc3QpLCJzZW1pZ3Jv │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=29 │ │ │ dG9TdHJpbmcoUG9seW5vbWlhbE9JQWxnZWJyYSk= │ │ │ #:len=1044 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheSBhIHBvbHlub21pYWwgT0kt │ │ │ YWxnZWJyYSBpbiBjb25kZW5zZWQgZm9ybSIsICJsaW5lbnVtIiA9PiAxNTE2LCBJbnB1dHMgPT4g │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/___Free__O__I__Module__Map.out │ │ │ @@ -7,44 +7,44 @@ │ │ │ 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}, {-5, -4, -3, -2, -5, -5, -4, -4, -3}) │ │ │ + 1: (e1, {5, 5}, {-3, -4}) │ │ │ + 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -2, -4, -5, -5, -3, -4}) │ │ │ │ │ │ o5 : OIResolution │ │ │ │ │ │ i6 : phi = C.dd_1 │ │ │ │ │ │ -o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ │ +o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ │ │ │ │ o6 : FreeOIModuleMap │ │ │ │ │ │ i7 : G = getBasisElements C_1 │ │ │ │ │ │ o7 = {1e1 , 1e1 } │ │ │ 5,{1, 2, 3, 4, 5},1 5,{1, 2, 3, 4, 5},2 │ │ │ │ │ │ o7 : List │ │ │ │ │ │ i8 : phi G#0 │ │ │ │ │ │ -o8 = x x e0 - 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, 5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ +o8 = -x e0 + 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 2,3 5,{1, 2, 4},1 │ │ │ │ │ │ 10 │ │ │ o8 : (QQ[x , x , x , x , x , x , x , x , x , x ]) in width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2} │ │ │ 2,5 2,4 2,3 2,2 2,1 1,5 1,4 1,3 1,2 1,1 │ │ │ │ │ │ i9 : phi G#1 │ │ │ │ │ │ -o9 = -x e0 + 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 2,3 5,{1, 2, 4},1 │ │ │ +o9 = x x e0 - 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, 5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ │ │ │ 10 │ │ │ o9 : (QQ[x , x , x , x , x , x , x , x , x , x ]) in width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2} │ │ │ 2,5 2,4 2,3 2,2 2,1 1,5 1,4 1,3 1,2 1,1 │ │ │ │ │ │ i10 : │ │ ├── ./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}, {-4, -3, -2, -5, -4, -3, -5, -4, -5}) │ │ │ + 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-4, -3, -4, -5, -3, -2, -4, -5, -5}) │ │ │ │ │ │ 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 0.0877009s (cpu); 0.0877049s (thread); 0s (gc) │ │ │ + -- used 0.0778809s (cpu); 0.0778813s (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 0.224314s (cpu); 0.152688s (thread); 0s (gc) │ │ │ + -- used 0.184811s (cpu); 0.109546s (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 0.459073s (cpu); 0.290457s (thread); 0s (gc) │ │ │ + -- used 0.419349s (cpu); 0.263685s (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 0.0985988s (cpu); 0.0985983s (thread); 0s (gc) │ │ │ + -- used 0.0801653s (cpu); 0.0801659s (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) │ │ │ @@ -21,11 +21,11 @@ │ │ │ Differential: Source: (e0, {2}, {-2}) Target: (e, {1, 1}, {0, 0}) │ │ │ 1: Module: Basis symbol: e1 │ │ │ Basis element widths: {4, 4} │ │ │ Degree shifts: {-4, -4} │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ │ Monomial order: Lex │ │ │ Differential: Source: (e1, {4, 4}, {-4, -4}) Target: (e0, {2}, {-2}) │ │ │ - Basis element images: {-x x e0 + x x e0 + x x e0 - x x e0 , 0} │ │ │ - 2,3 1,1 4,{2, 4},1 2,4 1,1 4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ + Basis element images: {0, -x x e0 + x x e0 + x x e0 - x x e0 } │ │ │ + 2,3 1,1 4,{2, 4},1 2,4 1,1 4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ │ │ i7 : │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_describe_lp__Free__O__I__Module__Map_rp.out │ │ │ @@ -10,18 +10,21 @@ │ │ │ │ │ │ i5 : C = oiRes({b}, 2); │ │ │ │ │ │ i6 : phi = C.dd_1; │ │ │ │ │ │ i7 : describe phi │ │ │ │ │ │ -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 │ │ │ +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 │ │ │ ------------------------------------------------------------------------ │ │ │ - - 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 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 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 │ │ │ + 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 │ │ │ │ │ │ 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 0.0855533s (cpu); 0.0855538s (thread); 0s (gc) │ │ │ + -- used 0.133181s (cpu); 0.133181s (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/_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 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 │ │ │ +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, │ │ │ ------------------------------------------------------------------------ │ │ │ - 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 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 x e0 + x x e0 } │ │ │ - 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ + + x e0 - x e0 } │ │ │ + 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},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 0.357846s (cpu); 0.284765s (thread); 0s (gc) │ │ │ + -- used 0.310358s (cpu); 0.240604s (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 0.167028s (cpu); 0.0579109s (thread); 0s (gc) │ │ │ + -- used 0.12668s (cpu); 0.046857s (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 0.0295377s (cpu); 0.0295415s (thread); 0s (gc) │ │ │ + -- used 0.0298772s (cpu); 0.0298769s (thread); 0s (gc) │ │ │ │ │ │ o10 = {x e + x e , x x e + x x e , │ │ │ 1,1 1,{1},1 2,1 1,{1},2 1,2 1,1 2,{2},2 2,2 2,1 2,{1, 2},3 │ │ │ ----------------------------------------------------------------------- │ │ │ x x x e - x x x e } │ │ │ 2,3 2,2 1,1 3,{2, 3},3 2,3 2,1 1,2 3,{1, 3},3 │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_net_lp__O__I__Resolution_rp.out │ │ │ @@ -5,15 +5,15 @@ │ │ │ i2 : F = makeFreeOIModule(e, {1,1}, P); │ │ │ │ │ │ i3 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2); │ │ │ │ │ │ i5 : time C = oiRes({b}, 1); │ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc) │ │ │ + -- used 0.189228s (cpu); 0.107306s (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 0.0272125s (cpu); 0.0272119s (thread); 0s (gc) │ │ │ + -- used 0.0291909s (cpu); 0.0291896s (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 0.652096s (cpu); 0.338501s (thread); 0s (gc) │ │ │ + -- used 0.536966s (cpu); 0.28093s (thread); 0s (gc) │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ 1: (e1, {4}, {-4}) │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ o5 : OIResolution │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/example-output/_reduce__O__I__G__B.out │ │ │ @@ -9,15 +9,15 @@ │ │ │ i4 : installGeneratorsInWidth(F, 2); │ │ │ │ │ │ i5 : use F_1; b1 = x_(2,1)*e_(1,{1},2)+x_(1,1)*e_(1,{1},2); │ │ │ │ │ │ i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2); │ │ │ │ │ │ i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal) │ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc) │ │ │ + -- used 0.12385s (cpu); 0.12385s (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 │ │ │ @@ -83,25 +83,25 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    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}, {-5, -4, -3, -2, -5, -5, -4, -4, -3})
│ │ │ +     1: (e1, {5, 5}, {-3, -4})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -2, -4, -5, -5, -3, -4})
│ │ │  
│ │ │  o5 : OIResolution
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : phi = C.dd_1
│ │ │  
│ │ │ -o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2})
│ │ │ +o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2})
│ │ │  
│ │ │  o6 : FreeOIModuleMap
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : G = getBasisElements C_1
│ │ │ @@ -112,28 +112,28 @@
│ │ │  o7 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : phi G#0
│ │ │  
│ │ │ -o8 = x   x   e0              - 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, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │ +o8 = -x   e0              + 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    2,3  5,{1, 2, 4},1
│ │ │  
│ │ │                                                                       10
│ │ │  o8 : (QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ])   in width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}
│ │ │            2,5   2,4   2,3   2,2   2,1   1,5   1,4   1,3   1,2   1,1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : phi G#1
│ │ │  
│ │ │ -o9 = -x   e0              + 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    2,3  5,{1, 2, 4},1
│ │ │ +o9 = x   x   e0              - 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, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │  
│ │ │                                                                       10
│ │ │  o9 : (QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ])   in width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2}
│ │ │            2,5   2,4   2,3   2,2   2,1   1,5   1,4   1,3   1,2   1,1
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -16,45 +16,45 @@ │ │ │ │ i1 : P = makePolynomialOIAlgebra(2, x, QQ); │ │ │ │ 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}, {-5, -4, -3, -2, -5, -5, -4, -4, -3}) │ │ │ │ + 1: (e1, {5, 5}, {-3, -4}) │ │ │ │ + 2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-3, -4, -5, -2, -4, -5, -5, -3, -4}) │ │ │ │ │ │ │ │ o5 : OIResolution │ │ │ │ i6 : phi = C.dd_1 │ │ │ │ │ │ │ │ -o6 = Source: (e1, {5, 5}, {-4, -3}) Target: (e0, {3}, {-2}) │ │ │ │ +o6 = Source: (e1, {5, 5}, {-3, -4}) Target: (e0, {3}, {-2}) │ │ │ │ │ │ │ │ o6 : FreeOIModuleMap │ │ │ │ i7 : G = getBasisElements C_1 │ │ │ │ │ │ │ │ o7 = {1e1 , 1e1 } │ │ │ │ 5,{1, 2, 3, 4, 5},1 5,{1, 2, 3, 4, 5},2 │ │ │ │ │ │ │ │ o7 : List │ │ │ │ i8 : phi G#0 │ │ │ │ │ │ │ │ -o8 = x x e0 - 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, │ │ │ │ -5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ │ +o8 = -x e0 + 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 2,3 │ │ │ │ +5,{1, 2, 4},1 │ │ │ │ │ │ │ │ 10 │ │ │ │ o8 : (QQ[x , x , x , x , x , x , x , x , x , x ]) in │ │ │ │ width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2} │ │ │ │ 2,5 2,4 2,3 2,2 2,1 1,5 1,4 1,3 1,2 1,1 │ │ │ │ i9 : phi G#1 │ │ │ │ │ │ │ │ -o9 = -x e0 + 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 2,3 │ │ │ │ -5,{1, 2, 4},1 │ │ │ │ +o9 = x x e0 - 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, │ │ │ │ +5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ │ │ │ │ │ 10 │ │ │ │ o9 : (QQ[x , x , x , x , x , x , x , x , x , x ]) in │ │ │ │ width 5, degrees {2, 2, 2, 2, 2, 2, 2, 2, 2, 2} │ │ │ │ 2,5 2,4 2,3 2,2 2,1 1,5 1,4 1,3 1,2 1,1 │ │ │ │ ********** FFuunnccttiioonnss aanndd mmeetthhooddss rreettuurrnniinngg aann oobbjjeecctt ooff ccllaassss FFrreeeeOOIIMMoodduulleeMMaapp:: ********** │ │ │ │ * _g_e_t_S_c_h_r_e_y_e_r_M_a_p -- get the Schreyer map of a free OI-module if it exists │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Free__O__I__Module__Map_sp__Vector__In__Width.html │ │ │ @@ -100,15 +100,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ 
    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}, {-4, -3, -2, -5, -4, -3, -5, -4, -5})
│ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-4, -3, -4, -5, -3, -2, -4, -5, -5})
│ │ │  
│ │ │  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}, {-4, -3, -2, -5, -4, -3, -5, -4, -5})
│ │ │ │ +     2: (e2, {6, 6, 6, 6, 6, 6, 6, 6, 6}, {-4, -3, -4, -5, -3, -2, -4, -5, -5})
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -79,15 +79,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1)
│ │ │ - -- used 0.0877009s (cpu); 0.0877049s (thread); 0s (gc)
│ │ │ + -- used 0.0778809s (cpu); 0.0778813s (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 0.0877009s (cpu); 0.0877049s (thread); 0s (gc)
│ │ │ │ + -- used 0.0778809s (cpu); 0.0778813s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  i6 : C.dd_0
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___O__I__Resolution_sp_us_sp__Z__Z.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.224314s (cpu); 0.152688s (thread); 0s (gc)
    
│ │ │ + -- used 0.184811s (cpu); 0.109546s (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 0.224314s (cpu); 0.152688s (thread); 0s (gc)
│ │ │ │ + -- used 0.184811s (cpu); 0.109546s (thread); 0s (gc)
│ │ │ │  i6 : C_0
│ │ │ │  
│ │ │ │  o6 = Basis symbol: e0
│ │ │ │       Basis element widths: {2}
│ │ │ │       Degree shifts: {-2}
│ │ │ │       Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │ │       Monomial order: Lex
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/___Top__Nonminimal.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ + -- used 0.419349s (cpu); 0.263685s (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 0.459073s (cpu); 0.290457s (thread); 0s (gc)
│ │ │ │ + -- used 0.419349s (cpu); 0.263685s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd TTooppNNoonnmmiinniimmaall:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe__Full.html
│ │ │ @@ -95,15 +95,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0985988s (cpu); 0.0985983s (thread); 0s (gc)
    
│ │ │ + -- used 0.0801653s (cpu); 0.0801659s (thread); 0s (gc)
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : describeFull C
│ │ │  
│ │ │  o6 = 0: Module: Basis symbol: e0
│ │ │ @@ -114,16 +114,16 @@
│ │ │       Differential: Source: (e0, {2}, {-2}) Target: (e, {1, 1}, {0, 0})
│ │ │       1: Module: Basis symbol: e1
│ │ │                  Basis element widths: {4, 4}
│ │ │                  Degree shifts: {-4, -4}
│ │ │                  Polynomial OI-algebra: (2, x, QQ, RowUpColUp)
│ │ │                  Monomial order: Lex
│ │ │       Differential: Source: (e1, {4, 4}, {-4, -4}) Target: (e0, {2}, {-2})
│ │ │ -                   Basis element images: {-x   x   e0           + x   x   e0           + x   x   e0           - x   x   e0          , 0}
│ │ │ -                                            2,3 1,1  4,{2, 4},1    2,4 1,1  4,{2, 3},1    2,3 1,2  4,{1, 4},1    2,4 1,2  4,{1, 3},1
    │ │ │ + Basis element images: {0, -x x e0 + x x e0 + x x e0 - x x e0 } │ │ │ + 2,3 1,1 4,{2, 4},1 2,4 1,1 4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use describeFull:

    │ │ │ ├── html2text {} │ │ │ │ @@ -14,33 +14,33 @@ │ │ │ │ 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 0.0985988s (cpu); 0.0985983s (thread); 0s (gc) │ │ │ │ + -- used 0.0801653s (cpu); 0.0801659s (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 │ │ │ │ Differential: Source: (e0, {2}, {-2}) Target: (e, {1, 1}, {0, 0}) │ │ │ │ 1: Module: Basis symbol: e1 │ │ │ │ Basis element widths: {4, 4} │ │ │ │ Degree shifts: {-4, -4} │ │ │ │ Polynomial OI-algebra: (2, x, QQ, RowUpColUp) │ │ │ │ Monomial order: Lex │ │ │ │ Differential: Source: (e1, {4, 4}, {-4, -4}) Target: (e0, {2}, {-2}) │ │ │ │ - Basis element images: {-x x e0 + x x e0 │ │ │ │ -+ x x e0 - x x e0 , 0} │ │ │ │ - 2,3 1,1 4,{2, 4},1 2,4 1,1 4, │ │ │ │ -{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ + Basis element images: {0, -x x e0 + x x e0 │ │ │ │ ++ x x e0 - x x e0 } │ │ │ │ + 2,3 1,1 4,{2, 4},1 2,4 1,1 │ │ │ │ +4,{2, 3},1 2,3 1,2 4,{1, 4},1 2,4 1,2 4,{1, 3},1 │ │ │ │ ********** WWaayyss ttoo uussee ddeessccrriibbeeFFuullll:: ********** │ │ │ │ * describeFull(OIResolution) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _d_e_s_c_r_i_b_e_F_u_l_l is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:2639:0. │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__Free__O__I__Module__Map_rp.html │ │ │ @@ -107,23 +107,26 @@ │ │ │ 
    i6 : phi = C.dd_1;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : describe phi
│ │ │  
│ │ │ -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
│ │ │ +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  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     - 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   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   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
    │ │ │ + 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 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -18,21 +18,24 @@ │ │ │ │ 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}, {-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 │ │ │ │ +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 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - - 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 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 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 │ │ │ │ + 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 │ │ │ │ ********** 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- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/OIGroebnerBases.m2:1979:0. │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_describe_lp__O__I__Resolution_rp.html │ │ │ @@ -96,15 +96,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
    │ │ │ + -- used 0.133181s (cpu); 0.133181s (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 0.0855533s (cpu); 0.0855538s (thread); 0s (gc)
│ │ │ │ + -- used 0.133181s (cpu); 0.133181s (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/_image_lp__Free__O__I__Module__Map_rp.html
│ │ │ @@ -107,22 +107,22 @@
│ │ │                
    i6 : phi = C.dd_1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : image phi
│ │ │  
│ │ │ -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  
│ │ │ +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,
│ │ │       ------------------------------------------------------------------------
│ │ │ -     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   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   x   e0              + x   x   e0             }
│ │ │ -      2,3 1,2  5,{1, 4, 5},1    2,4 1,2  5,{1, 3, 5},1
│ │ │ +          + x   e0              - x   e0             }
│ │ │ +     4},1    2,3  5,{1, 2, 5},1    2,3  5,{1, 2, 4},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 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 │ │ │ │ +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, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - 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 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 x e0 + x x e0 } │ │ │ │ - 2,3 1,2 5,{1, 4, 5},1 2,4 1,2 5,{1, 3, 5},1 │ │ │ │ + + x e0 - x e0 } │ │ │ │ + 4},1 2,3 5,{1, 2, 5},1 2,3 5,{1, 2, 4},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 │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ │ │ 
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time C = oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.357846s (cpu); 0.284765s (thread); 0s (gc)
│ │ │ + -- used 0.310358s (cpu); 0.240604s (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 0.357846s (cpu); 0.284765s (thread); 0s (gc) │ │ │ │ + -- used 0.310358s (cpu); 0.240604s (thread); 0s (gc) │ │ │ │ │ │ │ │ o5 = 0: (e0, {2}, {-2}) │ │ │ │ 1: (e1, {4}, {-4}) │ │ │ │ 2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5}) │ │ │ │ │ │ │ │ o5 : OIResolution │ │ │ │ i6 : isComplex C │ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_is__O__I__G__B.html │ │ │ @@ -121,15 +121,15 @@ │ │ │ │ │ │ o10 = false     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ + -- used 0.12668s (cpu); 0.046857s (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 0.167028s (cpu); 0.0579109s (thread); 0s (gc)
│ │ │ │ + -- used 0.12668s (cpu); 0.046857s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o11 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_minimize__O__I__G__B.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │              
│ │ │                
    i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : time B = oiGB {b1, b2}
│ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ + -- used 0.0298772s (cpu); 0.0298769s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │        -----------------------------------------------------------------------
│ │ │        x   x   x   e           - x   x   x   e          }
│ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : installGeneratorsInWidth(F, 3);
│ │ │ │  i6 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i8 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i10 : time B = oiGB {b1, b2}
│ │ │ │ - -- used 0.0295377s (cpu); 0.0295415s (thread); 0s (gc)
│ │ │ │ + -- used 0.0298772s (cpu); 0.0298769s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │          1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        x   x   x   e           - x   x   x   e          }
│ │ │ │         2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_net_lp__O__I__Resolution_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time C = oiRes({b}, 1);
│ │ │ - -- used 0.368596s (cpu); 0.179956s (thread); 0s (gc)
    
│ │ │ + -- used 0.189228s (cpu); 0.107306s (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 0.368596s (cpu); 0.179956s (thread); 0s (gc)
│ │ │ │ + -- used 0.189228s (cpu); 0.107306s (thread); 0s (gc)
│ │ │ │  i6 : net C
│ │ │ │  
│ │ │ │  o6 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4, 4}, {-4, -4})
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _n_e_t_(_O_I_R_e_s_o_l_u_t_i_o_n_) -- display an OI-resolution
│ │ │ │  ===============================================================================
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__G__B.html
│ │ │ @@ -117,15 +117,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time oiGB {b1, b2}
│ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ + -- used 0.0291909s (cpu); 0.0291896s (thread); 0s (gc)
│ │ │  
│ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   x   e           - x   x   x   e          }
│ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  i1 : P = makePolynomialOIAlgebra(2, x, QQ);
│ │ │ │  i2 : F = makeFreeOIModule(e, {1,1,2}, P);
│ │ │ │  i3 : installGeneratorsInWidth(F, 1);
│ │ │ │  i4 : installGeneratorsInWidth(F, 2);
│ │ │ │  i5 : use F_1; b1 = x_(1,1)*e_(1,{1},1)+x_(2,1)*e_(1,{1},2);
│ │ │ │  i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},2)+x_(2,2)*x_(2,1)*e_(2,{1,2},3);
│ │ │ │  i9 : time oiGB {b1, b2}
│ │ │ │ - -- used 0.0272125s (cpu); 0.0272119s (thread); 0s (gc)
│ │ │ │ + -- used 0.0291909s (cpu); 0.0291896s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o9 = {x   e        + x   e       , x   x   e        + x   x   e          ,
│ │ │ │         1,1 1,{1},1    2,1 1,{1},2   1,2 1,1 2,{2},2    2,2 2,1 2,{1, 2},3
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       x   x   x   e           - x   x   x   e          }
│ │ │ │        2,3 2,2 1,1 3,{2, 3},3    2,3 2,1 1,2 3,{1, 3},3
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_oi__Res.html
│ │ │ @@ -112,15 +112,15 @@
│ │ │              
│ │ │                
    i4 : b = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(2,1)*e_(2,{1},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time oiRes({b}, 2, TopNonminimal => true)
│ │ │ - -- used 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ + -- used 0.536966s (cpu); 0.28093s (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 0.652096s (cpu); 0.338501s (thread); 0s (gc)
│ │ │ │ + -- used 0.536966s (cpu); 0.28093s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 0: (e0, {2}, {-2})
│ │ │ │       1: (e1, {4}, {-4})
│ │ │ │       2: (e2, {4, 5, 5, 5, 5, 5}, {-4, -5, -5, -5, -5, -5})
│ │ │ │  
│ │ │ │  o5 : OIResolution
│ │ │ │  ********** WWaayyss ttoo uussee ooiiRReess:: **********
│ │ ├── ./usr/share/doc/Macaulay2/OIGroebnerBases/html/_reduce__O__I__G__B.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │              
│ │ │                
    i7 : use F_2; b2 = x_(1,2)*x_(1,1)*e_(2,{2},1)+x_(2,2)*x_(1,2)*e_(2,{2},2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time B = oiGB({b1, b2}, Strategy => FastNonminimal)
│ │ │ - -- used 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ + -- used 0.12385s (cpu); 0.12385s (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 0.133994s (cpu); 0.133994s (thread); 0s (gc)
│ │ │ │ + -- used 0.12385s (cpu); 0.12385s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  Z3JhZGVkTW9kdWxlKE1vZHVsZSk=
│ │ │  #:len=294
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODEsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KGdyYWRlZE1vZHVsZSxNb2R1bGUpLCJncmFkZWRNb2R1
│ │ ├── ./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)
│ │ │ - -- 1.66263s elapsed
│ │ │ + -- 1.64492s 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_*)
│ │ │ - -- 1.80662s elapsed
│ │ │ + -- 1.18591s 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)
│ │ │ - -- 1.7181s elapsed
│ │ │ + -- 1.69112s 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
│ │ │ @@ -36,22 +36,19 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ + -- used 1.12401s (cpu); 0.983389s (thread); 0s (gc)
│ │ │ + -- used 0.515617s (cpu); 0.445249s (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
│ │ │ @@ -94,28 +94,28 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.66263s elapsed
│ │ │ + -- 1.64492s 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_*)
│ │ │ - -- 1.80662s elapsed
│ │ │ + -- 1.18591s 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)
│ │ │ │ - -- 1.66263s elapsed
│ │ │ │ + -- 1.64492s 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_*)
│ │ │ │ - -- 1.80662s elapsed
│ │ │ │ + -- 1.18591s elapsed
│ │ │ │  
│ │ │ │        1      35      140      385      819      1080      819      385      140
│ │ │ │  35      1
│ │ │ │  o4 = S  <-- S   <-- S    <-- S    <-- S    <-- S     <-- S    <-- S    <-- S
│ │ │ │  <-- S   <-- S  <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_betti_lp..._cm__Minimize_eq_gt..._rp.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o2 : PolynomialRing
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime C = res(I, FastNonminimal => true)
│ │ │ - -- 1.7181s elapsed
│ │ │ + -- 1.69112s 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)
│ │ │ │ - -- 1.7181s elapsed
│ │ │ │ + -- 1.69112s elapsed
│ │ │ │  
│ │ │ │        1      35      241      841      1781      2464      2294      1432
│ │ │ │  576      135      14
│ │ │ │  o3 = S  <-- S   <-- S    <-- S    <-- S     <-- S     <-- S     <-- S     <-- S
│ │ │ │  <-- S    <-- S   <-- 0
│ │ ├── ./usr/share/doc/Macaulay2/OldChainComplexes/html/_computing_spresolutions.html
│ │ │ @@ -117,22 +117,19 @@
│ │ │            << res M << endl << endl;
│ │ │            break;
│ │ │            ) else (
│ │ │            << "-- computation interrupted" << endl;
│ │ │            status M.cache.resolution;
│ │ │            << "-- continuing the computation" << endl;
│ │ │            ))
│ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ + -- used 1.12401s (cpu); 0.983389s (thread); 0s (gc)
│ │ │ + -- used 0.515617s (cpu); 0.445249s (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 {}
│ │ │ │ @@ -50,22 +50,19 @@
│ │ │ │            << res M << endl << endl;
│ │ │ │            break;
│ │ │ │            ) else (
│ │ │ │            << "-- computation interrupted" << endl;
│ │ │ │            status M.cache.resolution;
│ │ │ │            << "-- continuing the computation" << endl;
│ │ │ │            ))
│ │ │ │ - -- used 0.906586s (cpu); 0.748056s (thread); 0s (gc)
│ │ │ │ - -- used 1.12461s (cpu); 0.850193s (thread); 0s (gc)
│ │ │ │ - -- used 0.0418703s (cpu); 0.0433256s (thread); 0s (gc)
│ │ │ │ + -- used 1.12401s (cpu); 0.983389s (thread); 0s (gc)
│ │ │ │ + -- used 0.515617s (cpu); 0.445249s (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/OnlineLookup/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  b2Vpcw==
│ │ │  #:len=753
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT0VJUyBsb29rdXAiLCBEZXNjcmlwdGlv
│ │ │  biA9PiAoRElWe1BBUkF7VEVYeyJUaGlzIGZ1bmN0aW9uIGxvb2tzIHVwIHRoZSBhcmd1bWVudCAo
│ │ ├── ./usr/share/doc/Macaulay2/OpenMath/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=8
│ │ │  T3Blbk1hdGg=
│ │ │  #:len=478
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3Blbk1hdGggc3VwcG9ydCIsICJsaW5l
│ │ │  bnVtIiA9PiA4NSwgU2VlQWxzbyA9PiBESVZ7SEVBREVSMnsiU2VlIGFsc28ifSxVTHtMSXtUT0h7
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  b3NjUmluZyguLi4sQ29lZmZpY2llbnRSaW5nPT4uLi4p
│ │ │  #:len=277
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvc2NSaW5nLENvZWZmaWNpZW50UmluZ10sIm9zY1Jp
│ │ ├── ./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}
│ │ │            );
│ │ │ - -- .284219s elapsed
│ │ │ - -- .401541s elapsed
│ │ │ - -- .867569s elapsed
│ │ │ - -- .245534s elapsed
│ │ │ - -- .354798s elapsed
│ │ │ - -- .429779s elapsed
│ │ │ - -- .666427s elapsed
│ │ │ - -- .418819s elapsed
│ │ │ - -- .478799s elapsed
│ │ │ - -- .409628s elapsed
│ │ │ - -- .282237s elapsed
│ │ │ + -- .20308s elapsed
│ │ │ + -- .250248s elapsed
│ │ │ + -- .378741s elapsed
│ │ │ + -- .197857s elapsed
│ │ │ + -- .223621s elapsed
│ │ │ + -- .250067s elapsed
│ │ │ + -- .458254s elapsed
│ │ │ + -- .351212s elapsed
│ │ │ + -- .369474s elapsed
│ │ │ + -- .28027s elapsed
│ │ │ + -- .188614s 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}
│ │ │            );
│ │ │ - -- .539129s elapsed
│ │ │ - -- .645006s elapsed
│ │ │ - -- 1.05918s elapsed
│ │ │ - -- 1.23488s elapsed
│ │ │ - -- .813801s elapsed
│ │ │ - -- 1.01888s elapsed
│ │ │ - -- 1.18556s elapsed
│ │ │ - -- 1.19237s elapsed
│ │ │ - -- .809367s elapsed
│ │ │ - -- .688287s elapsed
│ │ │ - -- .34871s elapsed
│ │ │ - -- .470565s elapsed
│ │ │ - -- .796532s elapsed
│ │ │ - -- .592092s elapsed
│ │ │ - -- .870412s elapsed
│ │ │ - -- 1.45151s elapsed
│ │ │ - -- 1.08533s elapsed
│ │ │ - -- .929525s elapsed
│ │ │ - -- 1.53034s elapsed
│ │ │ - -- 1.33678s elapsed
│ │ │ - -- .760184s elapsed
│ │ │ - -- 1.28518s elapsed
│ │ │ - -- 1.71666s elapsed
│ │ │ - -- 1.22854s elapsed
│ │ │ - -- .674943s elapsed
│ │ │ - -- .894164s elapsed
│ │ │ - -- 1.57127s elapsed
│ │ │ - -- .743101s elapsed
│ │ │ - -- .756756s elapsed
│ │ │ - -- 1.13138s elapsed
│ │ │ - -- 1.25775s elapsed
│ │ │ - -- 1.00643s elapsed
│ │ │ - -- .580285s elapsed
│ │ │ - -- 1.47228s elapsed
│ │ │ - -- 1.02334s elapsed
│ │ │ - -- 1.53993s elapsed
│ │ │ - -- .942582s elapsed
│ │ │ - -- 1.21686s elapsed
│ │ │ - -- 1.59056s elapsed
│ │ │ - -- 1.01279s elapsed
│ │ │ - -- .781942s elapsed
│ │ │ - -- 1.63071s elapsed
│ │ │ - -- 1.80895s elapsed
│ │ │ - -- 2.00814s elapsed
│ │ │ - -- 1.15234s elapsed
│ │ │ - -- 1.32068s elapsed
│ │ │ - -- 1.58339s elapsed
│ │ │ - -- 1.20196s elapsed
│ │ │ - -- 1.10322s elapsed
│ │ │ - -- 1.215s elapsed
│ │ │ - -- 1.22183s elapsed
│ │ │ - -- .920406s elapsed
│ │ │ - -- .900219s elapsed
│ │ │ - -- 1.32942s elapsed
│ │ │ - -- .609367s elapsed
│ │ │ - -- 1.44447s elapsed
│ │ │ - -- 1.35121s elapsed
│ │ │ - -- 1.47404s elapsed
│ │ │ - -- .758204s elapsed
│ │ │ - -- .533039s elapsed
│ │ │ - -- .40806s elapsed
│ │ │ + -- .316237s elapsed
│ │ │ + -- .370322s elapsed
│ │ │ + -- .764898s elapsed
│ │ │ + -- .938243s elapsed
│ │ │ + -- .565708s elapsed
│ │ │ + -- .695381s elapsed
│ │ │ + -- .825957s elapsed
│ │ │ + -- .831219s elapsed
│ │ │ + -- .57875s elapsed
│ │ │ + -- .595979s elapsed
│ │ │ + -- .278365s elapsed
│ │ │ + -- .355189s elapsed
│ │ │ + -- .405891s elapsed
│ │ │ + -- .519177s elapsed
│ │ │ + -- .673817s elapsed
│ │ │ + -- 1.01805s elapsed
│ │ │ + -- .733166s elapsed
│ │ │ + -- .709782s elapsed
│ │ │ + -- .98038s elapsed
│ │ │ + -- .818091s elapsed
│ │ │ + -- .63529s elapsed
│ │ │ + -- .742332s elapsed
│ │ │ + -- 1.10426s elapsed
│ │ │ + -- 1.01625s elapsed
│ │ │ + -- .397908s elapsed
│ │ │ + -- .543744s elapsed
│ │ │ + -- 1.01968s elapsed
│ │ │ + -- .549338s elapsed
│ │ │ + -- .459191s elapsed
│ │ │ + -- .653864s elapsed
│ │ │ + -- .809097s elapsed
│ │ │ + -- .635191s elapsed
│ │ │ + -- .44622s elapsed
│ │ │ + -- .835689s elapsed
│ │ │ + -- .634804s elapsed
│ │ │ + -- .83954s elapsed
│ │ │ + -- .7494s elapsed
│ │ │ + -- .955647s elapsed
│ │ │ + -- 1.00728s elapsed
│ │ │ + -- .643989s elapsed
│ │ │ + -- .572942s elapsed
│ │ │ + -- .908814s elapsed
│ │ │ + -- 1.10631s elapsed
│ │ │ + -- 1.40448s elapsed
│ │ │ + -- .904718s elapsed
│ │ │ + -- .905454s elapsed
│ │ │ + -- 1.13155s elapsed
│ │ │ + -- .967963s elapsed
│ │ │ + -- .799505s elapsed
│ │ │ + -- .847803s elapsed
│ │ │ + -- .894674s elapsed
│ │ │ + -- .635332s elapsed
│ │ │ + -- .697491s elapsed
│ │ │ + -- .836636s elapsed
│ │ │ + -- .504753s elapsed
│ │ │ + -- .942339s elapsed
│ │ │ + -- 1.01034s elapsed
│ │ │ + -- 1.06347s elapsed
│ │ │ + -- .612756s elapsed
│ │ │ + -- .382879s elapsed
│ │ │ + -- .297488s 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
│ │ │ - -- .859s elapsed
│ │ │ + -- .83s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -60,15 +60,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .901s elapsed
│ │ │ + -- .874s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/___S__C__T_spgraphs_spwith_spexotic_spsolutions.out
│ │ │ @@ -44,19 +44,19 @@
│ │ │  
│ │ │  i5 : printingPrecision = 3
│ │ │  
│ │ │  o5 = 3
│ │ │  
│ │ │  i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .646s elapsed
│ │ │ + -- .569s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .578s elapsed
│ │ │ - -- .694s elapsed
│ │ │ - -- .859s elapsed
│ │ │ + -- .549s elapsed
│ │ │ + -- .677s elapsed
│ │ │ + -- .866s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -69,20 +69,20 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .966s elapsed
│ │ │ - -- 1.19s elapsed
│ │ │ + -- .998s elapsed
│ │ │ + -- 1.03s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.28s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.07s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ + -- 1.23s elapsed
│ │ │ + -- 1.15s elapsed
│ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ - -- 1.21s elapsed
│ │ │ + -- 1.24s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.03s elapsed
│ │ │ + -- 1.12s elapsed
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_get__Linearly__Stable__Solutions.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1729328129346969841
│ │ │  
│ │ │  i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
│ │ │  
│ │ │  i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .215945s elapsed
│ │ │ + -- .186387s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/example-output/_show__Exotic__Solutions.out
│ │ │ @@ -7,15 +7,15 @@
│ │ │             2 => {1, 3}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 3}
│ │ │  
│ │ │  o1 : Graph
│ │ │  
│ │ │  i2 : showExoticSolutions G
│ │ │ - -- .722313s elapsed
│ │ │ + -- .784514s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -48,14 +48,14 @@
│ │ │             2 => {1, 3, 4}
│ │ │             3 => {2, 4}
│ │ │             4 => {0, 2, 3}
│ │ │  
│ │ │  o3 : Graph
│ │ │  
│ │ │  i4 : showExoticSolutions G
│ │ │ - -- .986724s elapsed
│ │ │ + -- 1.02869s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Checking_spthe_spcodimension_spand_spirreducible_spdecomposition_spof_spthe_sp__I__G_spideal.html
│ │ │ @@ -300,25 +300,25 @@
│ │ │            
│ │ │              
│ │ │                
    i16 : for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .284219s elapsed
│ │ │ - -- .401541s elapsed
│ │ │ - -- .867569s elapsed
│ │ │ - -- .245534s elapsed
│ │ │ - -- .354798s elapsed
│ │ │ - -- .429779s elapsed
│ │ │ - -- .666427s elapsed
│ │ │ - -- .418819s elapsed
│ │ │ - -- .478799s elapsed
│ │ │ - -- .409628s elapsed
│ │ │ - -- .282237s elapsed
    
│ │ │ + -- .20308s elapsed
│ │ │ + -- .250248s elapsed
│ │ │ + -- .378741s elapsed
│ │ │ + -- .197857s elapsed
│ │ │ + -- .223621s elapsed
│ │ │ + -- .250067s elapsed
│ │ │ + -- .458254s elapsed
│ │ │ + -- .351212s elapsed
│ │ │ + -- .369474s elapsed
│ │ │ + -- .28027s elapsed
│ │ │ + -- .188614s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : netList oo
│ │ │  
│ │ │        +---------------+---------------+
│ │ │ @@ -385,75 +385,75 @@
│ │ │            
│ │ │              
│ │ │                
    i23 : allcomps = for G in Gs list (
│ │ │            IG = oscQuadrics(G, R);
│ │ │            elapsedTime comps := decompose IG;
│ │ │            {comps/codim, comps/degree}
│ │ │            );
│ │ │ - -- .539129s elapsed
│ │ │ - -- .645006s elapsed
│ │ │ - -- 1.05918s elapsed
│ │ │ - -- 1.23488s elapsed
│ │ │ - -- .813801s elapsed
│ │ │ - -- 1.01888s elapsed
│ │ │ - -- 1.18556s elapsed
│ │ │ - -- 1.19237s elapsed
│ │ │ - -- .809367s elapsed
│ │ │ - -- .688287s elapsed
│ │ │ - -- .34871s elapsed
│ │ │ - -- .470565s elapsed
│ │ │ - -- .796532s elapsed
│ │ │ - -- .592092s elapsed
│ │ │ - -- .870412s elapsed
│ │ │ - -- 1.45151s elapsed
│ │ │ - -- 1.08533s elapsed
│ │ │ - -- .929525s elapsed
│ │ │ - -- 1.53034s elapsed
│ │ │ - -- 1.33678s elapsed
│ │ │ - -- .760184s elapsed
│ │ │ - -- 1.28518s elapsed
│ │ │ - -- 1.71666s elapsed
│ │ │ - -- 1.22854s elapsed
│ │ │ - -- .674943s elapsed
│ │ │ - -- .894164s elapsed
│ │ │ - -- 1.57127s elapsed
│ │ │ - -- .743101s elapsed
│ │ │ - -- .756756s elapsed
│ │ │ - -- 1.13138s elapsed
│ │ │ - -- 1.25775s elapsed
│ │ │ - -- 1.00643s elapsed
│ │ │ - -- .580285s elapsed
│ │ │ - -- 1.47228s elapsed
│ │ │ - -- 1.02334s elapsed
│ │ │ - -- 1.53993s elapsed
│ │ │ - -- .942582s elapsed
│ │ │ - -- 1.21686s elapsed
│ │ │ - -- 1.59056s elapsed
│ │ │ - -- 1.01279s elapsed
│ │ │ - -- .781942s elapsed
│ │ │ - -- 1.63071s elapsed
│ │ │ - -- 1.80895s elapsed
│ │ │ - -- 2.00814s elapsed
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│ │ │ - -- 1.22183s elapsed
│ │ │ - -- .920406s elapsed
│ │ │ - -- .900219s elapsed
│ │ │ - -- 1.32942s elapsed
│ │ │ - -- .609367s elapsed
│ │ │ - -- 1.44447s elapsed
│ │ │ - -- 1.35121s elapsed
│ │ │ - -- 1.47404s elapsed
│ │ │ - -- .758204s elapsed
│ │ │ - -- .533039s elapsed
│ │ │ - -- .40806s elapsed
    
│ │ │ + -- .316237s elapsed
│ │ │ + -- .370322s elapsed
│ │ │ + -- .764898s elapsed
│ │ │ + -- .938243s elapsed
│ │ │ + -- .565708s elapsed
│ │ │ + -- .695381s elapsed
│ │ │ + -- .825957s elapsed
│ │ │ + -- .831219s elapsed
│ │ │ + -- .57875s elapsed
│ │ │ + -- .595979s elapsed
│ │ │ + -- .278365s elapsed
│ │ │ + -- .355189s elapsed
│ │ │ + -- .405891s elapsed
│ │ │ + -- .519177s elapsed
│ │ │ + -- .673817s elapsed
│ │ │ + -- 1.01805s elapsed
│ │ │ + -- .733166s elapsed
│ │ │ + -- .709782s elapsed
│ │ │ + -- .98038s elapsed
│ │ │ + -- .818091s elapsed
│ │ │ + -- .63529s elapsed
│ │ │ + -- .742332s elapsed
│ │ │ + -- 1.10426s elapsed
│ │ │ + -- 1.01625s elapsed
│ │ │ + -- .397908s elapsed
│ │ │ + -- .543744s elapsed
│ │ │ + -- 1.01968s elapsed
│ │ │ + -- .549338s elapsed
│ │ │ + -- .459191s elapsed
│ │ │ + -- .653864s elapsed
│ │ │ + -- .809097s elapsed
│ │ │ + -- .635191s elapsed
│ │ │ + -- .44622s elapsed
│ │ │ + -- .835689s elapsed
│ │ │ + -- .634804s elapsed
│ │ │ + -- .83954s elapsed
│ │ │ + -- .7494s elapsed
│ │ │ + -- .955647s elapsed
│ │ │ + -- 1.00728s elapsed
│ │ │ + -- .643989s elapsed
│ │ │ + -- .572942s elapsed
│ │ │ + -- .908814s elapsed
│ │ │ + -- 1.10631s elapsed
│ │ │ + -- 1.40448s elapsed
│ │ │ + -- .904718s elapsed
│ │ │ + -- .905454s elapsed
│ │ │ + -- 1.13155s elapsed
│ │ │ + -- .967963s elapsed
│ │ │ + -- .799505s elapsed
│ │ │ + -- .847803s elapsed
│ │ │ + -- .894674s elapsed
│ │ │ + -- .635332s elapsed
│ │ │ + -- .697491s elapsed
│ │ │ + -- .836636s elapsed
│ │ │ + -- .504753s elapsed
│ │ │ + -- .942339s elapsed
│ │ │ + -- 1.01034s elapsed
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│ │ │ + -- .612756s elapsed
│ │ │ + -- .382879s elapsed
│ │ │ + -- .297488s 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}
│ │ │ │            );
│ │ │ │ - -- .284219s elapsed
│ │ │ │ - -- .401541s elapsed
│ │ │ │ - -- .867569s elapsed
│ │ │ │ - -- .245534s elapsed
│ │ │ │ - -- .354798s elapsed
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│ │ │ │ - -- .666427s elapsed
│ │ │ │ - -- .418819s elapsed
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│ │ │ │ - -- .409628s elapsed
│ │ │ │ - -- .282237s elapsed
│ │ │ │ + -- .20308s elapsed
│ │ │ │ + -- .250248s elapsed
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│ │ │ │ + -- .369474s elapsed
│ │ │ │ + -- .28027s elapsed
│ │ │ │ + -- .188614s 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}
│ │ │ │            );
│ │ │ │ - -- .539129s elapsed
│ │ │ │ - -- .645006s elapsed
│ │ │ │ - -- 1.05918s elapsed
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│ │ │ │ + -- 1.10426s elapsed
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│ │ │ │ + -- 1.01968s elapsed
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│ │ │ │ + -- .635191s elapsed
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│ │ │ │ + -- .7494s elapsed
│ │ │ │ + -- .955647s elapsed
│ │ │ │ + -- 1.00728s elapsed
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│ │ │ │ + -- .572942s elapsed
│ │ │ │ + -- .908814s elapsed
│ │ │ │ + -- 1.10631s elapsed
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│ │ │ │ + -- .904718s elapsed
│ │ │ │ + -- .905454s elapsed
│ │ │ │ + -- 1.13155s elapsed
│ │ │ │ + -- .967963s elapsed
│ │ │ │ + -- .799505s elapsed
│ │ │ │ + -- .847803s elapsed
│ │ │ │ + -- .894674s elapsed
│ │ │ │ + -- .635332s elapsed
│ │ │ │ + -- .697491s elapsed
│ │ │ │ + -- .836636s elapsed
│ │ │ │ + -- .504753s elapsed
│ │ │ │ + -- .942339s elapsed
│ │ │ │ + -- 1.01034s elapsed
│ │ │ │ + -- 1.06347s elapsed
│ │ │ │ + -- .612756s elapsed
│ │ │ │ + -- .382879s elapsed
│ │ │ │ + -- .297488s elapsed
│ │ │ │  i24 : netList ({{"codimensions", "degrees"}} | allcomps)
│ │ │ │  
│ │ │ │        +------------------------+------------------------+
│ │ │ │  o24 = |codimensions            |degrees                 |
│ │ │ │        +------------------------+------------------------+
│ │ │ │        |{3, 5, 5}               |{2, 4, 6}               |
│ │ │ │        +------------------------+------------------------+
│ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___Example_sp4.2_co_spa_sp__K5_spand_sppentagon_spglued_spalong_span_spedge.html
│ │ │ @@ -115,15 +115,15 @@
│ │ │  
│ │ │  o5 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : elapsedTime stablesolsPent = showExoticSolutions Pent
│ │ │ - -- .859s elapsed
│ │ │ + -- .83s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +----+-----+-----+----+-----+-----+-----+-----+
│ │ │  |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951|
│ │ │ @@ -136,15 +136,15 @@
│ │ │  +---+---+---+---+
│ │ │  |72 |144|216|288|
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |288|216|144|72 |
│ │ │  +---+---+---+---+
│ │ │ - -- .901s elapsed
│ │ │ + -- .874s elapsed
│ │ │  
│ │ │  o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0,
│ │ │       ------------------------------------------------------------------------
│ │ │       0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}}
│ │ │  
│ │ │  o6 : List
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -43,15 +43,15 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ .98, .98, .101, -.98, -.298, .393, .201, .201, .201, -.995, -.201, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ .954}} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : elapsedTime stablesolsPent = showExoticSolutions Pent │ │ │ │ - -- .859s elapsed │ │ │ │ + -- .83s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +----+-----+-----+----+-----+-----+-----+-----+ │ │ │ │ |.309|-.809|-.809|.309|.951 |.588 |-.588|-.951| │ │ │ │ @@ -64,15 +64,15 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |72 |144|216|288| │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ |288|216|144|72 | │ │ │ │ +---+---+---+---+ │ │ │ │ - -- .901s elapsed │ │ │ │ + -- .874s elapsed │ │ │ │ │ │ │ │ o6 = {{.309, -.809, -.809, .309, .951, .588, -.588, -.951}, {1, 1, 1, 1, 0, │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 0, 0, 0}, {.309, -.809, -.809, .309, -.951, -.588, .588, .951}} │ │ │ │ │ │ │ │ o6 : List │ │ │ │ Computing the (linearly) stable solutions for K5C5 takes a minute or two: │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/___S__C__T_spgraphs_spwith_spexotic_spsolutions.html │ │ │ @@ -120,19 +120,19 @@ │ │ │ o5 = 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : for G in Gs list showExoticSolutions G;
│ │ │  warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, 80, 82, 85, 86, 87, 88, 89, 90}
│ │ │ - -- .646s elapsed
│ │ │ + -- .569s elapsed
│ │ │  warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, 91, 94, 95, 97}
│ │ │ - -- .578s elapsed
│ │ │ - -- .694s elapsed
│ │ │ - -- .859s elapsed
│ │ │ + -- .549s elapsed
│ │ │ + -- .677s elapsed
│ │ │ + -- .866s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {2, 3}} --
│ │ │                                                  1 => {3, 4}
│ │ │                                                  2 => {0, 4}
│ │ │                                                  3 => {0, 1}
│ │ │                                                  4 => {2, 1}
│ │ │  +-----+----+----+-----+-----+-----+-----+-----+
│ │ │  |1    |1   |1   |1    |0    |0    |0    |0    |
│ │ │ @@ -145,25 +145,25 @@
│ │ │  +---+---+---+---+
│ │ │  |0  |0  |0  |0  |
│ │ │  +---+---+---+---+
│ │ │  |216|72 |288|144|
│ │ │  +---+---+---+---+
│ │ │  |144|288|72 |216|
│ │ │  +---+---+---+---+
│ │ │ - -- .966s elapsed
│ │ │ - -- 1.19s elapsed
│ │ │ + -- .998s elapsed
│ │ │ + -- 1.03s elapsed
│ │ │  warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, 47, 48, 53, 59, 60, 61}
│ │ │ - -- 1.27s elapsed
│ │ │ + -- 1.28s elapsed
│ │ │  warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34}
│ │ │ - -- 1.07s elapsed
│ │ │ - -- 1.05s elapsed
│ │ │ + -- 1.23s elapsed
│ │ │ + -- 1.15s elapsed
│ │ │  warning: some solutions are not regular: {26, 27, 30, 31, 33}
│ │ │ - -- 1.21s elapsed
│ │ │ + -- 1.24s elapsed
│ │ │  warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, 75, 76, 77}
│ │ │ - -- 1.03s elapsed
    │ │ │ + -- 1.12s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,21 +48,21 @@ │ │ │ │ i5 : printingPrecision = 3 │ │ │ │ │ │ │ │ o5 = 3 │ │ │ │ i6 : for G in Gs list showExoticSolutions G; │ │ │ │ warning: some solutions are not regular: {36, 41, 42, 43, 47, 48, 50, 51, 53, │ │ │ │ 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 74, 75, 76, 77, 79, │ │ │ │ 80, 82, 85, 86, 87, 88, 89, 90} │ │ │ │ - -- .646s elapsed │ │ │ │ + -- .569s elapsed │ │ │ │ warning: some solutions are not regular: {49, 50, 53, 56, 57, 58, 59, 60, 61, │ │ │ │ 62, 63, 64, 67, 68, 69, 70, 71, 72, 73, 75, 77, 78, 80, 82, 83, 84, 85, 86, 88, │ │ │ │ 91, 94, 95, 97} │ │ │ │ - -- .578s elapsed │ │ │ │ - -- .694s elapsed │ │ │ │ - -- .859s elapsed │ │ │ │ + -- .549s elapsed │ │ │ │ + -- .677s elapsed │ │ │ │ + -- .866s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {2, 3}} -- │ │ │ │ 1 => {3, 4} │ │ │ │ 2 => {0, 4} │ │ │ │ 3 => {0, 1} │ │ │ │ 4 => {2, 1} │ │ │ │ +-----+----+----+-----+-----+-----+-----+-----+ │ │ │ │ |1 |1 |1 |1 |0 |0 |0 |0 | │ │ │ │ @@ -75,24 +75,24 @@ │ │ │ │ +---+---+---+---+ │ │ │ │ |0 |0 |0 |0 | │ │ │ │ +---+---+---+---+ │ │ │ │ |216|72 |288|144| │ │ │ │ +---+---+---+---+ │ │ │ │ |144|288|72 |216| │ │ │ │ +---+---+---+---+ │ │ │ │ - -- .966s elapsed │ │ │ │ - -- 1.19s elapsed │ │ │ │ + -- .998s elapsed │ │ │ │ + -- 1.03s elapsed │ │ │ │ warning: some solutions are not regular: {28, 30, 35, 37, 38, 40, 43, 44, 46, │ │ │ │ 47, 48, 53, 59, 60, 61} │ │ │ │ - -- 1.27s elapsed │ │ │ │ + -- 1.28s elapsed │ │ │ │ warning: some solutions are not regular: {16, 17, 20, 21, 22, 23, 24, 26, 27, │ │ │ │ 28, 29, 30, 31, 32, 33, 34} │ │ │ │ - -- 1.07s elapsed │ │ │ │ - -- 1.05s elapsed │ │ │ │ + -- 1.23s elapsed │ │ │ │ + -- 1.15s elapsed │ │ │ │ warning: some solutions are not regular: {26, 27, 30, 31, 33} │ │ │ │ - -- 1.21s elapsed │ │ │ │ + -- 1.24s elapsed │ │ │ │ warning: some solutions are not regular: {38, 44, 46, 49, 52, 53, 63, 70, 74, │ │ │ │ 75, 76, 77} │ │ │ │ - -- 1.03s elapsed │ │ │ │ + -- 1.12s elapsed │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/Oscillators/Documentation.m2:812:0. │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_get__Linearly__Stable__Solutions.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ 
    i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}});
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : getLinearlyStableSolutions(G)
│ │ │  warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21}
│ │ │ - -- .215945s elapsed
│ │ │ + -- .186387s elapsed
│ │ │  
│ │ │  o2 = {{1, 1, 1, 0, 0, 0}}
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -19,15 +19,15 @@ │ │ │ │ of each oscillator is given by the Kuramoto model. The linear stability of a │ │ │ │ solution is determined by the eigenvalues of the Jacobian matrix of the system │ │ │ │ evaluated at the solution. │ │ │ │ i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}}); │ │ │ │ i2 : getLinearlyStableSolutions(G) │ │ │ │ warning: some solutions are not regular: {4, 5, 7, 8, 9, 10, 12, 13, 14, 15, │ │ │ │ 16, 17, 18, 19, 21} │ │ │ │ - -- .215945s elapsed │ │ │ │ + -- .186387s elapsed │ │ │ │ │ │ │ │ o2 = {{1, 1, 1, 0, 0, 0}} │ │ │ │ │ │ │ │ o2 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _f_i_n_d_R_e_a_l_S_o_l_u_t_i_o_n_s -- find real solutions, at least one per component for │ │ │ │ well-conditioned systems │ │ ├── ./usr/share/doc/Macaulay2/Oscillators/html/_show__Exotic__Solutions.html │ │ │ @@ -98,15 +98,15 @@ │ │ │ │ │ │ o1 : Graph │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : showExoticSolutions G
│ │ │ - -- .722313s elapsed
│ │ │ + -- .784514s elapsed
│ │ │  -- found extra exotic solutions for graph Graph{0 => {1, 4}} --
│ │ │                                                  1 => {0, 2}
│ │ │                                                  2 => {1, 3}
│ │ │                                                  3 => {2, 4}
│ │ │                                                  4 => {0, 3}
│ │ │  +-------+--------+--------+-------+--------+--------+--------+--------+
│ │ │  |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057|
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o3 : Graph
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : showExoticSolutions G
│ │ │ - -- .986724s elapsed
│ │ │ + -- 1.02869s elapsed
│ │ │  
│ │ │  o4 = {{1, 1, 1, 1, 0, 0, 0, 0}}
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ │ │ │ │ o1 : Graph │ │ │ │ i2 : showExoticSolutions G │ │ │ │ - -- .722313s elapsed │ │ │ │ + -- .784514s elapsed │ │ │ │ -- found extra exotic solutions for graph Graph{0 => {1, 4}} -- │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 3} │ │ │ │ +-------+--------+--------+-------+--------+--------+--------+--------+ │ │ │ │ |.309017|-.809017|-.809017|.309017|.951057 |.587785 |-.587785|-.951057| │ │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ 1 => {0, 2} │ │ │ │ 2 => {1, 3, 4} │ │ │ │ 3 => {2, 4} │ │ │ │ 4 => {0, 2, 3} │ │ │ │ │ │ │ │ o3 : Graph │ │ │ │ i4 : showExoticSolutions G │ │ │ │ - -- .986724s elapsed │ │ │ │ + -- 1.02869s 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/PHCpack/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ ZmFjdG9yV2l0bmVzc1NldCguLi4sVmVyYm9zZT0+Li4uKQ== │ │ │ #:len=595 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uIHRvIHNwZWNpZnkgd2hldGhl │ │ │ ciBhZGRpdGlvbmFsIG91dHB1dCBpcyB3YW50ZWQiLCAibGluZW51bSIgPT4gNDEyLCAiZmlsZW5h │ │ ├── ./usr/share/doc/Macaulay2/PackageCitations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ UGFja2FnZUNpdGF0aW9ucw== │ │ │ #:len=1928 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZhY2lsaXRhdGluZyBj │ │ │ aXRhdGlvbiBvZiBNYWNhdWxheTIgcGFja2FnZXMiLCAibGluZW51bSIgPT4gMjY1LCBTZWVBbHNv │ │ ├── ./usr/share/doc/Macaulay2/PackageTemplate/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ c2Vjb25kRnVuY3Rpb24= │ │ │ #:len=379 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBzaWxseSBzZWNvbmQgZnVuY3Rpb24i │ │ │ LCBEZXNjcmlwdGlvbiA9PiAoIlRoaXMgZnVuY3Rpb24gaXMgcHJvdmlkZWQgYnkgdGhlIHBhY2th │ │ ├── ./usr/share/doc/Macaulay2/Padic/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ aW52ZXJzZShQYWRpY051bWJlcik= │ │ │ #:len=1089 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibXVsdGlwbGljYXRpdmUgaW52ZXJzZSBv │ │ │ ZiBhIHAtYWRpYyBudW1iZXIiLCAibGluZW51bSIgPT4gNzU4LCBJbnB1dHMgPT4ge1NQQU57VFR7 │ │ ├── ./usr/share/doc/Macaulay2/Parametrization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ cGFyYW1ldHJpemVDb25pYw== │ │ │ #:len=1904 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiT3B0aW9uIHdoZXRoZXIgdG8gcmF0aW9u │ │ │ YWxseSBwYXJhbWV0cml6ZSBjb25pY3MuIiwgImxpbmVudW0iID0+IDE2NzYsICJmaWxlbmFtZSIg │ │ ├── ./usr/share/doc/Macaulay2/Parsing/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ Y2hhckFuYWx5emVy │ │ │ #:len=752 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBsZXhpY2FsIGFuYWx5emVyIHRoYXQg │ │ │ cHJvdmlkZXMgY2hhcmFjdGVycyBmcm9tIGEgc3RyaW5nIG9uZSBhdCBhIHRpbWUiLCAibGluZW51 │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ YW1iaWVudA== │ │ │ #:len=256 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZXhwb3J0ZWQgZnJvbSB0aGUgTkNBTGdl │ │ │ YnJhIHBhY2thZ2UuIiwgRGVzY3JpcHRpb24gPT4ge30sICJsaW5lbnVtIiA9PiAxNTksIHN5bWJv │ │ ├── ./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 │ │ │ - -- .0157139s elapsed │ │ │ + -- .00795355s 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 │ │ │ - -- .717745s elapsed │ │ │ + -- .479132s elapsed │ │ │ │ │ │ o20 : Ideal of S │ │ │ │ │ │ i21 : dim I │ │ │ │ │ │ o21 = 5 │ │ ├── ./usr/share/doc/Macaulay2/PathSignatures/html/___A_spfamily_spof_sppaths_spon_spa_spcone.html │ │ │ @@ -213,20 +213,20 @@ │ │ │ │ │ │ │ │ │ 
    i20 : I = sub(ideal flatten values componentsOfKernel(2, m, Grading => matrix {toList(9:1)}), S);
│ │ │  warning: computation begun over finite field. resulting polynomials may not lie in the ideal
│ │ │  computing total degree: 1
│ │ │  number of monomials = 9
│ │ │  number of distinct multidegrees = 1
│ │ │ - -- .0157139s elapsed
│ │ │ + -- .00795355s 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
│ │ │ - -- .717745s elapsed
│ │ │ + -- .479132s 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
│ │ │ │ - -- .0157139s elapsed
│ │ │ │ + -- .00795355s 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
│ │ │ │ - -- .717745s elapsed
│ │ │ │ + -- .479132s elapsed
│ │ │ │  
│ │ │ │  o20 : Ideal of S
│ │ │ │  i21 : dim I
│ │ │ │  
│ │ │ │  o21 = 5
│ │ │ │  i22 : isPrime I
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  cmFuZG9tRXh0ZW5zaW9uKE1hdHJpeCxNYXRyaXgp
│ │ │  #:len=301
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzIxMywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocmFuZG9tRXh0ZW5zaW9uLE1hdHJpeCxNYXRyaXgp
│ │ ├── ./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);
│ │ │ - -- .260086s elapsed
│ │ │ + -- .162138s 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 ();
│ │ │ - -- .898827s elapsed
│ │ │ + -- .536472s 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
│ │ │ - -- 30.2671s elapsed
│ │ │ + -- 14.6197s 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);
│ │ │ - -- 1.14431s elapsed
│ │ │ + -- .638495s 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);
│ │ │ - -- 2.91837s elapsed
│ │ │ + -- 1.97581s elapsed
│ │ │  
│ │ │  i16 : betti freeResolution Ulr3
│ │ │  
│ │ │                0  1  2
│ │ │  o16 = total: 12 24 12
│ │ │            0: 12 24 12
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/___Lab__Book__Protocol.html
│ │ │ @@ -133,15 +133,15 @@
│ │ │              
│ │ │                
    i4 : kk= ZZ/101;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : elapsedTime (S,qq,R,u, M1,M2, Mu1, Mu2)=randomNicePencil(kk,g);
│ │ │ - -- .260086s elapsed
    
│ │ │ + -- .162138s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : M=cliffordModule(Mu1,Mu2,R)
│ │ │  
│ │ │  o6 = CliffordModule{...6...}
│ │ │ @@ -177,15 +177,15 @@
│ │ │            m12=randomExtension(m1.yAction,m2.yAction);
│ │ │            V = vectorBundleOnE m12;
│ │ │            Ul=tensorProduct(Mor,V);
│ │ │            Ul1=tensorProduct(Mor1,V);
│ │ │            d0=unique degrees target Ul.yAction;
│ │ │            d1=unique degrees target Ul1.yAction;
│ │ │            #d1 >=3 or #d0 >=3) do ();
│ │ │ - -- .898827s elapsed
    │ │ │ + -- .536472s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : betti Ul.yAction, betti Ul1.yAction
│ │ │  
│ │ │                 0  1          0  1
│ │ │ @@ -198,15 +198,15 @@
│ │ │  
│ │ │  o12 : Sequence
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : elapsedTime Ul = tensorProduct(M,V); -- the heaviest part computing the actions of generators
│ │ │ - -- 30.2671s elapsed
    │ │ │ + -- 14.6197s 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);
│ │ │ │ - -- .260086s elapsed
│ │ │ │ + -- .162138s 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 ();
│ │ │ │ - -- .898827s elapsed
│ │ │ │ + -- .536472s 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
│ │ │ │ - -- 30.2671s elapsed
│ │ │ │ + -- 14.6197s elapsed
│ │ │ │  i14 : M1Ul=sum(#Ul.oddOperators,i->S_i*sub(Ul.oddOperators_i,S));
│ │ │ │  
│ │ │ │                32      32
│ │ │ │  o14 : Matrix S   <-- S
│ │ │ │  i15 : r=2
│ │ │ │  
│ │ │ │  o15 = 2
│ │ ├── ./usr/share/doc/Macaulay2/PencilsOfQuadrics/html/_search__Ulrich.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o11 : CliffordModule
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : elapsedTime Ulr = searchUlrich(M,S);
│ │ │ - -- 1.14431s elapsed
    │ │ │ + -- .638495s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : betti freeResolution Ulr
│ │ │  
│ │ │               0  1 2
│ │ │ @@ -190,15 +190,15 @@
│ │ │  
│ │ │  o14 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : elapsedTime Ulr3 = searchUlrich(M,S,3);
│ │ │ - -- 2.91837s elapsed
    │ │ │ + -- 1.97581s 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);
│ │ │ │ - -- 1.14431s elapsed
│ │ │ │ + -- .638495s 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);
│ │ │ │ - -- 2.91837s elapsed
│ │ │ │ + -- 1.97581s elapsed
│ │ │ │  i16 : betti freeResolution Ulr3
│ │ │ │  
│ │ │ │                0  1  2
│ │ │ │  o16 = total: 12 24 12
│ │ │ │            0: 12 24 12
│ │ │ │  
│ │ │ │  o16 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/Permanents/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  cnlzZXI=
│ │ │  #:len=2107
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBwZXJtYW5lbnQgdXNpbmcg
│ │ │  UnlzZXIncyBmb3JtdWxhIiwgImxpbmVudW0iID0+IDUyMywgSW5wdXRzID0+IHtTUEFOe1RUeyJN
│ │ ├── ./usr/share/doc/Macaulay2/Permutations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  ZGVzY2VudHMoUGVybXV0YXRpb24p
│ │ │  #:len=276
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkZXNjZW50cyxQZXJtdXRhdGlvbiksImRlc2NlbnRz
│ │ ├── ./usr/share/doc/Macaulay2/PhylogeneticTrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  cGh5bG9Ub3JpY1F1YWRz
│ │ │  #:len=2848
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgcXVhZHJhdGljIGlu
│ │ │  dmFyaWFudHMgb2YgYSBncm91cC1iYXNlZCBwaHlsb2dlbmV0aWMgbW9kZWwiLCAibGluZW51bSIg
│ │ ├── ./usr/share/doc/Macaulay2/PieriMaps/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZGlzcGxheUxSSW1hZ2UoTGlzdCk=
│ │ │  #:len=258
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│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhkaXNwbGF5TFJJbWFnZSxMaXN0KSwiZGlzcGxheUxS
│ │ ├── ./usr/share/doc/Macaulay2/PlaneCurveLinearSeries/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  YWRkaXRpb24oSWRlYWwsSWRlYWwsSWRlYWwp
│ │ │  #:len=295
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDYzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRpdGlvbixJZGVhbCxJZGVhbCxJZGVhbCksImFk
│ │ ├── ./usr/share/doc/Macaulay2/Points/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  cmFuZG9tUG9pbnRzTWF0KFJpbmcsWlop
│ │ │  #:len=255
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODgzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhyYW5kb21Qb2ludHNNYXQsUmluZyxaWiksInJhbmRv
│ │ ├── ./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);
│ │ │ - -- 2.34235s elapsed
│ │ │ + -- 1.27437s elapsed
│ │ │  
│ │ │  i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.80041s elapsed
│ │ │ + -- 3.47236s elapsed
│ │ │  
│ │ │  i12 : G==H
│ │ │  
│ │ │  o12 = true
│ │ │  
│ │ │  i13 :
│ │ ├── ./usr/share/doc/Macaulay2/Points/html/_affine__Fat__Points.html
│ │ │ @@ -182,21 +182,21 @@
│ │ │  
│ │ │  o9 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime (Q,inG,G) = affineFatPoints(M,mults,R);
│ │ │ - -- 2.34235s elapsed
    │ │ │ + -- 1.27437s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R);
│ │ │ - -- 5.80041s elapsed
    │ │ │ + -- 3.47236s 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); │ │ │ │ - -- 2.34235s elapsed │ │ │ │ + -- 1.27437s elapsed │ │ │ │ i11 : elapsedTime H = affineFatPointsByIntersection(M,mults,R); │ │ │ │ - -- 5.80041s elapsed │ │ │ │ + -- 3.47236s 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/Polyhedra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ bWF4Q29uZXM= │ │ │ #:len=1185 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZGlzcGxheXMgdGhlIGdlbmVyYXRpbmcg │ │ │ Q29uZXMgb2YgYSBGYW4iLCAibGluZW51bSIgPT4gODQzLCBJbnB1dHMgPT4ge1NQQU57VFR7IkYi │ │ ├── ./usr/share/doc/Macaulay2/Polymake/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ UG9seW1ha2U= │ │ │ #:len=610 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYSBwYWNrYWdlIGZvciBpbnRlcmZhY2lu │ │ │ ZyB3aXRoIHBvbHltYWtlIiwgRGVzY3JpcHRpb24gPT4gKEVNeyJQb2x5bWFrZSJ9LCIgaXMgYSBw │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ c3RhbmRhcmRSb29rTnVtYmVy │ │ │ #:len=1159 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU3RhbmRhcmQgcm9vayBudW1iZXIgb2Yg │ │ │ YSBjb2xsZWN0aW9uIG9mIGNlbGxzIiwgImxpbmVudW0iID0+IDE1MzQsIElucHV0cyA9PiB7U1BB │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/___Polyomino__Ideals.out │ │ │ @@ -5,18 +5,18 @@ │ │ │ o1 = {{1, 1}, {2, 1}, {2, 2}} │ │ │ │ │ │ o1 : CollectionOfCells │ │ │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 2,2 1,1 2,1 1,2 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ + 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 3,3 2,1 3,1 2,3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x ) │ │ │ - 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 │ │ │ + 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ i3 : polyoVertices Q │ │ │ │ │ │ o3 = {(1, 1), (1, 2), (3, 1), (3, 2), (3, 3), (2, 1), (2, 2), (2, 3)} │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/example-output/_polyo__Ideal.out │ │ │ @@ -5,44 +5,44 @@ │ │ │ o1 = {{1, 1}, {2, 1}, {2, 2}} │ │ │ │ │ │ o1 : CollectionOfCells │ │ │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 3,3 2,1 3,1 2,3 │ │ │ + 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x ) │ │ │ - 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ + 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ i3 : Q= cellCollection {{1, 1}, {2, 1}, {3, 1}, {3, 2}, {3, 3}, {2, 3}, {1, 3}, {1, 2}}; │ │ │ │ │ │ i4 : I = polyoIdeal Q │ │ │ │ │ │ o4 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ - 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 │ │ │ + 2,3 1,2 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 2,3 1,2 2,2 1,3 2,4 1,1 │ │ │ + 4,3 3,2 4,2 3,3 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 │ │ │ + 4,1 1,2 2,4 1,2 2,2 1,4 4,4 3,2 4,2 3,4 2,4 1,3 │ │ │ ------------------------------------------------------------------------ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ - 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 │ │ │ + 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 3,1 1,2 4,4 2,3 4,3 2,4 │ │ │ ------------------------------------------------------------------------ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ - 4,4 3,2 4,2 3,4 2,4 1,3 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 │ │ │ + 2,2 1,1 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 │ │ │ ------------------------------------------------------------------------ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ - 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 │ │ │ + 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 │ │ │ ------------------------------------------------------------------------ │ │ │ x x ) │ │ │ - 4,1 3,2 │ │ │ + 4,1 2,2 │ │ │ │ │ │ o4 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ] │ │ │ 4,4 4,3 4,2 4,1 3,4 3,3 3,2 3,1 2,4 2,3 2,2 2,1 1,4 1,3 1,2 1,1 │ │ │ │ │ │ i5 : │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/html/_polyo__Ideal.html │ │ │ @@ -89,18 +89,18 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : I = polyoIdeal Q
│ │ │  
│ │ │  o2 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2   3,3 2,1    3,1 2,3 
│ │ │ +             3,3 2,1    3,1 2,3   3,2 2,1    3,1 2,2   2,2 1,1    2,1 1,2 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   )
│ │ │ -      3,2 2,1    3,1 2,2   2,2 1,1    2,1 1,2
│ │ │ +      3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2
│ │ │  
│ │ │  o2 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
    │ │ │ │ │ │ │ │ │ │ │ │

    │ │ │ @@ -110,33 +110,33 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : I = polyoIdeal Q
│ │ │  
│ │ │  o4 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             2,3 1,1    2,1 1,3   4,3 3,1    4,1 3,3   4,4 1,3    4,3 1,4 
│ │ │ +             2,3 1,2    2,2 1,3   2,4 1,1    2,1 1,4   4,4 3,1    4,1 3,4 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      3,4 2,3    3,3 2,4   4,2 2,1    4,1 2,2   2,3 1,2    2,2 1,3   2,4 1,1
│ │ │ +      4,3 3,2    4,2 3,3   3,4 1,3    3,3 1,4   3,2 2,1    3,1 2,2   4,2 1,1
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        2,1 1,4   4,4 3,1    4,1 3,4   4,3 3,2    4,2 3,3   3,4 1,3  
│ │ │ +        4,1 1,2   2,4 1,2    2,2 1,4   4,4 3,2    4,2 3,4   2,4 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   , x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -      3,3 1,4   3,2 2,1    3,1 2,2   4,2 1,1    4,1 1,2   2,4 1,2    2,2 1,4 
│ │ │ +      2,3 1,4   4,4 3,3    4,3 3,4   3,2 1,1    3,1 1,2   4,4 2,3    4,3 2,4 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ -      4,4 3,2    4,2 3,4   2,4 1,3    2,3 1,4   4,4 3,3    4,3 3,4   3,2 1,1
│ │ │ +      2,2 1,1    2,1 1,2   4,2 3,1    4,1 3,2   2,3 1,1    2,1 1,3   4,3 3,1
│ │ │       ------------------------------------------------------------------------
│ │ │       - x   x   , x   x    - x   x   , x   x    - x   x   , x   x    -
│ │ │ -        3,1 1,2   4,4 2,3    4,3 2,4   2,2 1,1    2,1 1,2   4,2 3,1  
│ │ │ +        4,1 3,3   4,4 1,3    4,3 1,4   3,4 2,3    3,3 2,4   4,2 2,1  
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x   )
│ │ │ -      4,1 3,2
│ │ │ +      4,1 2,2
│ │ │  
│ │ │  o4 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    4,4   4,3   4,2   4,1   3,4   3,3   3,2   3,1   2,4   2,3   2,2   2,1   1,4   1,3   1,2   1,1
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -27,47 +27,47 @@ │ │ │ │ │ │ │ │ o1 = {{1, 1}, {2, 1}, {2, 2}} │ │ │ │ │ │ │ │ o1 : CollectionOfCells │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 3,3 2,1 3,1 2,3 │ │ │ │ + 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x ) │ │ │ │ - 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ + 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ │ │ │ │ │ │ │ │ i3 : Q= cellCollection {{1, 1}, {2, 1}, {3, 1}, {3, 2}, {3, 3}, {2, 3}, {1, 3}, │ │ │ │ {1, 2}}; │ │ │ │ i4 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o4 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 2,3 1,1 2,1 1,3 4,3 3,1 4,1 3,3 4,4 1,3 4,3 1,4 │ │ │ │ + 2,3 1,2 2,2 1,3 2,4 1,1 2,1 1,4 4,4 3,1 4,1 3,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 3,4 2,3 3,3 2,4 4,2 2,1 4,1 2,2 2,3 1,2 2,2 1,3 2,4 1,1 │ │ │ │ + 4,3 3,2 4,2 3,3 3,4 1,3 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 2,1 1,4 4,4 3,1 4,1 3,4 4,3 3,2 4,2 3,3 3,4 1,3 │ │ │ │ + 4,1 1,2 2,4 1,2 2,2 1,4 4,4 3,2 4,2 3,4 2,4 1,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x , x x - x x , x x - x x , x x - x x , │ │ │ │ - 3,3 1,4 3,2 2,1 3,1 2,2 4,2 1,1 4,1 1,2 2,4 1,2 2,2 1,4 │ │ │ │ + 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 3,1 1,2 4,4 2,3 4,3 2,4 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x , x x - x x , x x │ │ │ │ - 4,4 3,2 4,2 3,4 2,4 1,3 2,3 1,4 4,4 3,3 4,3 3,4 3,2 1,1 │ │ │ │ + 2,2 1,1 2,1 1,2 4,2 3,1 4,1 3,2 2,3 1,1 2,1 1,3 4,3 3,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - x x , x x - x x , x x - x x , x x - │ │ │ │ - 3,1 1,2 4,4 2,3 4,3 2,4 2,2 1,1 2,1 1,2 4,2 3,1 │ │ │ │ + 4,1 3,3 4,4 1,3 4,3 1,4 3,4 2,3 3,3 2,4 4,2 2,1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ - 4,1 3,2 │ │ │ │ + 4,1 2,2 │ │ │ │ │ │ │ │ o4 : Ideal of QQ[x , x , x , x , x , x , x , x , x , x , x │ │ │ │ , x , x , x , x , x ] │ │ │ │ 4,4 4,3 4,2 4,1 3,4 3,3 3,2 3,1 2,4 2,3 │ │ │ │ 2,2 2,1 1,4 1,3 1,2 1,1 │ │ │ │ ********** WWaayyss ttoo uussee ppoollyyooIIddeeaall:: ********** │ │ │ │ * polyoIdeal(CollectionOfCells) │ │ ├── ./usr/share/doc/Macaulay2/PolyominoIdeals/html/index.html │ │ │ @@ -67,18 +67,18 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : I = polyoIdeal Q
│ │ │  
│ │ │  o2 = ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │ -             2,2 1,1    2,1 1,2   3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2 
│ │ │ +             3,3 2,2    3,2 2,3   3,2 1,1    3,1 1,2   3,3 2,1    3,1 2,3 
│ │ │       ------------------------------------------------------------------------
│ │ │       x   x    - x   x   , x   x    - x   x   )
│ │ │ -      3,3 2,1    3,1 2,3   3,2 2,1    3,1 2,2
│ │ │ +      3,2 2,1    3,1 2,2   2,2 1,1    2,1 1,2
│ │ │  
│ │ │  o2 : Ideal of QQ[x   , x   , x   , x   , x   , x   , x   , x   ]
│ │ │                    3,3   3,2   3,1   2,3   2,2   2,1   1,2   1,1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -55,18 +55,18 @@ │ │ │ │ │ │ │ │ o1 = {{1, 1}, {2, 1}, {2, 2}} │ │ │ │ │ │ │ │ o1 : CollectionOfCells │ │ │ │ i2 : I = polyoIdeal Q │ │ │ │ │ │ │ │ o2 = ideal (x x - x x , x x - x x , x x - x x , │ │ │ │ - 2,2 1,1 2,1 1,2 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 │ │ │ │ + 3,3 2,2 3,2 2,3 3,2 1,1 3,1 1,2 3,3 2,1 3,1 2,3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x - x x , x x - x x ) │ │ │ │ - 3,3 2,1 3,1 2,3 3,2 2,1 3,1 2,2 │ │ │ │ + 3,2 2,1 3,1 2,2 2,2 1,1 2,1 1,2 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x , x , x , x , x , x , x , x ] │ │ │ │ 3,3 3,2 3,1 2,3 2,2 2,1 1,2 1,1 │ │ │ │ i3 : polyoVertices Q │ │ │ │ │ │ │ │ o3 = {(1, 1), (1, 2), (3, 1), (3, 2), (3, 3), (2, 1), (2, 2), (2, 3)} │ │ ├── ./usr/share/doc/Macaulay2/Posets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=17 │ │ │ bWF4aW1hbEFudGljaGFpbnM= │ │ │ #:len=1127 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYWxsIG1heGltYWwgYW50 │ │ │ aWNoYWlucyBvZiBhIHBvc2V0IiwgImxpbmVudW0iID0+IDQ5ODgsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./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 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc) │ │ │ + -- used 1.0911e-05s (cpu); 5.017e-06s (thread); 0s (gc) │ │ │ │ │ │ o7 = true │ │ │ │ │ │ i8 : time isDistributive P │ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc) │ │ │ + -- used 5.7764s (cpu); 3.55959s (thread); 0s (gc) │ │ │ │ │ │ o8 = true │ │ │ │ │ │ i9 : C' = dual C; │ │ │ │ │ │ i10 : time isDistributive C' │ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc) │ │ │ + -- used 6.117e-06s (cpu); 5.172e-06s (thread); 0s (gc) │ │ │ │ │ │ o10 = true │ │ │ │ │ │ i11 : peek C'.cache │ │ │ │ │ │ o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}} } │ │ │ coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}} │ │ ├── ./usr/share/doc/Macaulay2/Posets/example-output/_greene__Kleitman__Partition.out │ │ │ @@ -7,22 +7,22 @@ │ │ │ o2 = Partition{4, 2} │ │ │ │ │ │ o2 : Partition │ │ │ │ │ │ i3 : D = dominanceLattice 6; │ │ │ │ │ │ i4 : time greeneKleitmanPartition(D, Strategy => "antichains") │ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc) │ │ │ + -- used 0.378714s (cpu); 0.21263s (thread); 0s (gc) │ │ │ │ │ │ o4 = Partition{9, 2} │ │ │ │ │ │ o4 : Partition │ │ │ │ │ │ i5 : time greeneKleitmanPartition(D, Strategy => "chains") │ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc) │ │ │ + -- used 1.1413e-05s (cpu); 1.0547e-05s (thread); 0s (gc) │ │ │ │ │ │ o5 = Partition{9, 2} │ │ │ │ │ │ o5 : Partition │ │ │ │ │ │ i6 : greeneKleitmanPartition chain 10 │ │ ├── ./usr/share/doc/Macaulay2/Posets/html/___Precompute.html │ │ │ @@ -112,23 +112,23 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : time isDistributive C
│ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ + -- used 1.0911e-05s (cpu); 5.017e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o7 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time isDistributive P
│ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ + -- used 5.7764s (cpu); 3.55959s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = true
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -138,15 +138,15 @@ │ │ │ │ │ │ 
    i9 : C' = dual C;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time isDistributive C'
│ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ + -- used 6.117e-06s (cpu); 5.172e-06s (thread); 0s (gc)
│ │ │  
│ │ │  o10 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : peek C'.cache
│ │ │ ├── html2text {}
│ │ │ │ @@ -41,26 +41,26 @@
│ │ │ │  i5 : peek P.cache
│ │ │ │  
│ │ │ │  o5 = CacheTable{name => P}
│ │ │ │  i6 : C == P
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : time isDistributive C
│ │ │ │ - -- used 8.826e-06s (cpu); 6.002e-06s (thread); 0s (gc)
│ │ │ │ + -- used 1.0911e-05s (cpu); 5.017e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = true
│ │ │ │  i8 : time isDistributive P
│ │ │ │ - -- used 6.82863s (cpu); 4.33499s (thread); 0s (gc)
│ │ │ │ + -- used 5.7764s (cpu); 3.55959s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  We also know that the dual of a distributive lattice is again a distributive
│ │ │ │  lattice. Other information is copied when possible.
│ │ │ │  i9 : C' = dual C;
│ │ │ │  i10 : time isDistributive C'
│ │ │ │ - -- used 7.003e-06s (cpu); 6.472e-06s (thread); 0s (gc)
│ │ │ │ + -- used 6.117e-06s (cpu); 5.172e-06s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 = true
│ │ │ │  i11 : peek C'.cache
│ │ │ │  
│ │ │ │  o11 = CacheTable{connectedComponents => {{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}}
│ │ │ │  }
│ │ │ │                   coveringRelations => {{1, 0}, {2, 1}, {3, 2}, {4, 3}, {5, 4},
│ │ ├── ./usr/share/doc/Macaulay2/Posets/html/_greene__Kleitman__Partition.html
│ │ │ @@ -107,25 +107,25 @@
│ │ │              
│ │ │                
    i3 : D = dominanceLattice 6;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ + -- used 0.378714s (cpu); 0.21263s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = Partition{9, 2}
│ │ │  
│ │ │  o4 : Partition
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ + -- used 1.1413e-05s (cpu); 1.0547e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = Partition{9, 2}
│ │ │  
│ │ │  o5 : Partition
    
│ │ │              
│ │ │            
│ │ │          
│ │ │ ├── html2text {}
│ │ │ │ @@ -30,21 +30,21 @@
│ │ │ │  
│ │ │ │  o2 : Partition
│ │ │ │  The conjugate of $l$ has the same property, but with chains replaced by
│ │ │ │  _a_n_t_i_c_h_a_i_n_s. Because of this, it is often better to count via antichains instead
│ │ │ │  of chains. This can be done by passing "antichains" as the Strategy.
│ │ │ │  i3 : D = dominanceLattice 6;
│ │ │ │  i4 : time greeneKleitmanPartition(D, Strategy => "antichains")
│ │ │ │ - -- used 0.435352s (cpu); 0.261753s (thread); 0s (gc)
│ │ │ │ + -- used 0.378714s (cpu); 0.21263s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = Partition{9, 2}
│ │ │ │  
│ │ │ │  o4 : Partition
│ │ │ │  i5 : time greeneKleitmanPartition(D, Strategy => "chains")
│ │ │ │ - -- used 1.2754e-05s (cpu); 1.2283e-05s (thread); 0s (gc)
│ │ │ │ + -- used 1.1413e-05s (cpu); 1.0547e-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/PositivityToricBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=20
│ │ │  ZHJhd1BhcmxpYW1lbnQyRHRpa3o=
│ │ │  #:len=2626
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidmlzdWFsaXNlcyB0aGUgcGFybGlhbWVu
│ │ │  dCBvZiBwb2x5dG9wZXMgZm9yIGEgdmVjdG9yIGJ1bmRsZSBvbiBhIHRvcmljIHN1cmZhY2UgdXNp
│ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  cHJpbWFyeUNvbXBvbmVudCguLi4sU3RyYXRlZ3k9Pi4uLik=
│ │ │  #:len=316
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzcxLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1twcmltYXJ5Q29tcG9uZW50LFN0cmF0ZWd5XSwicHJp
│ │ ├── ./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 {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, 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,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, 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 {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, 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,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, 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)
│ │ │ - -- .217729s elapsed
│ │ │ + -- .127109s 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)
│ │ │ - -- .0174897s elapsed
│ │ │ + -- .0190079s 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)
│ │ │ - -- .0178098s elapsed
│ │ │ + -- .0182455s 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
│ │ │ - -- .0318276s elapsed
│ │ │ + -- .0333151s 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)
│ │ │ - -- .00758549s elapsed
│ │ │ + -- .00705053s 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
│ │ │ @@ -291,28 +291,28 @@
│ │ │  o19 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i20 : M1 = set apply(L1, I -> sort flatten entries gens I)
│ │ │  
│ │ │ -o20 = set {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, 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,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, 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 {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, 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,
│ │ │        -----------------------------------------------------------------------
│ │ │ -      {e, 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 {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, 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, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {e, 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 {{d, b, a}, {d, c, b, a}, {e, d, c, b, a}, {c, 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, │ │ │ │ ----------------------------------------------------------------------- │ │ │ │ - {e, 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 │ │ │ @@ -112,41 +112,41 @@ │ │ │ 3 │ │ │ o3 : R-module, quotient of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : elapsedTime kernelOfLocalization(M, I1)
│ │ │ - -- .217729s elapsed
│ │ │ + -- .127109s 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)
│ │ │ - -- .0174897s elapsed
│ │ │ + -- .0190079s 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)
│ │ │ - -- .0178098s elapsed
│ │ │ + -- .0182455s 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) │ │ │ │ - -- .217729s elapsed │ │ │ │ + -- .127109s 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) │ │ │ │ - -- .0174897s elapsed │ │ │ │ + -- .0190079s 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) │ │ │ │ - -- .0178098s elapsed │ │ │ │ + -- .0182455s elapsed │ │ │ │ │ │ │ │ o6 = subquotient (| 1 0 |, | x_2^2-x_1x_3 x_1x_2-x_0x_3 x_1^2-x_0x_2 0 │ │ │ │ 0 |) │ │ │ │ | 0 1 | | 0 0 0 x_1^3- │ │ │ │ x_0x_2^2 0 | │ │ │ │ | 0 0 | | 0 0 0 0 │ │ │ │ x_1^5-x_0x_2^4 | │ │ ├── ./usr/share/doc/Macaulay2/PrimaryDecomposition/html/_reg__Seq__In__Ideal.html │ │ │ @@ -107,15 +107,15 @@ │ │ │ │ │ │ o2 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime regSeqInIdeal I
│ │ │ - -- .0318276s elapsed
│ │ │ + -- .0333151s elapsed
│ │ │  
│ │ │  o3 = ideal (x x , x x  + x x , x x  + x x , x x  + x x )
│ │ │               2 7   3 6    0 7   2 5    0 7   1 4    0 7
│ │ │  
│ │ │  o3 : Ideal of R
    │ │ │ │ │ │ │ │ │ @@ -153,15 +153,15 @@ │ │ │ │ │ │ o6 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : elapsedTime regSeqInIdeal(I, 3, 3, 1)
│ │ │ - -- .00758549s elapsed
│ │ │ + -- .00705053s 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 │ │ │ │ - -- .0318276s elapsed │ │ │ │ + -- .0333151s 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) │ │ │ │ - -- .00758549s elapsed │ │ │ │ + -- .00705053s 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/Probability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ dERpc3RyaWJ1dGlvbihOdW1iZXIp │ │ │ #:len=261 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTIyOSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodERpc3RyaWJ1dGlvbixOdW1iZXIpLCJ0RGlzdHJp │ │ ├── ./usr/share/doc/Macaulay2/PseudomonomialPrimaryDecomposition/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ UHNldWRvbW9ub21pYWxQcmltYXJ5RGVjb21wb3NpdGlvbg== │ │ │ #:len=1413 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJpbWFyeSBkZWNvbXBvc2l0aW9uIG9m │ │ │ IGEgc3F1YXJlIGZyZWUgcHNldWRvbW9ub21pYWwgaWRlYWwiLCBEZXNjcmlwdGlvbiA9PiAoRElW │ │ ├── ./usr/share/doc/Macaulay2/Pullback/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ aW50ZXJuYWxVc2VEaXJlY3RTdW0oUmluZyxSaW5nKQ== │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY2LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpbnRlcm5hbFVzZURpcmVjdFN1bSxSaW5nLFJpbmcp │ │ ├── ./usr/share/doc/Macaulay2/PushForward/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ cHVzaEZ3ZChNb2R1bGUp │ │ │ #:len=267 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzgxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhwdXNoRndkLE1vZHVsZSksInB1c2hGd2QoTW9kdWxl │ │ ├── ./usr/share/doc/Macaulay2/Python/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=39 │ │ │ YWRkUHlUb00yRnVuY3Rpb24oTGlzdCxGdW5jdGlvbixTdHJpbmcp │ │ │ #:len=293 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDcxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhZGRQeVRvTTJGdW5jdGlvbixMaXN0LEZ1bmN0aW9u │ │ ├── ./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/M2-32830-0/0.py │ │ │ +o1 = /tmp/M2-44325-0/0.py │ │ │ │ │ │ i2 : pyfile << "import math" << endl │ │ │ │ │ │ -o2 = /tmp/M2-32830-0/0.py │ │ │ +o2 = /tmp/M2-44325-0/0.py │ │ │ │ │ │ o2 : File │ │ │ │ │ │ i3 : pyfile << "x = math.sin(3.4)" << endl << close │ │ │ │ │ │ -o3 = /tmp/M2-32830-0/0.py │ │ │ +o3 = /tmp/M2-44325-0/0.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 0x7f5ec19a1f30> │ │ │ +o9 = at 0x7f4515bf5f30> │ │ │ │ │ │ o9 : PythonObject of class function │ │ │ │ │ │ i10 : x │ │ │ │ │ │ o10 = 5 │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_iterator_lp__Python__Object_rp.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f5ec1813cc0>
│ │ │ +o3 = <range_iterator object at 0x7f4515a67d80>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,15 +22,15 @@ │ │ │ │ i2 : x = builtins@@range 3 │ │ │ │ │ │ │ │ o2 = range(0, 3) │ │ │ │ │ │ │ │ o2 : PythonObject of class range │ │ │ │ i3 : i = iterator x │ │ │ │ │ │ │ │ -o3 = │ │ │ │ +o3 = │ │ │ │ │ │ │ │ o3 : PythonObject of class range_iterator │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _n_e_x_t_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- retrieve the next item from a python iterator │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_t_e_r_a_t_o_r_(_P_y_t_h_o_n_O_b_j_e_c_t_) -- get iterator of iterable python object │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Python/html/_next_lp__Python__Object_rp.html │ │ │ @@ -91,15 +91,15 @@ │ │ │ o2 : PythonObject of class range │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : i = iterator x
│ │ │  
│ │ │ -o3 = <range_iterator object at 0x7f5ec190e9d0>
│ │ │ +o3 = <range_iterator object at 0x7f4515b61e90>
│ │ │  
│ │ │  o3 : PythonObject of class range_iterator
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : next i
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,15 @@
│ │ │ │  i2 : x = builtins@@range 3
│ │ │ │  
│ │ │ │  o2 = range(0, 3)
│ │ │ │  
│ │ │ │  o2 : PythonObject of class range
│ │ │ │  i3 : i = iterator x
│ │ │ │  
│ │ │ │ -o3 = 
│ │ │ │ +o3 = 
│ │ │ │  
│ │ │ │  o3 : PythonObject of class range_iterator
│ │ │ │  i4 : next i
│ │ │ │  
│ │ │ │  o4 = 0
│ │ │ │  
│ │ │ │  o4 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_python__Run__Script.html
│ │ │ @@ -81,31 +81,31 @@
│ │ │            
    The return value is a Python dictionary containing all the variables defined in the global scope.
    
│ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : pyfile = temporaryFileName() | ".py"
│ │ │  
│ │ │ -o1 = /tmp/M2-32830-0/0.py
    │ │ │ +o1 = /tmp/M2-44325-0/0.py │ │ │
    │ │ │ 
    i2 : pyfile << "import math" << endl
│ │ │  
│ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ +o2 = /tmp/M2-44325-0/0.py
│ │ │  
│ │ │  o2 : File
    │ │ │
    │ │ │ 
    i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │  
│ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ +o3 = /tmp/M2-44325-0/0.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/M2-32830-0/0.py
│ │ │ │ +o1 = /tmp/M2-44325-0/0.py
│ │ │ │  i2 : pyfile << "import math" << endl
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-32830-0/0.py
│ │ │ │ +o2 = /tmp/M2-44325-0/0.py
│ │ │ │  
│ │ │ │  o2 : File
│ │ │ │  i3 : pyfile << "x = math.sin(3.4)" << endl << close
│ │ │ │  
│ │ │ │ -o3 = /tmp/M2-32830-0/0.py
│ │ │ │ +o3 = /tmp/M2-44325-0/0.py
│ │ │ │  
│ │ │ │  o3 : File
│ │ │ │  i4 : get pyfile
│ │ │ │  
│ │ │ │  o4 = import math
│ │ │ │       x = math.sin(3.4)
│ │ │ │  i5 : pythonRunScript oo
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_to__Python.html
│ │ │ @@ -186,15 +186,15 @@
│ │ │  o12 : FunctionClosure
    │ │ │
    │ │ │ 
    i13 : pysqrt = toPython m2sqrt
│ │ │  
│ │ │ -o13 = <built-in method m2sqrt of PyCapsule object at 0x7f5ec17e36f0>
│ │ │ +o13 = <built-in method m2sqrt of PyCapsule object at 0x7f4515a376f0>
│ │ │  
│ │ │  o13 : PythonObject of class builtin_function_or_method
    │ │ │
    │ │ │ 
    i14 : pysqrt 2
│ │ │ ├── html2text {}
│ │ │ │ @@ -72,15 +72,15 @@
│ │ │ │            sqrt x)
│ │ │ │  
│ │ │ │  o12 = m2sqrt
│ │ │ │  
│ │ │ │  o12 : FunctionClosure
│ │ │ │  i13 : pysqrt = toPython m2sqrt
│ │ │ │  
│ │ │ │ -o13 = 
│ │ │ │ +o13 = 
│ │ │ │  
│ │ │ │  o13 : PythonObject of class builtin_function_or_method
│ │ │ │  i14 : pysqrt 2
│ │ │ │  calling Macaulay2 code from Python!
│ │ │ │  
│ │ │ │  o14 = 1.4142135623730951
│ │ ├── ./usr/share/doc/Macaulay2/Python/html/_use_lp__Python__Context_rp.html
│ │ │ @@ -129,15 +129,15 @@
│ │ │                
    i8 : use ctx
    
│ │ │
    │ │ │ 
    i9 : f
│ │ │  
│ │ │ -o9 = <function <lambda> at 0x7f5ec19a1f30>
│ │ │ +o9 = <function <lambda> at 0x7f4515bf5f30>
│ │ │  
│ │ │  o9 : PythonObject of class function
    │ │ │
    │ │ │ 
    i10 : x
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │  
│ │ │ │  o7 = y
│ │ │ │  
│ │ │ │  o7 : Symbol
│ │ │ │  i8 : use ctx
│ │ │ │  i9 : f
│ │ │ │  
│ │ │ │ -o9 =  at 0x7f5ec19a1f30>
│ │ │ │ +o9 =  at 0x7f4515bf5f30>
│ │ │ │  
│ │ │ │  o9 : PythonObject of class function
│ │ │ │  i10 : x
│ │ │ │  
│ │ │ │  o10 = 5
│ │ │ │  
│ │ │ │  o10 : PythonObject of class int
│ │ ├── ./usr/share/doc/Macaulay2/QthPower/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  bWluaW1pemF0aW9u
│ │ │  #:len=2755
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hhbmdlIHRvIGEgYmV0dGVyIE5vZXRo
│ │ │  ZXIgbm9ybWFsaXphdGlvbiBzdWdnZXN0ZWQgYnkgdGhlIGluZHVjZWQgd2VpZ2h0cyIsICJsaW5l
│ │ ├── ./usr/share/doc/Macaulay2/QuadraticIdealExamplesByRoos/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=16
│ │ │  aGlnaGVyRGVwdGhUYWJsZQ==
│ │ │  #:len=793
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ3JlYXRlcyBoYXNodGFibGUgb2YgSmFu
│ │ │  LUVyaWsgUm9vcycgZXhhbXBsZXMgb2YgcXVhZHJhdGljIGlkZWFscyB3aXRoIHBvc2l0aXZlIGRl
│ │ ├── ./usr/share/doc/Macaulay2/Quasidegrees/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=27
│ │ │  cXVhc2lkZWdyZWVzTG9jYWxDb2hvbW9sb2d5
│ │ │  #:len=3881
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV0dXJucyB0aGUgcXVhc2lkZWdyZWUg
│ │ │  c2V0cyBvZiBsb2NhbCBjb2hvbW9sb2d5IG1vZHVsZXMiLCAibGluZW51bSIgPT4gNzg1LCBJbnB1
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=4
│ │ │  W1FRXQ==
│ │ │  #:len=544
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUXVhdGVybmFyeSBRdWFydGljIEZvcm1z
│ │ │  IGFuZCBHb3JlbnN0ZWluIHJpbmdzIChLYXB1c3RrYSwgS2FwdXN0a2EsIFJhbmVzdGFkLCBTY2hl
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/example-output/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.out
│ │ │ @@ -180,15 +180,15 @@
│ │ │  i21 : L = trim groebnerStratum F;
│ │ │  
│ │ │  o21 : Ideal of T
│ │ │  
│ │ │  i22 : assert(dim L == 18)
│ │ │  
│ │ │  i23 : elapsedTime isPrime L
│ │ │ - -- 3.65241s elapsed
│ │ │ + -- 2.02151s 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;
│ │ │ - -- 2.58155s elapsed
│ │ │ + -- 1.6336s elapsed
│ │ │  
│ │ │  i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
│ │ │  
│ │ │  i42 : compsL441/dim -- two components, of dimensions 14 and 16.
│ │ ├── ./usr/share/doc/Macaulay2/QuaternaryQuartics/html/___Hilbert_spscheme_spof_sp6_sppoints_spin_spprojective_sp3-space.html
│ │ │ @@ -349,15 +349,15 @@
│ │ │
    		│ │ │ 
    i22 : assert(dim L == 18)
    │ │ │
    │ │ │ 
    i23 : elapsedTime isPrime L
│ │ │ - -- 3.65241s elapsed
│ │ │ + -- 2.02151s elapsed
│ │ │  
│ │ │  o23 = true
    │ │ │
    │ │ │
    │ │ │

    The Schreyer resolution and minimal Betti numbers

    │ │ │ @@ -561,15 +561,15 @@ │ │ │ │ │ │ o39 : Ideal of T     │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i40 : elapsedTime compsL441 = decompose L441;
│ │ │ - -- 2.58155s elapsed
    │ │ │ + -- 1.6336s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i41 : #compsL441
│ │ │  
│ │ │  o41 = 2
    │ │ │ ├── html2text {} │ │ │ │ @@ -251,15 +251,15 @@ │ │ │ │ | 31 33 32 34 35 36 | │ │ │ │ +--------------------------------------------------------------+ │ │ │ │ i21 : L = trim groebnerStratum F; │ │ │ │ │ │ │ │ o21 : Ideal of T │ │ │ │ i22 : assert(dim L == 18) │ │ │ │ i23 : elapsedTime isPrime L │ │ │ │ - -- 3.65241s elapsed │ │ │ │ + -- 2.02151s 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; │ │ │ │ - -- 2.58155s elapsed │ │ │ │ + -- 1.6336s elapsed │ │ │ │ i41 : #compsL441 │ │ │ │ │ │ │ │ o41 = 2 │ │ │ │ i42 : compsL441/dim -- two components, of dimensions 14 and 16. │ │ │ │ │ │ │ │ o42 = {16, 14} │ │ ├── ./usr/share/doc/Macaulay2/QuillenSuslin/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ aG9ycm9ja3M= │ │ │ #:len=4664 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBsb2NhbCBzb2x1dGlv │ │ │ biB0byB0aGUgdW5pbW9kdWxhciByb3cgcHJvYmxlbSBvdmVyIGEgbG9jYWxpemF0aW9uIGF0IGEg │ │ ├── ./usr/share/doc/Macaulay2/RInterface/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ bmV3IFJPYmplY3QgZnJvbSBDQw== │ │ │ #:len=1343 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY3JlYXRlIGFuIFIgY29tcGxleCB2ZWN0 │ │ │ b3IgZnJvbSBhIGNvbXBsZXggbnVtYmVyIiwgImxpbmVudW0iID0+IDI1NSwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/___R__Value.out │ │ │ @@ -14,15 +14,15 @@ │ │ │ │ │ │ o3 = [1] 120 │ │ │ │ │ │ o3 : RObject of type double │ │ │ │ │ │ i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ} │ │ │ │ │ │ -o4 = │ │ │ +o4 = │ │ │ │ │ │ o4 : RObject of type environment │ │ │ │ │ │ i5 : RValue("choose(n, k)", Environment => env) │ │ │ │ │ │ o5 = [1] 120 │ │ ├── ./usr/share/doc/Macaulay2/RInterface/example-output/_new_sp__R__Object_spfrom_sp__Hash__Table.out │ │ │ @@ -1,12 +1,12 @@ │ │ │ -- -*- M2-comint -*- hash: 5466188237138105394 │ │ │ │ │ │ i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ} │ │ │ │ │ │ -o1 = │ │ │ +o1 = │ │ │ │ │ │ o1 : RObject of type environment │ │ │ │ │ │ i2 : RValue("x^y", Environment => env) │ │ │ │ │ │ o2 = [1] 25 │ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/___R__Value.html │ │ │ @@ -115,15 +115,15 @@ │ │ │

    The Environment option specifies the R environment in which to evaluate the code.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │  
│ │ │ -o4 = <environment: 0x7ff03f169410>
│ │ │ +o4 = <environment: 0x7f02d716acd0>
│ │ │  
│ │ │  o4 : RObject of type environment
    │ │ │
    │ │ │ 
    i5 : RValue("choose(n, k)", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -32,15 +32,15 @@
│ │ │ │  o3 = [1] 120
│ │ │ │  
│ │ │ │  o3 : RObject of type double
│ │ │ │  The Environment option specifies the R environment in which to evaluate the
│ │ │ │  code.
│ │ │ │  i4 : env = RObject hashTable {"n" => 10_ZZ, "k" => 3_ZZ}
│ │ │ │  
│ │ │ │ -o4 = 
│ │ │ │ +o4 = 
│ │ │ │  
│ │ │ │  o4 : RObject of type environment
│ │ │ │  i5 : RValue("choose(n, k)", Environment => env)
│ │ │ │  
│ │ │ │  o5 = [1] 120
│ │ │ │  
│ │ │ │  o5 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RInterface/html/_new_sp__R__Object_spfrom_sp__Hash__Table.html
│ │ │ @@ -76,15 +76,15 @@
│ │ │            
    Converts a Macaulay2 HashTable with String keys to an R environment, with each key-value pair becoming a variable binding.
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │  
│ │ │ -o1 = <environment: 0x7ff03ea9ce20>
│ │ │ +o1 = <environment: 0x7f02d6a9e6e0>
│ │ │  
│ │ │  o1 : RObject of type environment
    
│ │ │              
    
│ │ │                
    i2 : RValue("x^y", Environment => env)
│ │ │ ├── html2text {}
│ │ │ │ @@ -12,15 +12,15 @@
│ │ │ │      * Outputs:
│ │ │ │            o a _R_ _o_b_j_e_c_t, an R environment;
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Converts a Macaulay2 _H_a_s_h_T_a_b_l_e with _S_t_r_i_n_g keys to an R eennvviirroonnmmeenntt, with each
│ │ │ │  key-value pair becoming a variable binding.
│ │ │ │  i1 : env = RObject hashTable {"x" => 5_ZZ, "y" => 2_ZZ}
│ │ │ │  
│ │ │ │ -o1 = 
│ │ │ │ +o1 = 
│ │ │ │  
│ │ │ │  o1 : RObject of type environment
│ │ │ │  i2 : RValue("x^y", Environment => env)
│ │ │ │  
│ │ │ │  o2 = [1] 25
│ │ │ │  
│ │ │ │  o2 : RObject of type double
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  bWF4aW1hbEVudHJ5KENvbXBsZXgp
│ │ │  #:len=270
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDAwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYXhpbWFsRW50cnksQ29tcGxleCksIm1heGltYWxF
│ │ ├── ./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}, .00359595, .00152225)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00173482, .000733673)
│ │ │  
│ │ │  o3 : Sequence
│ │ │  
│ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00456981, .0350517)
│ │ │  
│ │ │  o4 : Sequence
│ │ │  
│ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ +o5 = {(.00615194, .0119868), (.00553839, .00421806), (.00605076, .00703751),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ +     (.00628534, .00946123), (.00642446, .0142638), (.00535809, .0135767),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ +     (.00471098, .0080782), (.00457119, .00728228), (.00358066, .00547853),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00953055, .0147996)}
│ │ │ +     (.00489852, .00762428)}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .0101318
│ │ │ +o6 = .005357033000000122
│ │ │  
│ │ │  o6 : RR (of precision 53)
│ │ │  
│ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .01648367429999995
│ │ │ +o7 = .008900738000000085
│ │ │  
│ │ │  o7 : RR (of precision 53)
│ │ │  
│ │ │  i8 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomComplexes/html/_test__Time__For__L__L__Lon__Syzygies.html
│ │ │ @@ -98,57 +98,57 @@
│ │ │  o2 : Sequence
    
│ │ │              
    
│ │ │                
    i3 : (m,t1,t2)=testTimeForLLLonSyzygies(r,n,Height=>11)
│ │ │  
│ │ │ -o3 = ({5, 2.91596e52, 9}, .00359595, .00152225)
│ │ │ +o3 = ({5, 2.91596e52, 9}, .00173482, .000733673)
│ │ │  
│ │ │  o3 : Sequence
    
│ │ │              
    
│ │ │                
    i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │  
│ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ +o4 = ({50, 2.30853e454, 98}, .00456981, .0350517)
│ │ │  
│ │ │  o4 : Sequence
    
│ │ │              
    
│ │ │                
    i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │  
│ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ +o5 = {(.00615194, .0119868), (.00553839, .00421806), (.00605076, .00703751),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ +     (.00628534, .00946123), (.00642446, .0142638), (.00535809, .0135767),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ +     (.00471098, .0080782), (.00457119, .00728228), (.00358066, .00547853),
│ │ │       ------------------------------------------------------------------------
│ │ │ -     (.00953055, .0147996)}
│ │ │ +     (.00489852, .00762428)}
│ │ │  
│ │ │  o5 : List
    
│ │ │              
    
│ │ │                
    i6 : 1/10*sum(L,t->t_0)
│ │ │  
│ │ │ -o6 = .0101318
│ │ │ +o6 = .005357033000000122
│ │ │  
│ │ │  o6 : RR (of precision 53)
    
│ │ │              
    
│ │ │                
    i7 : 1/10*sum(L,t->t_1)
│ │ │  
│ │ │ -o7 = .01648367429999995
│ │ │ +o7 = .008900738000000085
│ │ │  
│ │ │  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}, .00359595, .00152225)
│ │ │ │ +o3 = ({5, 2.91596e52, 9}, .00173482, .000733673)
│ │ │ │  
│ │ │ │  o3 : Sequence
│ │ │ │  i4 : (m,t1,t2)=testTimeForLLLonSyzygies(15,30,Height=>100)
│ │ │ │  
│ │ │ │ -o4 = ({50, 2.30853e454, 98}, .00949277, .0656054)
│ │ │ │ +o4 = ({50, 2.30853e454, 98}, .00456981, .0350517)
│ │ │ │  
│ │ │ │  o4 : Sequence
│ │ │ │  i5 : L=apply(10,c->(testTimeForLLLonSyzygies(15,30))_{1,2})
│ │ │ │  
│ │ │ │ -o5 = {(.0124466, .0228281), (.0113995, .00784108), (.0115798, .0132774),
│ │ │ │ +o5 = {(.00615194, .0119868), (.00553839, .00421806), (.00605076, .00703751),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.0119615, .0184515), (.00929783, .0257456), (.00955183, .0231329),
│ │ │ │ +     (.00628534, .00946123), (.00642446, .0142638), (.00535809, .0135767),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00861791, .0149704), (.00912547, .0138485), (.00780699, .00994154),
│ │ │ │ +     (.00471098, .0080782), (.00457119, .00728228), (.00358066, .00547853),
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │ -     (.00953055, .0147996)}
│ │ │ │ +     (.00489852, .00762428)}
│ │ │ │  
│ │ │ │  o5 : List
│ │ │ │  i6 : 1/10*sum(L,t->t_0)
│ │ │ │  
│ │ │ │ -o6 = .0101318
│ │ │ │ +o6 = .005357033000000122
│ │ │ │  
│ │ │ │  o6 : RR (of precision 53)
│ │ │ │  i7 : 1/10*sum(L,t->t_1)
│ │ │ │  
│ │ │ │ -o7 = .01648367429999995
│ │ │ │ +o7 = .008900738000000085
│ │ │ │  
│ │ │ │  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/RandomCurves/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  a25vd25VbmlyYXRpb25hbENvbXBvbmVudE9mU3BhY2VDdXJ2ZXM=
│ │ │  #:len=2280
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2sgd2hldGhlciB0aGVyZSBpcyBh
│ │ │  IHVuaXJhdGlvbmFsIGNvbnN0cnVjdGlvbiBmb3IgYSBjb21wb25lbnQgb2YgdGhlIEhpbGJlcnQg
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/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 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ + -- used 6.74343s (cpu); 5.14463s (thread); 0s (gc)
│ │ │  
│ │ │              0  1
│ │ │  o5 = total: 1 66
│ │ │           0: 1  .
│ │ │           1: . 66
│ │ │  
│ │ │  o5 : BettiTally
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/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 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ + -- used 1.83493s (cpu); 1.53187s (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/RandomCurves/html/_canonical__Curve.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │
    		│ │ │ 
    i4 : R=FF[x_0..x_(g-1)];
    │ │ │
    │ │ │ 
    i5 : time betti(I=(random canonicalCurve)(g,R))
│ │ │ - -- used 9.30535s (cpu); 6.17515s (thread); 0s (gc)
│ │ │ + -- used 6.74343s (cpu); 5.14463s (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 9.30535s (cpu); 6.17515s (thread); 0s (gc) │ │ │ │ + -- used 6.74343s (cpu); 5.14463s (thread); 0s (gc) │ │ │ │ │ │ │ │ 0 1 │ │ │ │ o5 = total: 1 66 │ │ │ │ 0: 1 . │ │ │ │ 1: . 66 │ │ │ │ │ │ │ │ o5 : BettiTally │ │ ├── ./usr/share/doc/Macaulay2/RandomCurves/html/_random__Curve__Genus14__Degree18in__P6.html │ │ │ @@ -98,15 +98,15 @@ │ │ │     		│ │ │ 
    i3 : S=FF[x_0..x_6];
    │ │ │
    │ │ │ 
    i4 : time I=randomCurveGenus14Degree18inP6(S);
│ │ │ - -- used 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ + -- used 1.83493s (cpu); 1.53187s (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 1.96317s (cpu); 1.48421s (thread); 0s (gc)
│ │ │ │ + -- used 1.83493s (cpu); 1.53187s (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/RandomCurvesOverVerySmallFiniteFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=46
│ │ │  c21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1KC4uLixQcmludGluZz0+Li4uKQ==
│ │ │  #:len=370
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTI4Mywgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc21vb3RoQ2Fub25pY2FsQ3VydmVHZW51czE1LFBy
│ │ ├── ./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 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ + -- used 1.17979s (cpu); 1.01985s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
│ │ │  
│ │ │  i2 : (dim ICan, genus ICan, degree ICan)
│ │ ├── ./usr/share/doc/Macaulay2/RandomCurvesOverVerySmallFiniteFields/html/_smooth__Canonical__Curve.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │            
    If the option Printing is set to true then printings about the current step in the construction are displayed.
    
│ │ │            
    
│ │ │          
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,15 +29,15 @@
│ │ │ │  For g<=10 the curves are constructed via plane models.
│ │ │ │  For g<=13 the curves are constructed via space models.
│ │ │ │  For g=14 the curves are constructed by Verra's method.
│ │ │ │  For g=15 the curves are constructed via matrix factorizations.
│ │ │ │  If the option Printing is set to true then printings about the current step in
│ │ │ │  the construction are displayed.
│ │ │ │  i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ │ + -- used 1.17979s (cpu); 1.01985s (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/RandomIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  cmFuZG9tU2hlbGxhYmxlSWRlYWxDaGFpbg==
│ │ │  #:len=1799
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiUHJvZHVjZXMgYSBjaGFpbiBvZiBpZGVh
│ │ │  bHMgZnJvbSBhIHJhbmRvbSBzaGVsbGluZyIsICJsaW5lbnVtIiA9PiA3NjAsIElucHV0cyA9PiB7
│ │ ├── ./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 1792812652
│ │ │ + -- setting random seed to 1792835011
│ │ │  
│ │ │ -o1 = 1792812652
│ │ │ +o1 = 1792835011
│ │ │  
│ │ │  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 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ + -- used 3.32858s (cpu); 1.64193s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 37 }
│ │ │ -           5 => 221
│ │ │ -           6 => 158
│ │ │ -           7 => 75
│ │ │ -           8 => 8
│ │ │ -           9 => 1
│ │ │ +o4 = Tally{3 => 2  }
│ │ │ +           4 => 53
│ │ │ +           5 => 205
│ │ │ +           6 => 175
│ │ │ +           7 => 60
│ │ │ +           8 => 5
│ │ │  
│ │ │  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 1792812658
│ │ │ + -- setting random seed to 1792835015
│ │ │  
│ │ │ -o1 = 1792812658
│ │ │ +o1 = 1792835015
│ │ │  
│ │ │  i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │  
│ │ │  o2 : QuotientRing
│ │ │  
│ │ │ @@ -15,13 +15,13 @@
│ │ │  
│ │ │  o3 = S
│ │ │  
│ │ │  o3 : PolynomialRing
│ │ │  
│ │ │  i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = b c
│ │ │ +        2
│ │ │ +o4 = a*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 1792812665
│ │ │ + -- setting random seed to 1792835018
│ │ │  
│ │ │ -o1 = 1792812665
│ │ │ +o1 = 1792835018
│ │ │  
│ │ │  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*b*c)
│ │ │  
│ │ │  o5 : Ideal of S
│ │ │  
│ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ +o6 = ideal (b*d, a*e, a*b, b*e, c*e)
│ │ │  
│ │ │  o6 : Ideal of S
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/example-output/_random__Square__Free__Step.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 10504911213508281315
│ │ │  
│ │ │  i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812666
│ │ │ + -- setting random seed to 1792835020
│ │ │  
│ │ │ -o1 = 1792812666
│ │ │ +o1 = 1792835020
│ │ │  
│ │ │  i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │  
│ │ │  o2 : PolynomialRing
│ │ │  
│ │ │ @@ -15,15 +15,15 @@
│ │ │  
│ │ │  o3 = monomialIdeal (a*b, a*d, b*c*d)
│ │ │  
│ │ │  o3 : MonomialIdeal of S
│ │ │  
│ │ │  i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 : setRandomSeed(1)
│ │ │   -- setting random seed to 1
│ │ │  
│ │ │  o5 = 1
│ │ │ @@ -37,15 +37,15 @@
│ │ │  i7 : J = monomialIdeal 0_S
│ │ │  
│ │ │  o7 = monomialIdeal ()
│ │ │  
│ │ │  o7 : MonomialIdeal of S
│ │ │  
│ │ │  i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
│ │ │ + -- used 3.35547s (cpu); 2.57361s (thread); 0s (gc)
│ │ │  
│ │ │  i9 : #T
│ │ │  
│ │ │  o9 = 168
│ │ │  
│ │ │  i10 : T
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Monomial.html
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Chooses a random monomial.
    
│ │ │          
    
│ │ │          
    
│ │ │                
    i1 : time ICan = smoothCanonicalCurve(11,5);
│ │ │ - -- used 1.34101s (cpu); 1.04498s (thread); 0s (gc)
│ │ │ + -- used 1.17979s (cpu); 1.01985s (thread); 0s (gc)
│ │ │  
│ │ │                ZZ
│ │ │  o1 : Ideal of --[t ..t  ]
│ │ │                 5  0   10
    
│ │ │              
    
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812658
│ │ │ + -- setting random seed to 1792835015
│ │ │  
│ │ │ -o1 = 1792812658
    
│ │ │ +o1 = 1792835015
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -103,16 +103,16 @@
│ │ │  o3 : PolynomialRing
    
│ │ │              
    
│ │ │                
    i4 : randomMonomial(3,S)
│ │ │  
│ │ │ -      2
│ │ │ -o4 = b c
│ │ │ +        2
│ │ │ +o4 = a*c
│ │ │  
│ │ │  o4 : S
    
│ │ │              
    
│ │ │        
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -11,31 +11,31 @@
│ │ │ │            o d, an _i_n_t_e_g_e_r, non-negative
│ │ │ │            o S, a _r_i_n_g, polynomial ring
│ │ │ │      * Outputs:
│ │ │ │            o m, a _r_i_n_g_ _e_l_e_m_e_n_t, monomial of S
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Chooses a random monomial.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812658
│ │ │ │ + -- setting random seed to 1792835015
│ │ │ │  
│ │ │ │ -o1 = 1792812658
│ │ │ │ +o1 = 1792835015
│ │ │ │  i2 : kk=ZZ/101
│ │ │ │  
│ │ │ │  o2 = kk
│ │ │ │  
│ │ │ │  o2 : QuotientRing
│ │ │ │  i3 : S=kk[a,b,c]
│ │ │ │  
│ │ │ │  o3 = S
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ │ │  i4 : randomMonomial(3,S)
│ │ │ │  
│ │ │ │ -      2
│ │ │ │ -o4 = b c
│ │ │ │ +        2
│ │ │ │ +o4 = a*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
│ │ │ @@ -76,17 +76,17 @@
│ │ │          
    
│ │ │            
    Choose a random square-free monomial ideal whose generators have the degrees specified by the list or sequence L.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812665
│ │ │ + -- setting random seed to 1792835018
│ │ │  
│ │ │ -o1 = 1792812665
    
│ │ │ +o1 = 1792835018
│ │ │              
    
│ │ │                
    i2 : kk=ZZ/101
│ │ │  
│ │ │  o2 = kk
│ │ │ @@ -113,24 +113,24 @@
│ │ │              
    
│ │ │                
    i5 : randomSquareFreeMonomialIdeal(L, S)
│ │ │  low degree gens generated everything
│ │ │  
│ │ │ -o5 = ideal(a*c*d)
│ │ │ +o5 = ideal(a*b*c)
│ │ │  
│ │ │  o5 : Ideal of S
    
│ │ │              
    
│ │ │                
    i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │  
│ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ +o6 = ideal (b*d, a*e, a*b, b*e, c*e)
│ │ │  
│ │ │  o6 : Ideal of S
    
│ │ │              
    
│ │ │        
    
│ │ │        
    
│ │ │ ├── html2text {}
│ │ │ │ @@ -13,17 +13,17 @@
│ │ │ │      * Outputs:
│ │ │ │            o I, an _i_d_e_a_l, square-free monomial ideal with generators of
│ │ │ │              specified degrees
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Choose a random square-free monomial ideal whose generators have the degrees
│ │ │ │  specified by the list or sequence L.
│ │ │ │  i1 : setRandomSeed(currentTime())
│ │ │ │ - -- setting random seed to 1792812665
│ │ │ │ + -- setting random seed to 1792835018
│ │ │ │  
│ │ │ │ -o1 = 1792812665
│ │ │ │ +o1 = 1792835018
│ │ │ │  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*b*c)
│ │ │ │  
│ │ │ │  o5 : Ideal of S
│ │ │ │  i6 : randomSquareFreeMonomialIdeal(5:2, S)
│ │ │ │  
│ │ │ │ -o6 = ideal (a*e, c*d, a*c, b*c, c*e)
│ │ │ │ +o6 = ideal (b*d, a*e, a*b, b*e, c*e)
│ │ │ │  
│ │ │ │  o6 : Ideal of S
│ │ │ │  ********** CCaavveeaatt **********
│ │ │ │  The ideal is constructed degree by degree, starting from the lowest degree
│ │ │ │  specified. If there are not enough monomials of the next specified degree that
│ │ │ │  are not already in the ideal, the function prints a warning and returns an
│ │ │ │  ideal containing all the generators of that degree.
│ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/_random__Square__Free__Step.html
│ │ │ @@ -84,17 +84,17 @@
│ │ │            
    With probability p the routine takes the Alexander dual of I; the default value of p is .05, and it can be set with the option AlexanderProbility.
    
│ │ │            
    Otherwise uses the Metropolis algorithm to produce a random walk on the space of square-free ideals. Note that there are a LOT of square-free ideals; these are the Dedekind numbers, and the sequence (with 1,2,3,4,5,6,7,8 variables) begins 3,6,20,168,7581, 7828354, 2414682040998, 56130437228687557907788. (see the Online Encyclopedia of Integer Sequences for more information). Given I in a polynomial ring S, we make a list ISocgens of the square-free minimal monomial generators of the socle of S/(squares+I) and a list of minimal generators Igens of I. A candidate "next" ideal J is formed as follows: We choose randomly from the union of these lists; if a socle element is chosen, it's added to I; if a minimal generator is chosen, it's replaced by the square-free part of the maximal ideal times it. the chance of making the given move is then 1/(#ISocgens+#Igens), and the chance of making the move back would be the similar quantity for J, so we make the move or not depending on whether random RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1].
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812666
│ │ │ + -- setting random seed to 1792835020
│ │ │  
│ │ │ -o1 = 1792812666
    
│ │ │ +o1 = 1792835020
│ │ │              
    
│ │ │                
    i2 : S=ZZ/2[vars(0..3)]
│ │ │  
│ │ │  o2 = S
│ │ │ @@ -111,15 +111,15 @@
│ │ │  o3 : MonomialIdeal of S
    
│ │ │              
    
│ │ │                
    i4 : randomSquareFreeStep J
│ │ │  
│ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}}
│ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}}
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │          
    
│ │ │            
    With 4 variables and 168 possible monomial ideals, a run of 5000 takes less than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000 takes about 2 seconds.
    
│ │ │ @@ -150,15 +150,15 @@
│ │ │  
│ │ │  o7 : MonomialIdeal of S
    │ │ │
    │ │ │ 
    i8 : time T=tally for t from 1 to 5000 list first (J=rsfs(J,AlexanderProbability => .01));
│ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc)
    │ │ │ + -- used 3.35547s (cpu); 2.57361s (thread); 0s (gc) │ │ │
    │ │ │ 
    i9 : #T
│ │ │  
│ │ │  o9 = 168
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,30 +35,30 @@ │ │ │ │ choose randomly from the union of these lists; if a socle element is chosen, │ │ │ │ it's added to I; if a minimal generator is chosen, it's replaced by the square- │ │ │ │ free part of the maximal ideal times it. the chance of making the given move is │ │ │ │ then 1/(#ISocgens+#Igens), and the chance of making the move back would be the │ │ │ │ similar quantity for J, so we make the move or not depending on whether random │ │ │ │ RR < (nJ+nSocJ)/(nI+nSocI) or not; here random RR is a random number in [0,1]. │ │ │ │ i1 : setRandomSeed(currentTime()) │ │ │ │ - -- setting random seed to 1792812666 │ │ │ │ + -- setting random seed to 1792835020 │ │ │ │ │ │ │ │ -o1 = 1792812666 │ │ │ │ +o1 = 1792835020 │ │ │ │ i2 : S=ZZ/2[vars(0..3)] │ │ │ │ │ │ │ │ o2 = S │ │ │ │ │ │ │ │ o2 : PolynomialRing │ │ │ │ i3 : J = monomialIdeal"ab,ad, bcd" │ │ │ │ │ │ │ │ o3 = monomialIdeal (a*b, a*d, b*c*d) │ │ │ │ │ │ │ │ o3 : MonomialIdeal of S │ │ │ │ i4 : randomSquareFreeStep J │ │ │ │ │ │ │ │ -o4 = {monomialIdeal (a*b, b*c, a*d), {a*b, b*c, a*d}, {c*d, b*d, a*c}} │ │ │ │ +o4 = {monomialIdeal (a*b, a*d), {a*b, a*d}, {b*c*d, a*c}} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ With 4 variables and 168 possible monomial ideals, a run of 5000 takes less │ │ │ │ than 6 seconds on a reasonably fast machine. With 10 variables a run of 1000 │ │ │ │ takes about 2 seconds. │ │ │ │ i5 : setRandomSeed(1) │ │ │ │ -- setting random seed to 1 │ │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ i7 : J = monomialIdeal 0_S │ │ │ │ │ │ │ │ o7 = monomialIdeal () │ │ │ │ │ │ │ │ o7 : MonomialIdeal of S │ │ │ │ i8 : time T=tally for t from 1 to 5000 list first (J=rsfs │ │ │ │ (J,AlexanderProbability => .01)); │ │ │ │ - -- used 3.64766s (cpu); 2.71005s (thread); 0s (gc) │ │ │ │ + -- used 3.35547s (cpu); 2.57361s (thread); 0s (gc) │ │ │ │ i9 : #T │ │ │ │ │ │ │ │ o9 = 168 │ │ │ │ i10 : T │ │ │ │ │ │ │ │ o10 = Tally{monomialIdeal () => 45 } │ │ │ │ monomialIdeal (a*b*c, a*b*d) => 33 │ │ ├── ./usr/share/doc/Macaulay2/RandomIdeals/html/index.html │ │ │ @@ -59,17 +59,17 @@ │ │ │
    │ │ │

    This package can be used to make experiments, trying many ideals, perhaps over small fields. For example...what would you expect the regularities of "typical" monomial ideals with 10 generators of degree 3 in 6 variables to be? Try a bunch of examples -- it's fast. Here we do only 500 -- this takes about a second on a fast machine -- but with a little patience, thousands can be done conveniently.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -77,22 +77,22 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : setRandomSeed(currentTime())
│ │ │ - -- setting random seed to 1792812652
│ │ │ + -- setting random seed to 1792835011
│ │ │  
│ │ │ -o1 = 1792812652
    │ │ │ +o1 = 1792835011 │ │ │
    │ │ │ 
    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 4.13607s (cpu); 2.14846s (thread); 0s (gc)
│ │ │ + -- used 3.32858s (cpu); 1.64193s (thread); 0s (gc)
│ │ │  
│ │ │ -o4 = Tally{4 => 37 }
│ │ │ -           5 => 221
│ │ │ -           6 => 158
│ │ │ -           7 => 75
│ │ │ -           8 => 8
│ │ │ -           9 => 1
│ │ │ +o4 = Tally{3 => 2  }
│ │ │ +           4 => 53
│ │ │ +           5 => 205
│ │ │ +           6 => 175
│ │ │ +           7 => 60
│ │ │ +           8 => 5
│ │ │  
│ │ │  o4 : Tally
    │ │ │
    │ │ │
    │ │ │

    How does this compare with the case of binomial ideals? or pure binomial ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger numbers of examples. Click the link "Finding Extreme Examples" below to see some other, more elaborate ways to search.

    │ │ │ ├── html2text {} │ │ │ │ @@ -9,28 +9,28 @@ │ │ │ │ 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 1792812652 │ │ │ │ + -- setting random seed to 1792835011 │ │ │ │ │ │ │ │ -o1 = 1792812652 │ │ │ │ +o1 = 1792835011 │ │ │ │ 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 4.13607s (cpu); 2.14846s (thread); 0s (gc) │ │ │ │ + -- used 3.32858s (cpu); 1.64193s (thread); 0s (gc) │ │ │ │ │ │ │ │ -o4 = Tally{4 => 37 } │ │ │ │ - 5 => 221 │ │ │ │ - 6 => 158 │ │ │ │ - 7 => 75 │ │ │ │ - 8 => 8 │ │ │ │ - 9 => 1 │ │ │ │ +o4 = Tally{3 => 2 } │ │ │ │ + 4 => 53 │ │ │ │ + 5 => 205 │ │ │ │ + 6 => 175 │ │ │ │ + 7 => 60 │ │ │ │ + 8 => 5 │ │ │ │ │ │ │ │ o4 : Tally │ │ │ │ How does this compare with the case of binomial ideals? or pure binomial │ │ │ │ ideals? We invite the reader to experiment, replacing "randomMonomialIdeal" │ │ │ │ above with "randomBinomialIdeal" or "randomPureBinomialIdeal", or taking larger │ │ │ │ numbers of examples. Click the link "Finding Extreme Examples" below to see │ │ │ │ some other, more elaborate ways to search. │ │ ├── ./usr/share/doc/Macaulay2/RandomMonomialIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ VmFyaWFibGVOYW1l │ │ │ #:len=1241 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAib3B0aW9uYWwgaW5wdXQgdG8gY2hvb3Nl │ │ │ IHRoZSBpbmRleGVkIHZhcmlhYmxlIG5hbWUgZm9yIHRoZSBwb2x5bm9taWFsIHJpbmciLCAibGlu │ │ ├── ./usr/share/doc/Macaulay2/RandomObjects/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ QXR0ZW1wdHM= │ │ │ #:len=502 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibnVtYmVyIG9mIGF0dGVtcHRzIGluIHRo │ │ │ ZSBjb25zdHJ1Y3Rpb24gb2YgYSByYW5kb20gb2JqZWN0IiwgImxpbmVudW0iID0+IDI2MiwgImZp │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ ZmluZEFOb25aZXJvTWlub3IoLi4uLFBvaW50Q2hlY2tBdHRlbXB0cz0+Li4uKQ== │ │ │ #:len=315 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4OCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEFOb25aZXJvTWlub3IsUG9pbnRDaGVja0F0 │ │ ├── ./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) │ │ │ - -- 2.00843s elapsed │ │ │ + -- 1.21773s elapsed │ │ │ │ │ │ o4 = 4 │ │ │ │ │ │ i5 : elapsedTime dim I │ │ │ - -- 3.91258s elapsed │ │ │ + -- 2.56381s 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)) ); │ │ │ - -- 2.88102s elapsed │ │ │ + -- 1.63997s elapsed │ │ │ │ │ │ i11 : dim J │ │ │ │ │ │ o11 = 1 │ │ │ │ │ │ i12 : i │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/example-output/_random__Points.out │ │ │ @@ -27,24 +27,24 @@ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable) │ │ │ - -- 4.05108s elapsed │ │ │ + -- 2.44429s 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) │ │ │ - -- 3.02229s elapsed │ │ │ + -- 1.88851s elapsed │ │ │ │ │ │ o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16, │ │ │ ------------------------------------------------------------------------ │ │ │ 11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}} │ │ │ │ │ │ o9 : List │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_dim__Via__Bezout.html │ │ │ @@ -100,23 +100,23 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │
    │ │ │ 
    i4 : elapsedTime dimViaBezout(I)
│ │ │ - -- 2.00843s elapsed
│ │ │ + -- 1.21773s elapsed
│ │ │  
│ │ │  o4 = 4
    │ │ │
    │ │ │ 
    i5 : elapsedTime dim I
│ │ │ - -- 3.91258s elapsed
│ │ │ + -- 2.56381s 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) │ │ │ │ - -- 2.00843s elapsed │ │ │ │ + -- 1.21773s elapsed │ │ │ │ │ │ │ │ o4 = 4 │ │ │ │ i5 : elapsedTime dim I │ │ │ │ - -- 3.91258s elapsed │ │ │ │ + -- 2.56381s elapsed │ │ │ │ │ │ │ │ o5 = 4 │ │ │ │ The user may set the MinimumFieldSize to ensure that the field being worked │ │ │ │ over is big enough. For instance, there are relatively few linear spaces over a │ │ │ │ field of characteristic 2, and this can cause incorrect results to be provided. │ │ │ │ If no size is provided, the function tries to guess a good size based on │ │ │ │ ambient ring. │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_extend__Ideal__By__Non__Zero__Minor.html │ │ │ @@ -160,15 +160,15 @@ │ │ │ │ │ │ o9 : Ideal of T │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : elapsedTime(while (i < 10) and dim J > 1 do (i = i+1; J = extendIdealByNonZeroMinor(4, M, J)) );
│ │ │ - -- 2.88102s elapsed
    │ │ │ + -- 1.63997s 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)) ); │ │ │ │ - -- 2.88102s elapsed │ │ │ │ + -- 1.63997s elapsed │ │ │ │ i11 : dim J │ │ │ │ │ │ │ │ o11 = 1 │ │ │ │ i12 : i │ │ │ │ │ │ │ │ o12 = 4 │ │ │ │ In this particular example, there tend to be about 5 associated primes when │ │ ├── ./usr/share/doc/Macaulay2/RandomPoints/html/_random__Points.html │ │ │ @@ -149,27 +149,27 @@ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
│ │ │ - -- 4.05108s elapsed
│ │ │ + -- 2.44429s 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)
│ │ │ - -- 3.02229s elapsed
│ │ │ + -- 1.88851s 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
    │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -66,24 +66,24 @@ │ │ │ │ first in rings with more variables. │ │ │ │ i6 : S=ZZ/103[y_0..y_30]; │ │ │ │ i7 : I=minors(2,random(S^3,S^{3:-1})); │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, │ │ │ │ DecompositionStrategy=>MultiplicationTable) │ │ │ │ - -- 4.05108s elapsed │ │ │ │ + -- 2.44429s 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) │ │ │ │ - -- 3.02229s elapsed │ │ │ │ + -- 1.88851s 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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=46 │ │ │ aXNCaXJhdGlvbmFsT250b0ltYWdlKC4uLixBc3N1bWVEb21pbmFudD0+Li4uKQ== │ │ │ #:len=321 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgwNiwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaXNCaXJhdGlvbmFsT250b0ltYWdlLEFzc3VtZURv │ │ ├── ./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 0.341115s (cpu); 0.337449s (thread); 0s (gc) │ │ │ + -- used 0.634507s (cpu); 0.354015s (thread); 0s (gc) │ │ │ │ │ │ 125 124 120 5 124 100 25 104 20 108 15 2 112 10 3 116 5 4 120 5 124 125 4 120 8 115 2 12 110 3 16 105 4 20 100 5 24 95 6 28 90 7 32 85 8 36 80 9 40 75 10 44 70 11 48 65 12 52 60 13 56 55 14 60 50 15 64 45 16 68 40 17 72 35 18 76 30 19 80 25 20 84 20 21 88 15 22 92 10 23 96 5 24 100 25 24 100 28 95 32 90 2 36 85 3 40 80 4 44 75 5 48 70 6 52 65 7 56 60 8 60 55 9 64 50 10 68 45 11 72 40 12 76 35 13 80 30 14 84 25 15 88 20 16 92 15 17 96 10 18 100 5 19 104 20 48 75 2 52 70 2 56 65 2 2 60 60 3 2 64 55 4 2 68 50 5 2 72 45 6 2 76 40 7 2 80 35 8 2 84 30 9 2 88 25 10 2 92 20 11 2 96 15 12 2 100 10 13 2 104 5 14 2 108 15 2 72 50 3 76 45 3 80 40 2 3 84 35 3 3 88 30 4 3 92 25 5 3 96 20 6 3 100 15 7 3 104 10 8 3 108 5 9 3 112 10 3 96 25 4 100 20 4 104 15 2 4 108 10 3 4 112 5 4 4 116 5 4 120 5 124 │ │ │ o14 = Proj Q - - - > Proj Q {x , x y, - x y + x z, x y - 5x y z + 10x y z - 10x y z + 5x y z - x z + x t, - y + 25x y z - 300x y z + 2300x y z - 12650x y z + 53130x y z - 177100x y z + 480700x y z - 1081575x y z + 2042975x y z - 3268760x y z + 4457400x y z - 5200300x y z + 5200300x y z - 4457400x y z + 3268760x y z - 2042975x y z + 1081575x y z - 480700x y z + 177100x y z - 53130x y z + 12650x y z - 2300x y z + 300x y z - 25x y z + x z - 5x y t + 100x y z*t - 950x y z t + 5700x y z t - 24225x y z t + 77520x y z t - 193800x y z t + 387600x y z t - 629850x y z t + 839800x y z t - 923780x y z t + 839800x y z t - 629850x y z t + 387600x y z t - 193800x y z t + 77520x y z t - 24225x y z t + 5700x y z t - 950x y z t + 100x y z t - 5x z t - 10x y t + 150x y z*t - 1050x y z t + 4550x y z t - 13650x y z t + 30030x y z t - 50050x y z t + 64350x y z t - 64350x y z t + 50050x y z t - 30030x y z t + 13650x y z t - 4550x y z t + 1050x y z t - 150x y z t + 10x z t - 10x y t + 100x y z*t - 450x y z t + 1200x y z t - 2100x y z t + 2520x y z t - 2100x y z t + 1200x y z t - 450x y z t + 100x y z t - 10x z t - 5x y t + 25x y z*t - 50x y z t + 50x y z t - 25x y z t + 5x z t - x t + x u} │ │ │ │ │ │ o14 : RationalMapping │ │ │ │ │ │ i15 : R=QQ[x,y,z,t]/(z-2*t); │ │ ├── ./usr/share/doc/Macaulay2/RationalMaps/html/_inverse__Of__Map.html │ │ │ @@ -194,15 +194,15 @@ │ │ │ │ │ │ o13 : RingMap Q <-- Q │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time inverseOfMap(phi,CheckBirational=>false, Verbosity=>0)
│ │ │ - -- used 0.341115s (cpu); 0.337449s (thread); 0s (gc)
│ │ │ + -- used 0.634507s (cpu); 0.354015s (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
    │ │ │ │ │ │ │ │ │ ├── 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 0.341115s (cpu); 0.337449s (thread); 0s (gc) │ │ │ │ + -- used 0.634507s (cpu); 0.354015s (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/RationalPoints/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ U29ydEdlbnM= │ │ │ #:len=246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg3LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJTb3J0R2VucyIsIlNvcnRHZW5zIiwiUmF0aW9uYWxQ │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ bmV0KFByb2plY3RpdmVQb2ludCk= │ │ │ #:len=205 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTcsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KG5ldCxQcm9q │ │ ├── ./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 0.00513051s (cpu); 0.00512835s (thread); 0s (gc) │ │ │ + -- used 0.00335677s (cpu); 0.00335515s (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 1.28715s (cpu); 1.09718s (thread); 0s (gc) │ │ │ + -- used 0.909634s (cpu); 0.720444s (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 0.282073s (cpu); 0.221929s (thread); 0s (gc) │ │ │ + -- used 0.261165s (cpu); 0.195917s (thread); 0s (gc) │ │ │ │ │ │ o35 = 31 │ │ │ │ │ │ i36 : │ │ ├── ./usr/share/doc/Macaulay2/RationalPoints2/html/_rational__Points.html │ │ │ @@ -183,15 +183,15 @@ │ │ │ o13 : Ideal of ---[u ..u ] │ │ │ 101 0 10 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time rationalPoints(I, Amount => true)
│ │ │ - -- used 0.00513051s (cpu); 0.00512835s (thread); 0s (gc)
│ │ │ + -- used 0.00335677s (cpu); 0.00335515s (thread); 0s (gc)
│ │ │  
│ │ │  o14 = 110462212541120451001
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Over number fields

    │ │ │ @@ -353,15 +353,15 @@ │ │ │ o31 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i32 : time rationalPoints(variety nodes, Split=>true, Verbose=>true);
│ │ │  -- base change to the field QQ[a]/(a^8-40*a^6+230*a^4-200*a^2+25)
│ │ │ - -- used 1.28715s (cpu); 1.09718s (thread); 0s (gc)
    │ │ │ + -- used 0.909634s (cpu); 0.720444s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i33 : #oo
│ │ │  
│ │ │  o33 = 31
    │ │ │ @@ -378,15 +378,15 @@ │ │ │ │ │ │ o34 : Ideal of GF 1048969271299456081[x..z, w] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i35 : time #rationalPoints(variety nodes', Split=>true, Verbose=>true)
│ │ │ - -- used 0.282073s (cpu); 0.221929s (thread); 0s (gc)
│ │ │ + -- used 0.261165s (cpu); 0.195917s (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 0.00513051s (cpu); 0.00512835s (thread); 0s (gc) │ │ │ │ + -- used 0.00335677s (cpu); 0.00335515s (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 1.28715s (cpu); 1.09718s (thread); 0s (gc) │ │ │ │ + -- used 0.909634s (cpu); 0.720444s (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 0.282073s (cpu); 0.221929s (thread); 0s (gc) │ │ │ │ + -- used 0.261165s (cpu); 0.195917s (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/ReactionNetworks/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ bW9kaWZpY2F0aW9uT2ZUd29TdWJzdHJhdGVzSCgp │ │ │ #:len=344 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20gezE6KG1vZGlmaWNhdGlvbk9mVHdvU3Vic3RyYXRlc0gp │ │ ├── ./usr/share/doc/Macaulay2/RealRoots/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=17 │ │ │ dHJhY2VDb3VudChJZGVhbCk= │ │ │ #:len=241 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTAyMSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodHJhY2VDb3VudCxJZGVhbCksInRyYWNlQ291bnQo │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ aXNMaW5lYXJUeXBlKE1vZHVsZSk= │ │ │ #:len=257 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE5OSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNMaW5lYXJUeXBlLE1vZHVsZSksImlzTGluZWFy │ │ ├── ./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 1.07953s (cpu); 0.841221s (thread); 0s (gc) │ │ │ + -- used 0.823867s (cpu); 0.665192s (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 1.37298s (cpu); 0.841764s (thread); 0s (gc) │ │ │ + -- used 1.0094s (cpu); 0.717346s (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 1.79026s (cpu); 1.2643s (thread); 0s (gc) │ │ │ + -- used 1.55316s (cpu); 1.2112s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ 0 3 │ │ │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc) │ │ │ + -- used 1.51762s (cpu); 1.22176s (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 0.0295188s (cpu); 0.0275558s (thread); 0s (gc) │ │ │ + -- used 0.0939164s (cpu); 0.0320954s (thread); 0s (gc) │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ 0 6 │ │ │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc) │ │ │ + -- used 0.146767s (cpu); 0.134904s (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 0.023238s (cpu); 0.0222073s (thread); 0s (gc) │ │ │ + -- used 0.0610944s (cpu); 0.0213664s (thread); 0s (gc) │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc) │ │ │ + -- used 0.0277814s (cpu); 0.0111184s (thread); 0s (gc) │ │ │ │ │ │ o10 : Ideal of S[w ..w ] │ │ │ 0 2 │ │ │ │ │ │ i11 : transpose gens I1 │ │ │ │ │ │ o11 = {-1, -3} | aw_1-bw_2 | │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/___Plane__Curve__Singularities.html │ │ │ @@ -592,15 +592,15 @@ │ │ │
    │ │ │

    We compute the singular locus once again:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -325,15 +325,15 @@ │ │ │ │ 2 2 2 2 2 2 2 │ │ │ │ - p w , p y - p , p w y - p p , p w - p ) │ │ │ │ 2 1 0 1 0 0 1 2 0 0 2 │ │ │ │ │ │ │ │ o47 : Ideal of B2 │ │ │ │ We compute the singular locus once again: │ │ │ │ i48 : time sing2 = ideal singularLocus strictTransform2; │ │ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc) │ │ │ │ + -- used 0.823867s (cpu); 0.665192s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o48 : Ideal of -----[p ..p , w ..w , x..y] │ │ │ │ 32003 0 2 0 1 │ │ │ │ i49 : saturate(sing2, sub(irrelTot, ring sing2)) │ │ │ │ │ │ │ │ o49 = ideal 1 │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_expected__Rees__Ideal.html │ │ │ @@ -156,15 +156,15 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i48 : time sing2 = ideal singularLocus strictTransform2;
│ │ │ - -- used 1.07953s (cpu); 0.841221s (thread); 0s (gc)
│ │ │ + -- used 0.823867s (cpu); 0.665192s (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 1.37298s (cpu); 0.841764s (thread); 0s (gc)
│ │ │ + -- used 1.0094s (cpu); 0.717346s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S[w ..w ]
│ │ │                   0   4
    │ │ │
    │ │ │ @@ -190,24 +190,24 @@ │ │ │ │ │ │ o11 : Ideal of S │ │ │
    │ │ │ 
    i12 : time reesIdeal (I, I_0);
│ │ │ - -- used 1.79026s (cpu); 1.2643s (thread); 0s (gc)
│ │ │ + -- used 1.55316s (cpu); 1.2112s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S[w ..w ]
│ │ │                    0   3
    │ │ │
    │ │ │ 
    i13 : time reesIdeal (I, I_0, Jacobian =>false);
│ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc)
│ │ │ + -- used 1.51762s (cpu); 1.22176s (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 1.37298s (cpu); 0.841764s (thread); 0s (gc) │ │ │ │ + -- used 1.0094s (cpu); 0.717346s (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 1.79026s (cpu); 1.2643s (thread); 0s (gc) │ │ │ │ + -- used 1.55316s (cpu); 1.2112s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ i13 : time reesIdeal (I, I_0, Jacobian =>false); │ │ │ │ - -- used 2.01763s (cpu); 1.4125s (thread); 0s (gc) │ │ │ │ + -- used 1.51762s (cpu); 1.22176s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of S[w ..w ] │ │ │ │ 0 3 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_y_m_m_e_t_r_i_c_A_l_g_e_b_r_a_I_d_e_a_l -- Ideal of the symmetric algebra of an ideal or │ │ │ │ module │ │ │ │ * _j_a_c_o_b_i_a_n_D_u_a_l -- Computes the 'jacobian dual', part of a method of finding │ │ ├── ./usr/share/doc/Macaulay2/ReesAlgebra/html/_rees__Ideal.html │ │ │ @@ -115,24 +115,24 @@ │ │ │ │ │ │ o3 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time V1 = reesIdeal i;
│ │ │ - -- used 0.0295188s (cpu); 0.0275558s (thread); 0s (gc)
│ │ │ + -- used 0.0939164s (cpu); 0.0320954s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time V2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc)
│ │ │ + -- used 0.146767s (cpu); 0.134904s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : Ideal of S[w ..w ]
│ │ │                   0   6
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ @@ -169,24 +169,24 @@ │ │ │ │ │ │ o8 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : time I1 = reesIdeal i;
│ │ │ - -- used 0.023238s (cpu); 0.0222073s (thread); 0s (gc)
│ │ │ + -- used 0.0610944s (cpu); 0.0213664s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of S[w ..w ]
│ │ │                   0   2
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : time I2 = reesIdeal(i,i_0);
│ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc)
│ │ │ + -- used 0.0277814s (cpu); 0.0111184s (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 0.0295188s (cpu); 0.0275558s (thread); 0s (gc) │ │ │ │ + -- used 0.0939164s (cpu); 0.0320954s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 : Ideal of S[w ..w ] │ │ │ │ 0 6 │ │ │ │ i5 : time V2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.138904s (cpu); 0.138178s (thread); 0s (gc) │ │ │ │ + -- used 0.146767s (cpu); 0.134904s (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 0.023238s (cpu); 0.0222073s (thread); 0s (gc) │ │ │ │ + -- used 0.0610944s (cpu); 0.0213664s (thread); 0s (gc) │ │ │ │ │ │ │ │ o9 : Ideal of S[w ..w ] │ │ │ │ 0 2 │ │ │ │ i10 : time I2 = reesIdeal(i,i_0); │ │ │ │ - -- used 0.00959305s (cpu); 0.00933152s (thread); 0s (gc) │ │ │ │ + -- used 0.0277814s (cpu); 0.0111184s (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/ReflexivePolytopesDB/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ S1NFbnRyeQ== │ │ │ #:len=2438 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gZW50cnkgZnJvbSB0aGUgS3JldXpl │ │ │ ci1Ta2Fya2UgZGF0YWJhc2Ugb2YgZGltZW5zaW9uIDMgYW5kIDQgcmVmbGV4aXZlIHBvbHl0b3Bl │ │ ├── ./usr/share/doc/Macaulay2/Regularity/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ UmVndWxhcml0eQ== │ │ │ #:len=797 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSBDYXN0ZWxudW92by1NdW1m │ │ │ b3JkIHJlZ3VsYXJpdHkgb2YgYSBob21vZ2VuZW91cyBpZGVhbCIsIERlc2NyaXB0aW9uID0+IChQ │ │ ├── ./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 = .3009249333333335 │ │ │ +o8 = .216566079916667 │ │ │ │ │ │ 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 = .04256394509756095 │ │ │ +o11 = .04311311826388886 │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ │ │ i12 : time regularity I2 │ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc) │ │ │ + -- used 0.00312079s (cpu); 0.00312537s (thread); 0s (gc) │ │ │ │ │ │ o12 = 4 │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Regularity/html/_m__Regularity.html │ │ │ @@ -181,15 +181,15 @@ │ │ │ o7 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : benchmark "mRegularity I1"
│ │ │  
│ │ │ -o8 = .3009249333333335
│ │ │ +o8 = .216566079916667
│ │ │  
│ │ │  o8 : RR (of precision 53)
    │ │ │ │ │ │ │ │ │ │ │ │

    This is an example where regularity is faster than mRegularity.

    │ │ │ │ │ │ @@ -209,23 +209,23 @@ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i11 : benchmark " mRegularity I2"
│ │ │  
│ │ │ -o11 = .04256394509756095
│ │ │ +o11 = .04311311826388886
│ │ │  
│ │ │  o11 : RR (of precision 53)
    │ │ │
    │ │ │ 
    i12 : time regularity I2  
│ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc)
│ │ │ + -- used 0.00312079s (cpu); 0.00312537s (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 = .3009249333333335 │ │ │ │ +o8 = .216566079916667 │ │ │ │ │ │ │ │ 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 = .04256394509756095 │ │ │ │ +o11 = .04311311826388886 │ │ │ │ │ │ │ │ o11 : RR (of precision 53) │ │ │ │ i12 : time regularity I2 │ │ │ │ - -- used 0.00348648s (cpu); 0.00348699s (thread); 0s (gc) │ │ │ │ + -- used 0.00312079s (cpu); 0.00312537s (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/RelativeCanonicalResolution/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ Y2Fub25pY2FsTXVsdGlwbGllcnM= │ │ │ #:len=1705 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNhbm9uaWNhbCBt │ │ │ dWx0aXBsaWVycyBvZiBhIHJhdGlvbmFsIGN1cnZlcyB3aXRoIG5vZGVzIiwgImxpbmVudW0iID0+ │ │ ├── ./usr/share/doc/Macaulay2/ResLengthThree/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=21 │ │ │ bXVsdFRhYmxlT25lVHdvKFJpbmcp │ │ │ #:len=273 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjYyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtdWx0VGFibGVPbmVUd28sUmluZyksIm11bHRUYWJs │ │ ├── ./usr/share/doc/Macaulay2/ResidualIntersections/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=15 │ │ │ Z2VuZXJpY1Jlc2lkdWFs │ │ │ #:len=1739 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgZ2VuZXJpYyByZXNpZHVh │ │ │ bCBpbnRlcnNlY3Rpb25zIG9mIGFuIGlkZWFsIiwgImxpbmVudW0iID0+IDg4OSwgSW5wdXRzID0+ │ │ ├── ./usr/share/doc/Macaulay2/ResolutionsOfStanleyReisnerRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=41 │ │ │ YmFyeWNlbnRyaWNTdWJkaXZpc2lvbihTaW1wbGljaWFsQ29tcGxleCk= │ │ │ #:len=381 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiYXJ5Y2VudHJpY1N1YmRpdmlzaW9uLFNpbXBsaWNp │ │ ├── ./usr/share/doc/Macaulay2/ResultantComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ bWl4ZWRTdWJkaXZpc2lvbg== │ │ │ #:len=1366 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBjb2hlcmVudCBtaXhl │ │ │ ZCBzdWItZGl2aXNpb24iLCAibGluZW51bSIgPT4gNDc5LCBJbnB1dHMgPT4ge1NQQU57VFR7Imdy │ │ ├── ./usr/share/doc/Macaulay2/Resultants/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ aHVyd2l0ekZvcm0oLi4uLFNpbmd1bGFyTG9jdXM9Pi4uLik= │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMyOSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaHVyd2l0ekZvcm0sU2luZ3VsYXJMb2N1c10sImh1 │ │ ├── ./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 0.187761s (cpu); 0.10659s (thread); 0s (gc) │ │ │ + -- used 0.13619s (cpu); 0.0719734s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (P1xP1xP1,P1xP1xP1') = cayleyTrick(P1xP1,1) │ │ │ - -- used 0.297065s (cpu); 0.151907s (thread); 0s (gc) │ │ │ + -- used 0.196954s (cpu); 0.0883871s (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 0.288499s (cpu); 0.151777s (thread); 0s (gc) │ │ │ + -- used 0.21949s (cpu); 0.106864s (thread); 0s (gc) │ │ │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc) │ │ │ + -- used 0.224647s (cpu); 0.117533s (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 0.0407572s (cpu); 0.040758s (thread); 0s (gc) │ │ │ + -- used 0.0423984s (cpu); 0.0423978s (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 0.0401433s (cpu); 0.0401425s (thread); 0s (gc) │ │ │ + -- used 0.0426096s (cpu); 0.0426104s (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 0.160178s (cpu); 0.106293s (thread); 0s (gc) │ │ │ + -- used 0.169639s (cpu); 0.102352s (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 0.0594214s (cpu); 0.0594176s (thread); 0s (gc) │ │ │ + -- used 0.0623613s (cpu); 0.0623675s (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 5.50368s (cpu); 4.92753s (thread); 0s (gc) │ │ │ + -- used 4.54286s (cpu); 4.20868s (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 1.1406s (cpu); 0.999558s (thread); 0s (gc) │ │ │ + -- used 0.973138s (cpu); 0.901726s (thread); 0s (gc) │ │ │ │ │ │ i5 : -- X-resultant of V │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc) │ │ │ + -- used 0.254184s (cpu); 0.193535s (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 0.341473s (cpu); 0.215038s (thread); 0s (gc) │ │ │ + -- used 0.24341s (cpu); 0.182054s (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 0.00924086s (cpu); 0.00924015s (thread); 0s (gc) │ │ │ + -- used 0.00962444s (cpu); 0.0096242s (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 0.0098778s (cpu); 0.0098795s (thread); 0s (gc) │ │ │ + -- used 0.011164s (cpu); 0.0111655s (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 0.44332s (cpu); 0.443247s (thread); 0s (gc) │ │ │ + -- used 0.377897s (cpu); 0.377897s (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 0.202374s (cpu); 0.134305s (thread); 0s (gc) │ │ │ + -- used 0.172913s (cpu); 0.10843s (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 0.310587s (cpu); 0.159166s (thread); 0s (gc) │ │ │ + -- used 0.281289s (cpu); 0.149719s (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 0.162855s (cpu); 0.0858796s (thread); 0s (gc) │ │ │ + -- used 0.127578s (cpu); 0.0720894s (thread); 0s (gc) │ │ │ │ │ │ ZZ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ 3331 0 9 │ │ │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc) │ │ │ + -- used 0.595438s (cpu); 0.52974s (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 0.141834s (cpu); 0.0752672s (thread); 0s (gc) │ │ │ + -- used 0.119086s (cpu); 0.0633871s (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 0.177092s (cpu); 0.069066s (thread); 0s (gc) │ │ │ + -- used 0.193864s (cpu); 0.0664178s (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 0.108209s (cpu); 0.0419396s (thread); 0s (gc) │ │ │ + -- used 0.102619s (cpu); 0.0350048s (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 0.115818s (cpu); 0.0462516s (thread); 0s (gc) │ │ │ + -- used 0.106946s (cpu); 0.0407097s (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 0.125887s (cpu); 0.0563778s (thread); 0s (gc) │ │ │ + -- used 0.111462s (cpu); 0.0527121s (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 0.00877231s (cpu); 0.00877035s (thread); 0s (gc) │ │ │ + -- used 0.00846686s (cpu); 0.00846591s (thread); 0s (gc) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ 3 2 3 7 2 5 7 2 10 │ │ │ @@ -56,12 +56,12 @@ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ o3 : -------------------------------------- │ │ │ p p - p p + p p │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ │ │ i4 : time isCoisotropic w' │ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc) │ │ │ + -- used 0.00736545s (cpu); 0.00736643s (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 0.0181849s (cpu); 0.0181858s (thread); 0s (gc) │ │ │ + -- used 0.0171607s (cpu); 0.017161s (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 0.00373056s (cpu); 0.00372864s (thread); 0s (gc) │ │ │ + -- used 0.00367841s (cpu); 0.00367712s (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 0.00252047s (cpu); 0.00251818s (thread); 0s (gc) │ │ │ + -- used 0.00225783s (cpu); 0.00225483s (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 0.00473681s (cpu); 0.00473472s (thread); 0s (gc) │ │ │ + -- used 0.00499304s (cpu); 0.00499201s (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 0.121088s (cpu); 0.0718194s (thread); 0s (gc) │ │ │ + -- used 0.106833s (cpu); 0.0480662s (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 0.148823s (cpu); 0.0922017s (thread); 0s (gc) │ │ │ + -- used 0.105147s (cpu); 0.0468495s (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 0.244058s (cpu); 0.244015s (thread); 0s (gc) │ │ │ + -- used 0.141418s (cpu); 0.141421s (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 0.562085s (cpu); 0.289584s (thread); 0s (gc) │ │ │ + -- used 0.498087s (cpu); 0.257168s (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 0.273183s (cpu); 0.131953s (thread); 0s (gc) │ │ │ + -- used 0.224784s (cpu); 0.102755s (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 0.302008s (cpu); 0.301983s (thread); 0s (gc) │ │ │ + -- used 0.289087s (cpu); 0.289088s (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 0.625765s (cpu); 0.559323s (thread); 0s (gc) │ │ │ + -- used 0.567787s (cpu); 0.513751s (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 0.123723s (cpu); 0.0492639s (thread); 0s (gc) │ │ │ + -- used 0.0910604s (cpu); 0.0366713s (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 3.07394s (cpu); 2.19358s (thread); 0s (gc) │ │ │ + -- used 2.20994s (cpu); 1.70119s (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 0.644433s (cpu); 0.456975s (thread); 0s (gc) │ │ │ + -- used 0.466228s (cpu); 0.344333s (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 0.0398049s (cpu); 0.039807s (thread); 0s (gc) │ │ │ + -- used 0.040897s (cpu); 0.0408965s (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 0.16271s (cpu); 0.0952426s (thread); 0s (gc) │ │ │ + -- used 0.141926s (cpu); 0.0872269s (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 0.117742s (cpu); 0.0528496s (thread); 0s (gc) │ │ │ + -- used 0.106118s (cpu); 0.0491561s (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 0.130131s (cpu); 0.0621371s (thread); 0s (gc) │ │ │ + -- used 0.114541s (cpu); 0.0521643s (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 0.190994s (cpu); 0.117998s (thread); 0s (gc) │ │ │ + -- used 0.163264s (cpu); 0.103452s (thread); 0s (gc) │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_cayley__Trick.html │ │ │ @@ -91,24 +91,24 @@ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2);
│ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc)
    │ │ │ + -- used 0.13619s (cpu); 0.0719734s (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 0.297065s (cpu); 0.151907s (thread); 0s (gc)
│ │ │ + -- used 0.196954s (cpu); 0.0883871s (thread); 0s (gc)
│ │ │  
│ │ │                                                                             
│ │ │  o4 = (ideal (x   x    - x   x   , x   x    - x   x   , x   x    - x   x   ,
│ │ │                0,3 1,2    0,2 1,3   1,0 1,1    1,2 1,3   0,3 1,1    0,1 1,3 
│ │ │       ------------------------------------------------------------------------
│ │ │                                                                              
│ │ │       x   x    - x   x   , x   x    - x   x   , x   x    - x   x   , x   x   
│ │ │ @@ -135,26 +135,26 @@
│ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -38,20 +38,20 @@ │ │ │ │ │ │ │ │ o2 = ideal(x x - x x ) │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Ideal of QQ[x ..x ] │ │ │ │ 0 3 │ │ │ │ i3 : time (P1xP1xP2,P1xP1xP2') = cayleyTrick(P1xP1,2); │ │ │ │ - -- used 0.187761s (cpu); 0.10659s (thread); 0s (gc) │ │ │ │ + -- used 0.13619s (cpu); 0.0719734s (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 0.297065s (cpu); 0.151907s (thread); 0s (gc) │ │ │ │ + -- used 0.196954s (cpu); 0.0883871s (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 0.288499s (cpu); 0.151777s (thread); 0s (gc) │ │ │ │ + -- used 0.21949s (cpu); 0.106864s (thread); 0s (gc) │ │ │ │ i6 : assert(oo == (P1xP1xP1,P1xP1xP1')) │ │ │ │ i7 : time cayleyTrick(P1xP1,2,Duality=>true); │ │ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc) │ │ │ │ + -- used 0.224647s (cpu); 0.117533s (thread); 0s (gc) │ │ │ │ i8 : assert(oo == (P1xP1xP2,P1xP1xP2')) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_u_a_l_V_a_r_i_e_t_y -- projective dual variety │ │ │ │ ********** WWaayyss ttoo uussee ccaayylleeyyTTrriicckk:: ********** │ │ │ │ * cayleyTrick(Ideal,ZZ) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_a_y_l_e_y_T_r_i_c_k is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Equations.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o2 : Ideal of P3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time cayleyTrick(P1xP1,1,Duality=>true);
│ │ │ - -- used 0.288499s (cpu); 0.151777s (thread); 0s (gc)
    │ │ │ + -- used 0.21949s (cpu); 0.106864s (thread); 0s (gc) │ │ │
    │ │ │ 
    i6 : assert(oo == (P1xP1xP1,P1xP1xP1'))
    │ │ │
    │ │ │ 
    i7 : time cayleyTrick(P1xP1,2,Duality=>true);
│ │ │ - -- used 0.340521s (cpu); 0.183633s (thread); 0s (gc)
    │ │ │ + -- used 0.224647s (cpu); 0.117533s (thread); 0s (gc) │ │ │
    │ │ │ 
    i8 : assert(oo == (P1xP1xP2,P1xP1xP2'))
    │ │ │
    │ │ │ 
    i3 : -- Chow equations of C
│ │ │       time eqsC = chowEquations chowForm C
│ │ │ - -- used 0.0407572s (cpu); 0.040758s (thread); 0s (gc)
│ │ │ + -- used 0.0423984s (cpu); 0.0423978s (thread); 0s (gc)
│ │ │  
│ │ │               2 2    2 2    2 2    4                2      2 2   2      
│ │ │  o3 = ideal (x x  + x x  + x x  + x , x x x x  + x x x  + x x , x x x  +
│ │ │               0 3    1 3    2 3    3   0 1 2 3    1 2 3    2 3   0 2 3  
│ │ │       ------------------------------------------------------------------------
│ │ │        2        3           2         2      3   3         2          2    2 2
│ │ │       x x x  + x x  - 2x x x  - 2x x x  - x x , x x  + 2x x x  - x x x  + x x 
│ │ │ @@ -167,15 +167,15 @@
│ │ │  o5 : Ideal of P3
    │ │ │
    │ │ │ 
    i6 : -- Chow equations of D
│ │ │       time eqsD = chowEquations chowForm D
│ │ │ - -- used 0.0401433s (cpu); 0.0401425s (thread); 0s (gc)
│ │ │ + -- used 0.0426096s (cpu); 0.0426104s (thread); 0s (gc)
│ │ │  
│ │ │               4      3 2     3        2 2     3      2   2   2 2      2   2 
│ │ │  o6 = ideal (x x  - x x , x x x  - x x x , x x x  - x x x , x x x  - x x x ,
│ │ │               2 3    1 3   1 2 3    0 1 3   0 2 3    0 1 3   1 2 3    0 1 3 
│ │ │       ------------------------------------------------------------------------
│ │ │            2      3 2   3        3 2   4         2         2 2       3    
│ │ │       x x x x  - x x , x x x  - x x , x x  - 4x x x x  + 3x x x , x x x  -
│ │ │ @@ -227,30 +227,30 @@
│ │ │  o9 : Ideal of P3
    │ │ │
    │ │ │ 
    i10 : -- tangential Chow forms of Q
│ │ │        time (W0,W1,W2) = (tangentialChowForm(Q,0),tangentialChowForm(Q,1),tangentialChowForm(Q,2))
│ │ │ - -- used 0.160178s (cpu); 0.106293s (thread); 0s (gc)
│ │ │ + -- used 0.169639s (cpu); 0.102352s (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 0.0594214s (cpu); 0.0594176s (thread); 0s (gc)
│ │ │ + -- used 0.0623613s (cpu); 0.0623675s (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 0.0407572s (cpu); 0.040758s (thread); 0s (gc) │ │ │ │ + -- used 0.0423984s (cpu); 0.0423978s (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 0.0401433s (cpu); 0.0401425s (thread); 0s (gc) │ │ │ │ + -- used 0.0426096s (cpu); 0.0426104s (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 0.160178s (cpu); 0.106293s (thread); 0s (gc) │ │ │ │ + -- used 0.169639s (cpu); 0.102352s (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 0.0594214s (cpu); 0.0594176s (thread); 0s (gc) │ │ │ │ + -- used 0.0623613s (cpu); 0.0623675s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 = true │ │ │ │ Note that chowEquations(W,0) is not the same as chowEquations W. │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwEEqquuaattiioonnss:: ********** │ │ │ │ * chowEquations(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_h_o_w_E_q_u_a_t_i_o_n_s is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 3331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- Chow form of V in Grass(2,5) (performing internal computations on an affine chart of the Grassmannian)
│ │ │       time ChowV = chowForm(V,AffineChartGrass=>{1,2,3})
│ │ │ - -- used 5.50368s (cpu); 4.92753s (thread); 0s (gc)
│ │ │ + -- used 4.54286s (cpu); 4.20868s (thread); 0s (gc)
│ │ │  
│ │ │        4               2              2     2               2            
│ │ │  o3 = x      + 2x     x     x      + x     x      - 2x     x     x      +
│ │ │        1,2,4     0,2,4 1,2,4 2,3,4    0,2,4 2,3,4     1,2,3 1,2,4 1,2,5  
│ │ │       ------------------------------------------------------------------------
│ │ │        2     2              2                                       
│ │ │       x     x      - x     x     x      + x     x     x     x      +
│ │ │ @@ -232,22 +232,22 @@
│ │ │         2,3,5 1,4,5    1,3,5 2,4,5    1,2,5 3,4,5   2,3,4 1,4,5    1,3,4 2,4,5    1,2,4 3,4,5   2,3,5 0,4,5    0,3,5 2,4,5    0,2,5 3,4,5   1,3,5 0,4,5    0,3,5 1,4,5    0,1,5 3,4,5   1,2,5 0,4,5    0,2,5 1,4,5    0,1,5 2,4,5   2,3,4 0,4,5    0,3,4 2,4,5    0,2,4 3,4,5   1,3,4 0,4,5    0,3,4 1,4,5    0,1,4 3,4,5   1,2,4 0,4,5    0,2,4 1,4,5    0,1,4 2,4,5   1,2,3 0,4,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 1,3,5    1,3,4 2,3,5    1,2,3 3,4,5   1,2,5 0,3,5    0,2,5 1,3,5    0,1,5 2,3,5   2,3,4 0,3,5    0,3,4 2,3,5    0,2,3 3,4,5   1,3,4 0,3,5    0,3,4 1,3,5    0,1,3 3,4,5   1,2,4 0,3,5    0,2,4 1,3,5    0,1,4 2,3,5    0,1,2 3,4,5   1,2,3 0,3,5    0,2,3 1,3,5    0,1,3 2,3,5   2,3,4 1,2,5    1,2,4 2,3,5    1,2,3 2,4,5   1,3,4 1,2,5    1,2,4 1,3,5    1,2,3 1,4,5   0,3,4 1,2,5    0,2,4 1,3,5    0,1,4 2,3,5    0,2,3 1,4,5    0,1,3 2,4,5    0,1,2 3,4,5   2,3,4 0,2,5    0,2,4 2,3,5    0,2,3 2,4,5   1,3,4 0,2,5    0,2,4 1,3,5    0,2,3 1,4,5    0,1,2 3,4,5   0,3,4 0,2,5    0,2,4 0,3,5    0,2,3 0,4,5   1,2,4 0,2,5    0,2,4 1,2,5    0,1,2 2,4,5   1,2,3 0,2,5    0,2,3 1,2,5    0,1,2 2,3,5   2,3,4 0,1,5    0,1,4 2,3,5    0,1,3 2,4,5    0,1,2 3,4,5   1,3,4 0,1,5    0,1,4 1,3,5    0,1,3 1,4,5   0,3,4 0,1,5    0,1,4 0,3,5    0,1,3 0,4,5   1,2,4 0,1,5    0,1,4 1,2,5    0,1,2 1,4,5   0,2,4 0,1,5    0,1,4 0,2,5    0,1,2 0,4,5   1,2,3 0,1,5    0,1,3 1,2,5    0,1,2 1,3,5   0,2,3 0,1,5    0,1,3 0,2,5    0,1,2 0,3,5   1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- equivalently (but faster)...
│ │ │       time assert(ChowV === chowForm f)
│ │ │ - -- used 1.1406s (cpu); 0.999558s (thread); 0s (gc)
    │ │ │ + -- used 0.973138s (cpu); 0.901726s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- X-resultant of V
│ │ │       time Xres = fromPluckerToStiefel dualize ChowV;
│ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc)
    │ │ │ + -- used 0.254184s (cpu); 0.193535s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : -- three generic ternary quadrics
│ │ │       F = genericPolynomials({2,2,2},ZZ/3331)
│ │ │  
│ │ │ @@ -262,15 +262,15 @@
│ │ │  o6 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : -- resultant of the three forms
│ │ │       time resF = resultant F;
│ │ │ - -- used 0.341473s (cpu); 0.215038s (thread); 0s (gc)
    │ │ │ + -- used 0.24341s (cpu); 0.182054s (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 5.50368s (cpu); 4.92753s (thread); 0s (gc) │ │ │ │ + -- used 4.54286s (cpu); 4.20868s (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 1.1406s (cpu); 0.999558s (thread); 0s (gc) │ │ │ │ + -- used 0.973138s (cpu); 0.901726s (thread); 0s (gc) │ │ │ │ i5 : -- X-resultant of V │ │ │ │ time Xres = fromPluckerToStiefel dualize ChowV; │ │ │ │ - -- used 0.325978s (cpu); 0.196045s (thread); 0s (gc) │ │ │ │ + -- used 0.254184s (cpu); 0.193535s (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 0.341473s (cpu); 0.215038s (thread); 0s (gc) │ │ │ │ + -- used 0.24341s (cpu); 0.182054s (thread); 0s (gc) │ │ │ │ i8 : assert(resF === sub(Xres,vars ring resF) and Xres === sub(resF,vars ring │ │ │ │ Xres)) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _h_u_r_w_i_t_z_F_o_r_m -- Hurwitz form of a projective variety │ │ │ │ ********** WWaayyss ttoo uussee cchhoowwFFoorrmm:: ********** │ │ │ │ * chowForm(Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_discriminant_lp__Ring__Element_rp.html │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ │ │ o2 : ZZ[a..c][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : time discriminant F
│ │ │ - -- used 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ + -- used 0.00962444s (cpu); 0.0096242s (thread); 0s (gc)
│ │ │  
│ │ │          2
│ │ │  o3 = - b  + 4a*c
│ │ │  
│ │ │  o3 : ZZ[a..c]
    │ │ │ │ │ │ │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o5 : ZZ[a..d][x..y] │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : time discriminant F
│ │ │ - -- used 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ + -- used 0.011164s (cpu); 0.0111655s (thread); 0s (gc)
│ │ │  
│ │ │          2 2       3     3                   2 2
│ │ │  o6 = - b c  + 4a*c  + 4b d - 18a*b*c*d + 27a d
│ │ │  
│ │ │  o6 : ZZ[a..d]
    │ │ │ │ │ │ │ │ │ @@ -170,15 +170,15 @@ │ │ │ │ │ │ o12 : R' │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time D=discriminant pencil
│ │ │ - -- used 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ + -- used 0.377897s (cpu); 0.377897s (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 0.00924086s (cpu); 0.00924015s (thread); 0s (gc)
│ │ │ │ + -- used 0.00962444s (cpu); 0.0096242s (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 0.0098778s (cpu); 0.0098795s (thread); 0s (gc)
│ │ │ │ + -- used 0.011164s (cpu); 0.0111655s (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 0.44332s (cpu); 0.443247s (thread); 0s (gc)
│ │ │ │ + -- used 0.377897s (cpu); 0.377897s (thread); 0s (gc)
│ │ │ │  
│ │ │ │             108      106 2       102 6      100 8       98 10       96 12
│ │ │ │  o13 = - 62t    + 19t   t  + 160t   t  + 91t   t  + 129t  t   + 117t  t   +
│ │ │ │             0        0   1       0   1      0   1       0  1        0  1
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │            94 14       92 16      90 18      88 20      86 22       84 24
│ │ │ │        161t  t   + 124t  t   - 82t  t   - 21t  t   - 49t  t   - 123t  t   +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_dual__Variety.html
│ │ │ @@ -95,28 +95,28 @@
│ │ │  o1 : Ideal of QQ[x ..x ]
│ │ │                    0   5
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time V' = dualVariety V
│ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc)
│ │ │ + -- used 0.172913s (cpu); 0.10843s (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 0.310587s (cpu); 0.159166s (thread); 0s (gc)
│ │ │ + -- used 0.281289s (cpu); 0.149719s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = true
    │ │ │ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ @@ -136,25 +136,25 @@ │ │ │ o4 : ----[a ..a ][x ..x ] │ │ │ 3331 0 9 0 2 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -31,24 +31,24 @@ │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ x x ) │ │ │ │ 0 3 │ │ │ │ │ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ │ 0 5 │ │ │ │ i2 : time V' = dualVariety V │ │ │ │ - -- used 0.202374s (cpu); 0.134305s (thread); 0s (gc) │ │ │ │ + -- used 0.172913s (cpu); 0.10843s (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 0.310587s (cpu); 0.159166s (thread); 0s (gc) │ │ │ │ + -- used 0.281289s (cpu); 0.149719s (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 0.162855s (cpu); 0.0858796s (thread); 0s (gc) │ │ │ │ + -- used 0.127578s (cpu); 0.0720894s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o5 : Ideal of ----[a ..a ] │ │ │ │ 3331 0 9 │ │ │ │ i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true); │ │ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc) │ │ │ │ + -- used 0.595438s (cpu); 0.52974s (thread); 0s (gc) │ │ │ │ │ │ │ │ ZZ │ │ │ │ o6 : Ideal of ----[x ..x ] │ │ │ │ 3331 0 9 │ │ │ │ i7 : discF == sub(Z,vars ring discF) and Z == sub(discF,vars ring Z) │ │ │ │ │ │ │ │ o7 = true │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_from__Plucker__To__Stiefel.html │ │ │ @@ -90,15 +90,15 @@ │ │ │ o1 : Ideal of QQ[x ..x ] │ │ │ 0 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : time discF = ideal discriminant F;
│ │ │ - -- used 0.162855s (cpu); 0.0858796s (thread); 0s (gc)
│ │ │ + -- used 0.127578s (cpu); 0.0720894s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o5 : Ideal of ----[a ..a ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i6 : time Z = dualVariety(veronese(2,3,ZZ/3331),AssumeOrdinary=>true);
│ │ │ - -- used 0.745903s (cpu); 0.673493s (thread); 0s (gc)
│ │ │ + -- used 0.595438s (cpu); 0.52974s (thread); 0s (gc)
│ │ │  
│ │ │                 ZZ
│ │ │  o6 : Ideal of ----[x ..x ]
│ │ │                3331  0   9
    │ │ │
    │ │ │ 
    i2 : time fromPluckerToStiefel dualize chowForm C
│ │ │ - -- used 0.141834s (cpu); 0.0752672s (thread); 0s (gc)
│ │ │ + -- used 0.119086s (cpu); 0.0633871s (thread); 0s (gc)
│ │ │  
│ │ │          3   3          2   2              2       2          2   3    
│ │ │  o2 = - x   x    + x   x   x   x    - x   x   x   x    + x   x   x    -
│ │ │          0,3 1,0    0,2 0,3 1,0 1,1    0,1 0,3 1,0 1,1    0,0 0,3 1,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        2       2               2   2                                   
│ │ │       x   x   x   x    + 2x   x   x   x    + x   x   x   x   x   x    -
│ │ │ @@ -143,15 +143,15 @@
│ │ │  o2 : QQ[x   ..x   ]
│ │ │           0,0   1,3
    │ │ │
    │ │ │ 
    i3 : time fromPluckerToStiefel(dualize chowForm C,AffineChartGrass=>{0,1})
│ │ │ - -- used 0.177092s (cpu); 0.069066s (thread); 0s (gc)
│ │ │ + -- used 0.193864s (cpu); 0.0664178s (thread); 0s (gc)
│ │ │  
│ │ │              3          2                         2                        
│ │ │  o3 = - x   x    + x   x   x    - x   x   x    + x   x    + 3x   x   x    -
│ │ │          0,3 1,2    0,2 1,2 1,3    0,2 0,3 1,2    0,2 1,3     0,3 1,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │             2      3      2
│ │ │       2x   x    + x    + x
│ │ │ @@ -184,15 +184,15 @@
│ │ │
    		│ │ │ 
    i5 : w = chowForm C;
    │ │ │
    │ │ │ 
    i6 : time U = apply(subsets(4,2),s->ideal fromPluckerToStiefel(w,AffineChartGrass=>s))
│ │ │ - -- used 0.108209s (cpu); 0.0419396s (thread); 0s (gc)
│ │ │ + -- used 0.102619s (cpu); 0.0350048s (thread); 0s (gc)
│ │ │  
│ │ │                     3          2          3                       2        
│ │ │  o6 = {ideal(- x   x    + x   x   x    - x    - 3x   x   x    + 2x   x    +
│ │ │                 0,3 1,2    0,2 1,2 1,3    0,2     0,2 0,3 1,2     0,2 1,3  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2      2            2   3               2        
│ │ │       x   x   x    - x   x    + x   ), ideal(x   x    - 2x   x   x   x    +
│ │ │ @@ -232,15 +232,15 @@
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │ 
    i7 : time apply(U,u->dim singularLocus u)
│ │ │ - -- used 0.115818s (cpu); 0.0462516s (thread); 0s (gc)
│ │ │ + -- used 0.106946s (cpu); 0.0407097s (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 0.141834s (cpu); 0.0752672s (thread); 0s (gc) │ │ │ │ + -- used 0.119086s (cpu); 0.0633871s (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 0.177092s (cpu); 0.069066s (thread); 0s (gc) │ │ │ │ + -- used 0.193864s (cpu); 0.0664178s (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 0.108209s (cpu); 0.0419396s (thread); 0s (gc) │ │ │ │ + -- used 0.102619s (cpu); 0.0350048s (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 0.115818s (cpu); 0.0462516s (thread); 0s (gc) │ │ │ │ + -- used 0.106946s (cpu); 0.0407097s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = {2, 2, 2, 2, 2, 2} │ │ │ │ │ │ │ │ o7 : List │ │ │ │ ********** WWaayyss ttoo uussee ffrroommPPlluucckkeerrTTooSSttiieeffeell:: ********** │ │ │ │ * fromPluckerToStiefel(Ideal) │ │ │ │ * fromPluckerToStiefel(Matrix) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_hurwitz__Form.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ o1 : Ideal of QQ[p ..p ] │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time hurwitzForm Q
│ │ │ - -- used 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ + -- used 0.111462s (cpu); 0.0527121s (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 0.125887s (cpu); 0.0563778s (thread); 0s (gc)
│ │ │ │ + -- used 0.111462s (cpu); 0.0527121s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2                                 2
│ │ │ │  o2 = 11966535p    + 14645610p   p    + 11354175p    + 1666980p   p    +
│ │ │ │                0,1            0,1 0,2            0,2           0,1 1,2
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │                                 2
│ │ │ │       4456620p   p    + 1127196p    + 54176850p   p    + 20326950p   p    +
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__Coisotropic.html
│ │ │ @@ -109,15 +109,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time isCoisotropic w
│ │ │ - -- used 0.00877231s (cpu); 0.00877035s (thread); 0s (gc)
│ │ │ + -- used 0.00846686s (cpu); 0.00846591s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = true
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- random quadric in G(1,3)
│ │ │ @@ -145,15 +145,15 @@
│ │ │           p   p    - p   p    + p   p
│ │ │            1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isCoisotropic w'
│ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc)
│ │ │ + -- used 0.00736545s (cpu); 0.00736643s (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 0.00877231s (cpu); 0.00877035s (thread); 0s (gc) │ │ │ │ + -- used 0.00846686s (cpu); 0.00846591s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : -- random quadric in G(1,3) │ │ │ │ w' = random(2,Grass(1,3)) │ │ │ │ │ │ │ │ 3 2 3 7 2 5 7 2 10 │ │ │ │ o3 = --p + -p p + -p + 5p p + -p p + -p + --p p │ │ │ │ @@ -72,15 +72,15 @@ │ │ │ │ │ │ │ │ QQ[p ..p , p , p , p , p ] │ │ │ │ 0,1 0,2 1,2 0,3 1,3 2,3 │ │ │ │ o3 : -------------------------------------- │ │ │ │ p p - p p + p p │ │ │ │ 1,2 0,3 0,2 1,3 0,1 2,3 │ │ │ │ i4 : time isCoisotropic w' │ │ │ │ - -- used 0.00769242s (cpu); 0.00769265s (thread); 0s (gc) │ │ │ │ + -- used 0.00736545s (cpu); 0.00736643s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = false │ │ │ │ ********** WWaayyss ttoo uussee iissCCooiissoottrrooppiicc:: ********** │ │ │ │ * isCoisotropic(RingElement) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_i_s_o_t_r_o_p_i_c is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_is__In__Coisotropic.html │ │ │ @@ -122,15 +122,15 @@ │ │ │ o3 : Ideal of -----[x ..x ] │ │ │ 33331 0 5 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time isInCoisotropic(L,I) -- whether L belongs to Z_1(V(I))
│ │ │ - -- used 0.0181849s (cpu); 0.0181858s (thread); 0s (gc)
│ │ │ + -- used 0.0171607s (cpu); 0.017161s (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 0.0181849s (cpu); 0.0181858s (thread); 0s (gc) │ │ │ │ + -- used 0.0171607s (cpu); 0.017161s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _t_a_n_g_e_n_t_i_a_l_C_h_o_w_F_o_r_m -- higher Chow forms of a projective variety │ │ │ │ * _p_l_u_c_k_e_r -- get the Plücker coordinates of a linear subspace │ │ │ │ ********** WWaayyss ttoo uussee iissIInnCCooiissoottrrooppiicc:: ********** │ │ │ │ * isInCoisotropic(Ideal,Ideal) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_macaulay__Formula.html │ │ │ @@ -92,15 +92,15 @@ │ │ │ │ │ │ o1 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ + -- used 0.00367841s (cpu); 0.00367712s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = (| a_0 a_1 a_2 a_3 a_4 a_5 0   0   0   0   0   0   0   0   0   0   0  
│ │ │        | 0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0   0  
│ │ │        | 0   0   a_0 0   a_1 a_2 0   a_3 a_4 a_5 0   0   0   0   0   0   0  
│ │ │        | 0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0   0  
│ │ │        | 0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0   0  
│ │ │        | 0   0   0   0   0   a_0 0   0   a_1 a_2 0   0   a_3 a_4 a_5 0   0  
│ │ │ @@ -163,15 +163,15 @@
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time (D,D') = macaulayFormula F
│ │ │ - -- used 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ + -- used 0.00225783s (cpu); 0.00225483s (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 0.00373056s (cpu); 0.00372864s (thread); 0s (gc)
│ │ │ │ + -- used 0.00367841s (cpu); 0.00367712s (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 0.00252047s (cpu); 0.00251818s (thread); 0s (gc)
│ │ │ │ + -- used 0.00225783s (cpu); 0.00225483s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = (| 9/2 9/4 3/4 7/4  7/9  7/10 0    0    0    0    0   0   0    0    0
│ │ │ │        | 0   9/2 0   9/4  3/4  0    7/4  7/9  7/10 0    0   0   0    0    0
│ │ │ │        | 0   0   9/2 0    9/4  3/4  0    7/4  7/9  7/10 0   0   0    0    0
│ │ │ │        | 0   0   0   9/2  0    0    9/4  3/4  0    0    7/4 7/9 7/10 0    0
│ │ │ │        | 0   0   0   0    9/2  0    0    9/4  3/4  0    0   7/4 7/9  7/10 0
│ │ │ │        | 0   0   0   0    0    9/2  0    0    9/4  3/4  0   0   7/4  7/9  7/10
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_plucker.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o3 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time p = plucker L
│ │ │ - -- used 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ + -- used 0.00499304s (cpu); 0.00499201s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = ideal (x    + 8480x   , x    - 6727x   , x    + 15777x   , x    +
│ │ │               2,4        3,4   1,4        3,4   0,4         3,4   2,3  
│ │ │       ------------------------------------------------------------------------
│ │ │       11656x   , x    - 14853x   , x    + 664x   , x    + 13522x   , x    +
│ │ │             3,4   1,3         3,4   0,3       3,4   1,2         3,4   0,2  
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o4 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : time L' = plucker p
│ │ │ - -- used 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ + -- used 0.106833s (cpu); 0.0480662s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = ideal (x  + 8480x  - 11656x , x  - 6727x  + 14853x , x  + 15777x  -
│ │ │               2        3         4   1        3         4   0         3  
│ │ │       ------------------------------------------------------------------------
│ │ │       664x )
│ │ │           4
│ │ │  
│ │ │ @@ -143,15 +143,15 @@
│ │ │  
│ │ │  o7 : Ideal of G'1'4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : time W = plucker Y; -- surface swept out by the lines of Y
│ │ │ - -- used 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ + -- used 0.105147s (cpu); 0.0468495s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of P4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : (codim W,degree W)
│ │ │ @@ -163,15 +163,15 @@
│ │ │            
│ │ │          
│ │ │          
    In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i10 : time Y' = plucker(W,1); -- variety of lines contained in W
│ │ │ - -- used 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ + -- used 0.141418s (cpu); 0.141421s (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 0.00473681s (cpu); 0.00473472s (thread); 0s (gc)
│ │ │ │ + -- used 0.00499304s (cpu); 0.00499201s (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 0.121088s (cpu); 0.0718194s (thread); 0s (gc)
│ │ │ │ + -- used 0.106833s (cpu); 0.0480662s (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 0.148823s (cpu); 0.0922017s (thread); 0s (gc)
│ │ │ │ + -- used 0.105147s (cpu); 0.0468495s (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 0.244058s (cpu); 0.244015s (thread); 0s (gc)
│ │ │ │ + -- used 0.141418s (cpu); 0.141421s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o10 : Ideal of G'1'4
│ │ │ │  i11 : assert(Y' == Y)
│ │ │ │  WWaarrnniinngg: Notice that, by default, the computation is done on a randomly chosen
│ │ │ │  affine chart on the Grassmannian. To change this behavior, you can use the
│ │ │ │  _A_f_f_i_n_e_C_h_a_r_t_G_r_a_s_s option.
│ │ │ │  ********** WWaayyss ttoo uussee pplluucckkeerr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp..._cm__Algorithm_eq_gt..._rp.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │  
│ │ │  o2 : List
│ │ │              
    
│ │ │                
    i3 : time resultant(F,Algorithm=>"Poisson2")
│ │ │ - -- used 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ + -- used 0.498087s (cpu); 0.257168s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o3 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -137,15 +137,15 @@
│ │ │  
│ │ │  o3 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i4 : time resultant(F,Algorithm=>"Macaulay2")
│ │ │ - -- used 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ + -- used 0.224784s (cpu); 0.102755s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o4 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -161,15 +161,15 @@
│ │ │  
│ │ │  o4 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i5 : time resultant(F,Algorithm=>"Poisson")
│ │ │ - -- used 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ + -- used 0.289087s (cpu); 0.289088s (thread); 0s (gc)
│ │ │  
│ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4   
│ │ │  o5 = - -----------------------------a  - -------------------------------a b -
│ │ │             2222549728809984000000            12700284164628480000000         
│ │ │       ------------------------------------------------------------------------
│ │ │       348237304382147063838108483692249 3 2  
│ │ │       ---------------------------------a b  -
│ │ │ @@ -185,15 +185,15 @@
│ │ │  
│ │ │  o5 : QQ[a..b]
    
│ │ │              
    
│ │ │                
    i6 : time resultant(F,Algorithm=>"Macaulay")
│ │ │ - -- used 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ + -- used 0.567787s (cpu); 0.513751s (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 0.562085s (cpu); 0.289584s (thread); 0s (gc)
│ │ │ │ + -- used 0.498087s (cpu); 0.257168s (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 0.273183s (cpu); 0.131953s (thread); 0s (gc)
│ │ │ │ + -- used 0.224784s (cpu); 0.102755s (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 0.302008s (cpu); 0.301983s (thread); 0s (gc)
│ │ │ │ + -- used 0.289087s (cpu); 0.289088s (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 0.625765s (cpu); 0.559323s (thread); 0s (gc)
│ │ │ │ + -- used 0.567787s (cpu); 0.513751s (thread); 0s (gc)
│ │ │ │  
│ │ │ │         21002161660529014459938925799 5   2085933800619238998825958079203 4
│ │ │ │  o6 = - -----------------------------a  - -------------------------------a b -
│ │ │ │             2222549728809984000000            12700284164628480000000
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │       348237304382147063838108483692249 3 2
│ │ │ │       ---------------------------------a b  -
│ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_resultant_lp__Matrix_rp.html
│ │ │ @@ -97,15 +97,15 @@
│ │ │  
│ │ │  o2 : List
    
│ │ │              
    
│ │ │                
    i3 : time resultant F
│ │ │ - -- used 0.123723s (cpu); 0.0492639s (thread); 0s (gc)
│ │ │ + -- used 0.0910604s (cpu); 0.0366713s (thread); 0s (gc)
│ │ │  
│ │ │            12         11 2         10 3         9 4          8 5          7 6
│ │ │  o3 = - 81t  u - 1701t  u  - 15309t  u  - 76545t u  - 229635t u  - 413343t u 
│ │ │       ------------------------------------------------------------------------
│ │ │                6 7          5 8       11          10 2         9 3  
│ │ │       - 413343t u  - 177147t u  + 567t  u + 10206t  u  + 76545t u  +
│ │ │       ------------------------------------------------------------------------
│ │ │ @@ -160,15 +160,15 @@
│ │ │  
│ │ │  o4 : List
    
│ │ │              
    
│ │ │                
    i5 : time resultant F
│ │ │ - -- used 3.07394s (cpu); 2.19358s (thread); 0s (gc)
│ │ │ + -- used 2.20994s (cpu); 1.70119s (thread); 0s (gc)
│ │ │  
│ │ │        6 3 2       5 2   2     2 4   2 2    3 3 3 2     2 4 2   2  
│ │ │  o5 = a b c  - 3a a b b c  + 3a a b b c  - a a b c  + 3a a b b c  -
│ │ │        2 3 0     1 2 3 4 0     1 2 3 4 0    1 2 4 0     1 2 3 5 0  
│ │ │       ------------------------------------------------------------------------
│ │ │         3 3       2     4 2 2   2     4 2   2 2     5     2 2    6 3 2  
│ │ │       6a a b b b c  + 3a a b b c  + 3a a b b c  - 3a a b b c  + a b c  -
│ │ │ @@ -1795,15 +1795,15 @@
│ │ │  
│ │ │  o6 : List
    
│ │ │              
    
│ │ │                
    i7 : time # terms resultant F
│ │ │ - -- used 0.644433s (cpu); 0.456975s (thread); 0s (gc)
│ │ │ + -- used 0.466228s (cpu); 0.344333s (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 0.123723s (cpu); 0.0492639s (thread); 0s (gc) │ │ │ │ + -- used 0.0910604s (cpu); 0.0366713s (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 3.07394s (cpu); 2.19358s (thread); 0s (gc) │ │ │ │ + -- used 2.20994s (cpu); 1.70119s (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 0.644433s (cpu); 0.456975s (thread); 0s (gc) │ │ │ │ + -- used 0.466228s (cpu); 0.344333s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 = 21894 │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _c_h_o_w_F_o_r_m -- Chow form of a projective variety │ │ │ │ * _d_i_s_c_r_i_m_i_n_a_n_t_(_R_i_n_g_E_l_e_m_e_n_t_) │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * resultant(List) │ │ ├── ./usr/share/doc/Macaulay2/Resultants/html/_tangential__Chow__Form.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ 0 4 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : -- 0-th associated hypersurface of S in G(1,4) (Chow form)
│ │ │       time tangentialChowForm(S,0)
│ │ │ - -- used 0.0398049s (cpu); 0.039807s (thread); 0s (gc)
│ │ │ + -- used 0.040897s (cpu); 0.0408965s (thread); 0s (gc)
│ │ │  
│ │ │        2                                                       2        
│ │ │  o3 = p   p    - p   p   p    - p   p   p    + p   p   p    + p   p    +
│ │ │        1,3 2,3    1,2 1,3 2,4    0,3 1,3 2,4    0,2 1,4 2,4    1,2 3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │        2
│ │ │       p   p    - 2p   p   p    - p   p   p
│ │ │ @@ -123,15 +123,15 @@
│ │ │         2,3 1,4    1,3 2,4    1,2 3,4   2,3 0,4    0,3 2,4    0,2 3,4   1,3 0,4    0,3 1,4    0,1 3,4   1,2 0,4    0,2 1,4    0,1 2,4   1,2 0,3    0,2 1,3    0,1 2,3
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : -- 1-th associated hypersurface of S in G(2,4)
│ │ │       time tangentialChowForm(S,1)
│ │ │ - -- used 0.16271s (cpu); 0.0952426s (thread); 0s (gc)
│ │ │ + -- used 0.141926s (cpu); 0.0872269s (thread); 0s (gc)
│ │ │  
│ │ │        2     2        2     2               3        2     2      
│ │ │  o4 = p     p      + p     p      - 2p     p      + p     p      -
│ │ │        1,2,3 1,2,4    0,2,4 1,2,4     0,2,3 1,2,4    0,2,4 0,3,4  
│ │ │       ------------------------------------------------------------------------
│ │ │               3         3               3            
│ │ │       4p     p      - 4p     p      - 2p     p      +
│ │ │ @@ -168,43 +168,43 @@
│ │ │         1,2,4 0,3,4    0,2,4 1,3,4    0,1,4 2,3,4   1,2,3 0,3,4    0,2,3 1,3,4    0,1,3 2,3,4   1,2,3 0,2,4    0,2,3 1,2,4    0,1,2 2,3,4   1,2,3 0,1,4    0,1,3 1,2,4    0,1,2 1,3,4   0,2,3 0,1,4    0,1,3 0,2,4    0,1,2 0,3,4
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : -- 2-th associated hypersurface of S in G(3,4) (parameterizing tangent hyperplanes to S)
│ │ │       time tangentialChowForm(S,2)
│ │ │ - -- used 0.117742s (cpu); 0.0528496s (thread); 0s (gc)
│ │ │ + -- used 0.106118s (cpu); 0.0491561s (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 0.130131s (cpu); 0.0621371s (thread); 0s (gc)
│ │ │ + -- used 0.114541s (cpu); 0.0521643s (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 0.190994s (cpu); 0.117998s (thread); 0s (gc)
    │ │ │ + -- used 0.163264s (cpu); 0.103452s (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 0.0398049s (cpu); 0.039807s (thread); 0s (gc) │ │ │ │ + -- used 0.040897s (cpu); 0.0408965s (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 0.16271s (cpu); 0.0952426s (thread); 0s (gc) │ │ │ │ + -- used 0.141926s (cpu); 0.0872269s (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 0.117742s (cpu); 0.0528496s (thread); 0s (gc) │ │ │ │ + -- used 0.106118s (cpu); 0.0491561s (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 0.130131s (cpu); 0.0621371s (thread); 0s (gc) │ │ │ │ + -- used 0.114541s (cpu); 0.0521643s (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 0.190994s (cpu); 0.117998s (thread); 0s (gc) │ │ │ │ + -- used 0.163264s (cpu); 0.103452s (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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ aXNFeHRlcm5hbE0yUGFyZW50 │ │ │ #:len=1043 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiaW5kaWNhdGUgaWYgdGhpcyBwcm9jZXNz │ │ │ IGlzIGEgcGFyZW50IHByb2Nlc3Mgb3Igbm90IiwgImxpbmVudW0iID0+IDg5NCwgImZpbGVuYW1l │ │ ├── ./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 63815 │ │ │ +process 63520 │ │ │ nofiles 512 │ │ │ vmemory(kbytes) unlimited │ │ │ locks unlimited │ │ │ rtprio 0 │ │ │ │ │ │ o1 = 0 │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/example-output/_run__External__M2.out │ │ │ @@ -1,129 +1,129 @@ │ │ │ -- -*- M2-comint -*- hash: 2927978066455787395 │ │ │ │ │ │ i1 : fn=temporaryFileName()|".m2" │ │ │ │ │ │ -o1 = /tmp/M2-24375-0/0.m2 │ │ │ +o1 = /tmp/M2-31178-0/0.m2 │ │ │ │ │ │ i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24375-0/1.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/1.m2" >"/tmp/M2-31178-0/1.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i7 : h │ │ │ │ │ │ o7 = HashTable{"answer file" => null} │ │ │ "exit code" => 0 │ │ │ "output file" => null │ │ │ "return code" => 0 │ │ │ "statistics" => null │ │ │ - "time used" => 5 │ │ │ + "time used" => 3 │ │ │ value => 16 │ │ │ │ │ │ o7 : HashTable │ │ │ │ │ │ i8 : h#value===4^2 │ │ │ │ │ │ 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/M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/2.m2" >"/tmp/M2-31178-0/2.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i11 : h │ │ │ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans} │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31178-0/2.ans} │ │ │ "exit code" => 27 │ │ │ - "output file" => /tmp/M2-24375-0/2.out │ │ │ + "output file" => /tmp/M2-31178-0/2.out │ │ │ "return code" => 6912 │ │ │ "statistics" => null │ │ │ - "time used" => 5 │ │ │ + "time used" => 3 │ │ │ value => null │ │ │ │ │ │ o11 : HashTable │ │ │ │ │ │ i12 : fileExists(h#"output file") │ │ │ │ │ │ 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/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 )) │ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/3.m2" >"/tmp/M2-31178-0/3.out" 2>&1 )) │ │ │ Killed │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ i15 : h │ │ │ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans} │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31178-0/3.ans} │ │ │ "exit code" => 0 │ │ │ - "output file" => /tmp/M2-24375-0/3.out │ │ │ + "output file" => /tmp/M2-31178-0/3.out │ │ │ "return code" => 9 │ │ │ "statistics" => null │ │ │ - "time used" => 2 │ │ │ + "time used" => 1 │ │ │ value => null │ │ │ │ │ │ o15 : HashTable │ │ │ │ │ │ i16 : if h#"output file" =!= null and fileExists(h#"output file") then get(h#"output file") │ │ │ │ │ │ o16 = │ │ │ │ │ │ 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/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1') >"/tmp/M2-24375-0/4.stat" 2>&1 )) │ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/4.m2" >"/tmp/M2-31178-0/4.out" 2>&1') >"/tmp/M2-31178-0/4.stat" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i19 : h#"statistics" │ │ │ │ │ │ -o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1" │ │ │ - User time (seconds): 7.49 │ │ │ - System time (seconds): 0.10 │ │ │ - Percent of CPU this job got: 93% │ │ │ - Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15 │ │ │ +o19 = Command being timed: "sh -c /usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/4.m2" >"/tmp/M2-31178-0/4.out" 2>&1" │ │ │ + User time (seconds): 6.93 │ │ │ + System time (seconds): 0.20 │ │ │ + Percent of CPU this job got: 120% │ │ │ + Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.93 │ │ │ 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): 255852 │ │ │ + Maximum resident set size (kbytes): 340228 │ │ │ Average resident set size (kbytes): 0 │ │ │ Major (requiring I/O) page faults: 0 │ │ │ - Minor (reclaiming a frame) page faults: 10158 │ │ │ - Voluntary context switches: 2973 │ │ │ - Involuntary context switches: 2382 │ │ │ + Minor (reclaiming a frame) page faults: 12075 │ │ │ + Voluntary context switches: 19121 │ │ │ + Involuntary context switches: 3275 │ │ │ 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/M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/6.m2" >"/tmp/M2-31178-0/6.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ o21 = true │ │ │ │ │ │ i22 : R=QQ[x,y]; │ │ │ │ │ │ i23 : v=coker random(R^2,R^{3:-1}) │ │ │ @@ -131,54 +131,54 @@ │ │ │ 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/M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/7.m2" >"/tmp/M2-31178-0/7.out" 2>&1 )) │ │ │ Finished running. │ │ │ RunExternalM2: expected answer file does not exist │ │ │ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans} │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31178-0/7.ans} │ │ │ "exit code" => 1 │ │ │ - "output file" => /tmp/M2-24375-0/7.out │ │ │ + "output file" => /tmp/M2-31178-0/7.out │ │ │ "return code" => 256 │ │ │ "statistics" => null │ │ │ - "time used" => 4 │ │ │ + "time used" => 2 │ │ │ value => null │ │ │ │ │ │ o24 : HashTable │ │ │ │ │ │ i25 : get(h#"output file") │ │ │ │ │ │ o25 = │ │ │ - i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by RunExternalM2 │ │ │ + i1 : -- Script /tmp/M2-31178-0/7.m2 automatically generated by RunExternalM2 │ │ │ needsPackage("RunExternalM2",Configuration=>{"isChild"=>true}); │ │ │ │ │ │ - i2 : load "/tmp/M2-24375-0/0.m2"; │ │ │ + i2 : load "/tmp/M2-31178-0/0.m2"; │ │ │ │ │ │ - i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.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/M2-31178-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}})))); │ │ │ stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects: │ │ │ R (of class Symbol) │ │ │ ^ 2 (of class ZZ) │ │ │ │ │ │ │ │ │ i26 : fn<"/tmp/M2-24375-0/8.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/8.m2" >"/tmp/M2-31178-0/8.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/M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 )) │ │ │ +Running (true && (/usr/bin/M2-binary --stop --no-debug --silent -q <"/tmp/M2-31178-0/9.m2" >"/tmp/M2-31178-0/9.out" 2>&1 )) │ │ │ Finished running. │ │ │ │ │ │ i30 : h#value │ │ │ │ │ │ o30 = QQ[x..y] │ │ │ │ │ │ o30 : PolynomialRing │ │ ├── ./usr/share/doc/Macaulay2/RunExternalM2/html/_resource_splimits.html │ │ │ @@ -80,15 +80,15 @@ │ │ │ time(seconds) 700 │ │ │ file(blocks) unlimited │ │ │ data(kbytes) unlimited │ │ │ stack(kbytes) 8192 │ │ │ coredump(blocks) unlimited │ │ │ memory(kbytes) 850000 │ │ │ locked memory(kbytes) 8192 │ │ │ -process 63815 │ │ │ +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 63815 │ │ │ │ +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 │ │ │ @@ -89,15 +89,15 @@ │ │ │

      For example, we can write a few functions to a temporary file:

      │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -120,28 +120,28 @@ │ │ │
    │ │ │

    and then call them:

    │ │ │
    │ │ │
    │ │ │ 
    i1 : fn=temporaryFileName()|".m2"
│ │ │  
│ │ │ -o1 = /tmp/M2-24375-0/0.m2
    │ │ │ +o1 = /tmp/M2-31178-0/0.m2 │ │ │
    │ │ │ 
    i2 : fn<</// square = (x) -> (stderr<<"Running"<<endl; sleep(1); x^2); ///<<endl;
    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : h=runExternalM2(fn,"square",(4));
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/1.m2" >"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/1.m2" >"/tmp/M2-31178-0/1.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i7 : h
│ │ │  
│ │ │  o7 = HashTable{"answer file" => null}
│ │ │                 "exit code" => 0
│ │ │                 "output file" => null
│ │ │                 "return code" => 0
│ │ │                 "statistics" => null
│ │ │ -               "time used" => 5
│ │ │ +               "time used" => 3
│ │ │                 value => 16
│ │ │  
│ │ │  o7 : HashTable
    │ │ │
    │ │ │ @@ -162,29 +162,29 @@ │ │ │

    │ │ │

    An abnormal program exit will have a nonzero exit code; also, the value will be null, the output file should exist, but the answer file may not exist unless the routine finished successfully.

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i10 : h=runExternalM2(fn,"justexit",());
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/2.m2" >"/tmp/M2-31178-0/2.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    │ │ │
    │ │ │ 
    i11 : h
│ │ │  
│ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31178-0/2.ans}
│ │ │                  "exit code" => 27
│ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ +                "output file" => /tmp/M2-31178-0/2.out
│ │ │                  "return code" => 6912
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 5
│ │ │ +                "time used" => 3
│ │ │                  value => null
│ │ │  
│ │ │  o11 : HashTable
    │ │ │
    │ │ │ @@ -204,30 +204,30 @@ │ │ │
    │ │ │

    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:

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ -Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ +Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/3.m2" >"/tmp/M2-31178-0/3.out" 2>&1 ))
│ │ │  Killed
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
    │ │ │
    │ │ │ 
    i15 : h
│ │ │  
│ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31178-0/3.ans}
│ │ │                  "exit code" => 0
│ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ +                "output file" => /tmp/M2-31178-0/3.out
│ │ │                  "return code" => 9
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 2
│ │ │ +                "time used" => 1
│ │ │                  value => null
│ │ │  
│ │ │  o15 : HashTable
    │ │ │
    │ │ │ @@ -246,40 +246,40 @@ │ │ │

    │ │ │

    We can get quite a lot of detail on the resources used with the KeepStatistics command:

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -293,15 +293,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ -Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1') >"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ +Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/4.m2" >"/tmp/M2-31178-0/4.out" 2>&1') >"/tmp/M2-31178-0/4.stat" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i19 : h#"statistics"
│ │ │  
│ │ │ -o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ -              User time (seconds): 7.49
│ │ │ -              System time (seconds): 0.10
│ │ │ -              Percent of CPU this job got: 93%
│ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ +o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/4.m2" >"/tmp/M2-31178-0/4.out" 2>&1"
│ │ │ +              User time (seconds): 6.93
│ │ │ +              System time (seconds): 0.20
│ │ │ +              Percent of CPU this job got: 120%
│ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.93
│ │ │                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): 255852
│ │ │ +              Maximum resident set size (kbytes): 340228
│ │ │                Average resident set size (kbytes): 0
│ │ │                Major (requiring I/O) page faults: 0
│ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ -              Voluntary context switches: 2973
│ │ │ -              Involuntary context switches: 2382
│ │ │ +              Minor (reclaiming a frame) page faults: 12075
│ │ │ +              Voluntary context switches: 19121
│ │ │ +              Involuntary context switches: 3275
│ │ │                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/M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/6.m2" >"/tmp/M2-31178-0/6.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o21 = true
    │ │ │
    │ │ │
    │ │ │ @@ -323,24 +323,24 @@ │ │ │ 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/M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/7.m2" >"/tmp/M2-31178-0/7.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  RunExternalM2: expected answer file does not exist
│ │ │  
│ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31178-0/7.ans}
│ │ │                  "exit code" => 1
│ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ +                "output file" => /tmp/M2-31178-0/7.out
│ │ │                  "return code" => 256
│ │ │                  "statistics" => null
│ │ │ -                "time used" => 4
│ │ │ +                "time used" => 2
│ │ │                  value => null
│ │ │  
│ │ │  o24 : HashTable
    │ │ │
    │ │ │
    │ │ │ @@ -348,20 +348,20 @@ │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i25 : get(h#"output file")
│ │ │  
│ │ │  o25 = 
│ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by RunExternalM2
│ │ │ +      i1 : -- Script /tmp/M2-31178-0/7.m2 automatically generated by RunExternalM2
│ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │  
│ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ +      i2 : load "/tmp/M2-31178-0/0.m2";
│ │ │  
│ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.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/M2-31178-0/7.ans",identity (cokernel(map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to objects:
│ │ │                    R (of class Symbol)
│ │ │              ^     2 (of class ZZ)
    │ │ │
    │ │ │
    │ │ │ @@ -372,15 +372,15 @@ │ │ │
    		│ │ │ 
    i26 : fn<<///R=QQ[x,y];///<<endl<<flush;
    │ │ │
    │ │ │ 
    i27 : (runExternalM2(fn,identity,v))#value===v
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/8.m2" >"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/8.m2" >"/tmp/M2-31178-0/8.out" 2>&1 ))
│ │ │  Finished running.
│ │ │  
│ │ │  o27 = true
    │ │ │
    │ │ │
    │ │ │ @@ -392,15 +392,15 @@ │ │ │
    		│ │ │ 
    i28 : v=R;
    │ │ │
    │ │ │ 
    i29 : h=runExternalM2(fn,identity,v);
│ │ │ -Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ +Running (true && (/usr/bin/M2-binary  --stop --no-debug --silent  -q  <"/tmp/M2-31178-0/9.m2" >"/tmp/M2-31178-0/9.out" 2>&1 ))
│ │ │  Finished running.
    │ │ │
    │ │ │ 
    i30 : h#value
│ │ │ ├── html2text {}
│ │ │ │ @@ -45,193 +45,193 @@
│ │ │ │  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/M2-24375-0/0.m2
│ │ │ │ +o1 = /tmp/M2-31178-0/0.m2
│ │ │ │  i2 : fn< (stderr<<"Running"< ( exit(27); ); ///< (stderr<<"Spinning!!"<"/tmp/M2-24375-0/1.out" 2>&1 ))
│ │ │ │ +M2-31178-0/1.m2" >"/tmp/M2-31178-0/1.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i7 : h
│ │ │ │  
│ │ │ │  o7 = HashTable{"answer file" => null}
│ │ │ │                 "exit code" => 0
│ │ │ │                 "output file" => null
│ │ │ │                 "return code" => 0
│ │ │ │                 "statistics" => null
│ │ │ │ -               "time used" => 5
│ │ │ │ +               "time used" => 3
│ │ │ │                 value => 16
│ │ │ │  
│ │ │ │  o7 : HashTable
│ │ │ │  i8 : h#value===4^2
│ │ │ │  
│ │ │ │  o8 = true
│ │ │ │  i9 : h#"exit code"===0
│ │ │ │  
│ │ │ │  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/
│ │ │ │ -M2-24375-0/2.m2" >"/tmp/M2-24375-0/2.out" 2>&1 ))
│ │ │ │ +M2-31178-0/2.m2" >"/tmp/M2-31178-0/2.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i11 : h
│ │ │ │  
│ │ │ │ -o11 = HashTable{"answer file" => /tmp/M2-24375-0/2.ans}
│ │ │ │ +o11 = HashTable{"answer file" => /tmp/M2-31178-0/2.ans}
│ │ │ │                  "exit code" => 27
│ │ │ │ -                "output file" => /tmp/M2-24375-0/2.out
│ │ │ │ +                "output file" => /tmp/M2-31178-0/2.out
│ │ │ │                  "return code" => 6912
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 5
│ │ │ │ +                "time used" => 3
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o11 : HashTable
│ │ │ │  i12 : fileExists(h#"output file")
│ │ │ │  
│ │ │ │  o12 = true
│ │ │ │  i13 : fileExists(h#"answer file")
│ │ │ │  
│ │ │ │  o13 = false
│ │ │ │  Here, we use _r_e_s_o_u_r_c_e_ _l_i_m_i_t_s to limit the routine to 2 seconds of computational
│ │ │ │  time, while the system is asked to use 10 seconds of computational time:
│ │ │ │  i14 : h=runExternalM2(fn,"spin",10,PreRunScript=>"ulimit -t 2");
│ │ │ │  Running (ulimit -t 2 && (/usr/bin/M2-binary  --stop --no-debug --silent  -
│ │ │ │ -q  <"/tmp/M2-24375-0/3.m2" >"/tmp/M2-24375-0/3.out" 2>&1 ))
│ │ │ │ +q  <"/tmp/M2-31178-0/3.m2" >"/tmp/M2-31178-0/3.out" 2>&1 ))
│ │ │ │  Killed
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  i15 : h
│ │ │ │  
│ │ │ │ -o15 = HashTable{"answer file" => /tmp/M2-24375-0/3.ans}
│ │ │ │ +o15 = HashTable{"answer file" => /tmp/M2-31178-0/3.ans}
│ │ │ │                  "exit code" => 0
│ │ │ │ -                "output file" => /tmp/M2-24375-0/3.out
│ │ │ │ +                "output file" => /tmp/M2-31178-0/3.out
│ │ │ │                  "return code" => 9
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 2
│ │ │ │ +                "time used" => 1
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o15 : HashTable
│ │ │ │  i16 : if h#"output file" =!= null and fileExists(h#"output file") then get
│ │ │ │  (h#"output file")
│ │ │ │  
│ │ │ │  o16 =
│ │ │ │  i17 : if h#"answer file" =!= null and fileExists(h#"answer file") then get
│ │ │ │  (h#"answer file")
│ │ │ │  We can get quite a lot of detail on the resources used with the _K_e_e_p_S_t_a_t_i_s_t_i_c_s
│ │ │ │  command:
│ │ │ │  i18 : h=runExternalM2(fn,"spin",3,KeepStatistics=>true);
│ │ │ │  Running (true && ( (/usr/bin/time --verbose sh -c '/usr/bin/M2-binary  --stop -
│ │ │ │ --no-debug --silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1')
│ │ │ │ ->"/tmp/M2-24375-0/4.stat" 2>&1 ))
│ │ │ │ +-no-debug --silent  -q  <"/tmp/M2-31178-0/4.m2" >"/tmp/M2-31178-0/4.out" 2>&1')
│ │ │ │ +>"/tmp/M2-31178-0/4.stat" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i19 : h#"statistics"
│ │ │ │  
│ │ │ │  o19 =         Command being timed: "sh -c /usr/bin/M2-binary  --stop --no-debug
│ │ │ │ ---silent  -q  <"/tmp/M2-24375-0/4.m2" >"/tmp/M2-24375-0/4.out" 2>&1"
│ │ │ │ -              User time (seconds): 7.49
│ │ │ │ -              System time (seconds): 0.10
│ │ │ │ -              Percent of CPU this job got: 93%
│ │ │ │ -              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:08.15
│ │ │ │ +--silent  -q  <"/tmp/M2-31178-0/4.m2" >"/tmp/M2-31178-0/4.out" 2>&1"
│ │ │ │ +              User time (seconds): 6.93
│ │ │ │ +              System time (seconds): 0.20
│ │ │ │ +              Percent of CPU this job got: 120%
│ │ │ │ +              Elapsed (wall clock) time (h:mm:ss or m:ss): 0:05.93
│ │ │ │                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): 255852
│ │ │ │ +              Maximum resident set size (kbytes): 340228
│ │ │ │                Average resident set size (kbytes): 0
│ │ │ │                Major (requiring I/O) page faults: 0
│ │ │ │ -              Minor (reclaiming a frame) page faults: 10158
│ │ │ │ -              Voluntary context switches: 2973
│ │ │ │ -              Involuntary context switches: 2382
│ │ │ │ +              Minor (reclaiming a frame) page faults: 12075
│ │ │ │ +              Voluntary context switches: 19121
│ │ │ │ +              Involuntary context switches: 3275
│ │ │ │                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/
│ │ │ │ -M2-24375-0/6.m2" >"/tmp/M2-24375-0/6.out" 2>&1 ))
│ │ │ │ +M2-31178-0/6.m2" >"/tmp/M2-31178-0/6.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/
│ │ │ │ -M2-24375-0/7.m2" >"/tmp/M2-24375-0/7.out" 2>&1 ))
│ │ │ │ +M2-31178-0/7.m2" >"/tmp/M2-31178-0/7.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  RunExternalM2: expected answer file does not exist
│ │ │ │  
│ │ │ │ -o24 = HashTable{"answer file" => /tmp/M2-24375-0/7.ans}
│ │ │ │ +o24 = HashTable{"answer file" => /tmp/M2-31178-0/7.ans}
│ │ │ │                  "exit code" => 1
│ │ │ │ -                "output file" => /tmp/M2-24375-0/7.out
│ │ │ │ +                "output file" => /tmp/M2-31178-0/7.out
│ │ │ │                  "return code" => 256
│ │ │ │                  "statistics" => null
│ │ │ │ -                "time used" => 4
│ │ │ │ +                "time used" => 2
│ │ │ │                  value => null
│ │ │ │  
│ │ │ │  o24 : HashTable
│ │ │ │  To view the error message:
│ │ │ │  i25 : get(h#"output file")
│ │ │ │  
│ │ │ │  o25 =
│ │ │ │ -      i1 : -- Script /tmp/M2-24375-0/7.m2 automatically generated by
│ │ │ │ +      i1 : -- Script /tmp/M2-31178-0/7.m2 automatically generated by
│ │ │ │  RunExternalM2
│ │ │ │             needsPackage("RunExternalM2",Configuration=>{"isChild"=>true});
│ │ │ │  
│ │ │ │ -      i2 : load "/tmp/M2-24375-0/0.m2";
│ │ │ │ +      i2 : load "/tmp/M2-31178-0/0.m2";
│ │ │ │  
│ │ │ │ -      i3 : runExternalM2ReturnAnswer("/tmp/M2-24375-0/7.ans",identity (cokernel
│ │ │ │ +      i3 : runExternalM2ReturnAnswer("/tmp/M2-31178-0/7.ans",identity (cokernel
│ │ │ │  (map(R^2,R^{3:{-1}},{{(9/2)*x+(9/4)*y, (7/9)*x+(7/10)*y, 7*x+(3/7)*y}, {(3/
│ │ │ │  4)*x+(7/4)*y, (7/10)*x+(7/3)*y, (6/7)*x+6*y}}))));
│ │ │ │        stdio:4:74:(3):[2]: error: no method for binary operator ^ applied to
│ │ │ │  objects:
│ │ │ │                    R (of class Symbol)
│ │ │ │              ^     2 (of class ZZ)
│ │ │ │  Keep in mind that the object you are passing must make sense in the context of
│ │ │ │  the file containing your function! For instance, here we need to define the
│ │ │ │  ring:
│ │ │ │  i26 : fn<"/tmp/M2-24375-0/8.out" 2>&1 ))
│ │ │ │ +M2-31178-0/8.m2" >"/tmp/M2-31178-0/8.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/
│ │ │ │ -M2-24375-0/9.m2" >"/tmp/M2-24375-0/9.out" 2>&1 ))
│ │ │ │ +M2-31178-0/9.m2" >"/tmp/M2-31178-0/9.out" 2>&1 ))
│ │ │ │  Finished running.
│ │ │ │  i30 : h#value
│ │ │ │  
│ │ │ │  o30 = QQ[x..y]
│ │ │ │  
│ │ │ │  o30 : PolynomialRing
│ │ │ │  i31 : v===h#value
│ │ ├── ./usr/share/doc/Macaulay2/SCMAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=22
│ │ │  Y2Fub25pY2FsTW9kdWxlKElkZWFsKQ==
│ │ │  #:len=266
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjYW5vbmljYWxNb2R1bGUsSWRlYWwpLCJjYW5vbmlj
│ │ ├── ./usr/share/doc/Macaulay2/SCSCP/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  UmVtb3RlT2JqZWN0IGFuZCBSZW1vdGVPYmplY3Q=
│ │ │  #:len=217
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTMwLCAidW5kb2N1bWVudGVkIiA9PiB0
│ │ │  cnVlLCBzeW1ib2wgRG9jdW1lbnRUYWcgPT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wg
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=21
│ │ │  dHJhbnNwb3NlKEdhdGVNYXRyaXgp
│ │ │  #:len=299
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjU1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0cmFuc3Bvc2UsR2F0ZU1hdHJpeCksInRyYW5zcG9z
│ │ ├── ./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 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ + -- used 0.000202361s (cpu); 0.00019382s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │  
│ │ │  i7 : ZZ[y];
│ │ │  
│ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ + -- used 3.61023s (cpu); 2.5346s (thread); 0s (gc)
│ │ │  
│ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLPexpressions/html/index.html
│ │ │ @@ -109,29 +109,29 @@
│ │ │  
│ │ │  o5 : InterpretedSLProgram
    │ │ │
    │ │ │ 
    i6 : time A = evaluate(slp,matrix{{1}});
│ │ │ - -- used 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ + -- used 0.000202361s (cpu); 0.00019382s (thread); 0s (gc)
│ │ │  
│ │ │                1       1
│ │ │  o6 : Matrix ZZ  <-- ZZ
    │ │ │
    │ │ │ 
    i7 : ZZ[y];
    │ │ │
    │ │ │ 
    i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ + -- used 3.61023s (cpu); 2.5346s (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 0.000243396s (cpu); 0.000238547s (thread); 0s (gc)
│ │ │ │ + -- used 0.000202361s (cpu); 0.00019382s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                1       1
│ │ │ │  o6 : Matrix ZZ  <-- ZZ
│ │ │ │  i7 : ZZ[y];
│ │ │ │  i8 : time B = sub((y+1)^(2^n),{y=>1})
│ │ │ │ - -- used 5.61088s (cpu); 3.68025s (thread); 0s (gc)
│ │ │ │ + -- used 3.61023s (cpu); 2.5346s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o8 = 104438888141315250669175271071662438257996424904738378038423348328395390
│ │ │ │       797155745684882681193499755834089010671443926283798757343818579360726323
│ │ │ │       608785136527794595697654370999834036159013438371831442807001185594622637
│ │ │ │       631883939771274567233468434458661749680790870580370407128404874011860911
│ │ │ │       446797778359802900668693897688178778594690563019026094059957945343282346
│ │ │ │       930302669644305902501597239986771421554169383555988529148631823791443449
│ │ ├── ./usr/share/doc/Macaulay2/SLnEquivariantMatrices/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  c2xFcXVpdmFyaWFudFZlY3RvckJ1bmRsZQ==
│ │ │  #:len=4962
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgYSBTTC1lcXVpdmFyaWFu
│ │ │  dCB2ZWN0b3IgYnVuZGxlIG92ZXIgc29tZSBwcm9qZWN0aXZlIHNwYWNlIiwgImxpbmVudW0iID0+
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  RXhhbXBsZSBmaXJzdCBvcmRlciBkZWZvcm1hdGlvbg==
│ │ │  #:len=3174
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRXhhbXBsZSBhY2Nlc3NpbmcgdGhlIGRh
│ │ │  dGEgc3RvcmVkIGluIGEgZmlyc3Qgb3JkZXIgZGVmb3JtYXRpb24uIiwgImxpbmVudW0iID0+IDQy
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/example-output/___Co__Complex.out
│ │ │ @@ -56,15 +56,15 @@
│ │ │  
│ │ │  o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i8 : cC=complement C
│ │ │  
│ │ │  o8 = 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  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
│ │ │  
│ │ │  i9 : dualize cC
│ │ │  
│ │ │  o9 = 2: v v v  v v v  v v v  v v v  v v v  v v v  v v v  v v v
│ │ ├── ./usr/share/doc/Macaulay2/SRdeformations/html/___Co__Complex.html
│ │ │ @@ -163,15 +163,15 @@
│ │ │
    │ │ │ 
    i8 : cC=complement C
│ │ │  
│ │ │  o8 = 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  
│ │ │ -         3 4 5  1 4 5  2 1 5  2 3 4  3 0 4  1 0 4  2 3 1  2 1 0
│ │ │ +         4 5 3  1 4 5  1 5 2  4 2 3  0 4 3  1 0 4  1 2 3  1 0 2
│ │ │  
│ │ │  o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
│ │ │       equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -82,15 +82,15 @@ │ │ │ │ 0 1 2 0 1 4 0 3 4 1 2 3 1 2 5 1 4 5 2 3 4 3 4 5 │ │ │ │ │ │ │ │ o7 : co-complex of dim 2 embedded in dim 5 (printing facets) │ │ │ │ equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 │ │ │ │ i8 : cC=complement C │ │ │ │ │ │ │ │ o8 = 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 │ │ │ │ - 3 4 5 1 4 5 2 1 5 2 3 4 3 0 4 1 0 4 2 3 1 2 1 0 │ │ │ │ + 4 5 3 1 4 5 1 5 2 4 2 3 0 4 3 1 0 4 1 2 3 1 0 2 │ │ │ │ │ │ │ │ o8 : co-complex of dim 2 embedded in dim 5 (printing facets) │ │ │ │ equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1 │ │ │ │ i9 : dualize cC │ │ │ │ │ │ │ │ o9 = 2: v v v v v v v v v v v v v v v v v v v v v v v v │ │ │ │ 3 4 5 2 3 5 1 2 5 0 4 5 0 3 4 0 2 3 0 1 5 0 1 2 │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=35 │ │ │ cHJvamVjdFRvQ29tcGxleChDb21wbGV4LEhhc2hUYWJsZSk= │ │ │ #:len=298 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTE4NCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsocHJvamVjdFRvQ29tcGxleCxDb21wbGV4LEhhc2hU │ │ ├── ./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) │ │ │ - -- .0570737s elapsed │ │ │ + -- .0297892s elapsed │ │ │ │ │ │ 6 19 19 7 3 │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o4 : Complex │ │ │ @@ -41,15 +41,15 @@ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ - -- .00246826s elapsed │ │ │ + -- .00217325s elapsed │ │ │ │ │ │ i8 : h │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ 0 => 3 │ │ │ 1 => 5 │ │ │ 2 => 2 │ │ │ @@ -85,15 +85,15 @@ │ │ │ i12 : maximalEntry complex errors │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ o12 : List │ │ │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ - -- .00494602s elapsed │ │ │ + -- .00435795s 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) │ │ │ - -- .0717096s elapsed │ │ │ + -- .0176315s elapsed │ │ │ │ │ │ 5 10 11 5 │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o4 : Complex │ │ │ @@ -41,25 +41,25 @@ │ │ │ 53 53 53 53 │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o6 : Complex │ │ │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ - -- .000541143s elapsed │ │ │ + -- .000466877s 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) │ │ │ - -- .00894283s elapsed │ │ │ + -- .00111759s 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) │ │ │ - -- .0727341s elapsed │ │ │ + -- .0184333s elapsed │ │ │ │ │ │ 6 10 13 8 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_euclidean__Distance.out │ │ │ @@ -18,15 +18,15 @@ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .0562647s elapsed │ │ │ + -- .0166791s elapsed │ │ │ │ │ │ 6 10 11 5 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/example-output/_project__To__Complex.out │ │ │ @@ -18,15 +18,15 @@ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ o4 : List │ │ │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ - -- .00316014s elapsed │ │ │ + -- .00315621s elapsed │ │ │ │ │ │ 6 10 11 5 │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Complex.html │ │ │ @@ -110,15 +110,15 @@ │ │ │ │ │ │ o3 : List │ │ │
    │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>4)
│ │ │ - -- .0570737s elapsed
│ │ │ + -- .0297892s elapsed
│ │ │  
│ │ │         6       19       19       7       3
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ  <-- ZZ
│ │ │                                          
│ │ │       0       1        2        3       4
│ │ │  
│ │ │  o4 : Complex
    │ │ │ @@ -145,15 +145,15 @@ │ │ │ │ │ │ o6 : Complex │ │ │
    │ │ │ 
    i7 : elapsedTime (h,U)=SVDComplex CR;
│ │ │ - -- .00246826s elapsed
    │ │ │ + -- .00217325s elapsed │ │ │
    │ │ │ 
    i8 : h
│ │ │  
│ │ │  o8 = HashTable{-1 => 1}
│ │ │ @@ -207,15 +207,15 @@
│ │ │  
│ │ │  o12 : List
    │ │ │
    │ │ │ 
    i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian);
│ │ │ - -- .00494602s elapsed
    │ │ │ + -- .00435795s 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) │ │ │ │ - -- .0570737s elapsed │ │ │ │ + -- .0297892s elapsed │ │ │ │ │ │ │ │ 6 19 19 7 3 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ │ │ o4 : Complex │ │ │ │ @@ -60,15 +60,15 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 53 │ │ │ │ │ │ │ │ -1 0 1 2 3 │ │ │ │ │ │ │ │ o6 : Complex │ │ │ │ i7 : elapsedTime (h,U)=SVDComplex CR; │ │ │ │ - -- .00246826s elapsed │ │ │ │ + -- .00217325s elapsed │ │ │ │ i8 : h │ │ │ │ │ │ │ │ o8 = HashTable{-1 => 1} │ │ │ │ 0 => 3 │ │ │ │ 1 => 5 │ │ │ │ 2 => 2 │ │ │ │ 3 => 1 │ │ │ │ @@ -99,15 +99,15 @@ │ │ │ │ 1)*Sigma.dd_ell*transpose U_ell); │ │ │ │ i12 : maximalEntry complex errors │ │ │ │ │ │ │ │ o12 = {8.43769e-15, 6.39488e-14, 1.06581e-13, 9.76996e-15} │ │ │ │ │ │ │ │ o12 : List │ │ │ │ i13 : elapsedTime (hL,U)=SVDComplex(CR,Strategy=>Laplacian); │ │ │ │ - -- .00494602s elapsed │ │ │ │ + -- .00435795s elapsed │ │ │ │ i14 : hL === h │ │ │ │ │ │ │ │ o14 = true │ │ │ │ i15 : SigmaL =source U; │ │ │ │ i16 : for i from min CR+1 to max CR list maximalEntry(SigmaL.dd_i -Sigma.dd_i) │ │ │ │ │ │ │ │ o16 = {1.77636e-14, 6.39488e-14, 8.52651e-14, 3.55271e-15} │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/___S__V__D__Homology.html │ │ │ @@ -112,15 +112,15 @@ │ │ │ │ │ │ o3 : List │ │ │
    │ │ │ 
    i4 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0717096s elapsed
│ │ │ + -- .0176315s elapsed
│ │ │  
│ │ │         5       10       11       5
│ │ │  o4 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o4 : Complex
    │ │ │ @@ -147,28 +147,28 @@ │ │ │ │ │ │ o6 : Complex │ │ │
    │ │ │ 
    i7 : elapsedTime (h,h1)=SVDHomology CR
│ │ │ - -- .000541143s elapsed
│ │ │ + -- .000466877s 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)
│ │ │ - -- .00894283s elapsed
│ │ │ + -- .00111759s 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) │ │ │ │ - -- .0717096s elapsed │ │ │ │ + -- .0176315s elapsed │ │ │ │ │ │ │ │ 5 10 11 5 │ │ │ │ o4 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o4 : Complex │ │ │ │ @@ -63,24 +63,24 @@ │ │ │ │ o6 = RR <-- RR <-- RR <-- RR │ │ │ │ 53 53 53 53 │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o6 : Complex │ │ │ │ i7 : elapsedTime (h,h1)=SVDHomology CR │ │ │ │ - -- .000541143s elapsed │ │ │ │ + -- .000466877s 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) │ │ │ │ - -- .00894283s elapsed │ │ │ │ + -- .00111759s elapsed │ │ │ │ │ │ │ │ o8 = (HashTable{0 => 1}, HashTable{0 => (, 1.71747, -1.72291e-14) }) │ │ │ │ 1 => 3 1 => (1.71747, 922.381, 2.51496e-13) │ │ │ │ 2 => 5 2 => (922.381, 73.5901, 1.81323e-13) │ │ │ │ 3 => 2 3 => (73.5901, , 2.82914e-13) │ │ │ │ │ │ │ │ o8 : Sequence │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_common__Entries.html │ │ │ @@ -115,15 +115,15 @@ │ │ │ │ │ │ o4 : List │ │ │
    │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true)
│ │ │ - -- .0727341s elapsed
│ │ │ + -- .0184333s elapsed
│ │ │  
│ │ │         6       10       13       8
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -34,15 +34,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,5} │ │ │ │ │ │ │ │ o4 = {4, 3, 5} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>100,ZeroMean=>true) │ │ │ │ - -- .0727341s elapsed │ │ │ │ + -- .0184333s elapsed │ │ │ │ │ │ │ │ 6 10 13 8 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_euclidean__Distance.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o4 : List │ │ │
    │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .0562647s elapsed
│ │ │ + -- .0166791s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .0562647s elapsed │ │ │ │ + -- .0166791s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SVDComplexes/html/_project__To__Complex.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o4 : List │ │ │
    │ │ │ 
    i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true)
│ │ │ - -- .00316014s elapsed
│ │ │ + -- .00315621s elapsed
│ │ │  
│ │ │         6       10       11       5
│ │ │  o5 = ZZ  <-- ZZ   <-- ZZ   <-- ZZ
│ │ │                                  
│ │ │       0       1        2        3
│ │ │  
│ │ │  o5 : Complex
    │ │ │ ├── html2text {} │ │ │ │ @@ -29,15 +29,15 @@ │ │ │ │ o3 : List │ │ │ │ i4 : r={4,3,3} │ │ │ │ │ │ │ │ o4 = {4, 3, 3} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ i5 : elapsedTime C=randomChainComplex(h,r,Height=>5,ZeroMean=>true) │ │ │ │ - -- .00316014s elapsed │ │ │ │ + -- .00315621s elapsed │ │ │ │ │ │ │ │ 6 10 11 5 │ │ │ │ o5 = ZZ <-- ZZ <-- ZZ <-- ZZ │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o5 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SagbiGbDetection/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ d2VpZ2h0VmVjdG9yc1JlYWxpemluZ1NBR0JJ │ │ │ #:len=1819 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIG1haW4gZnVuY3Rpb24gZm9yIGRl │ │ │ dGVjdGluZyBTQUdCSSBiYXNlcyIsICJsaW5lbnVtIiA9PiAyMTUsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/Saturation/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ c2F0dXJhdGUoSWRlYWwsTGlzdCk= │ │ │ #:len=258 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzYXR1cmF0ZSxJZGVhbCxMaXN0KSwic2F0dXJhdGUo │ │ ├── ./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 0.412639s (cpu); 0.315292s (thread); 0s (gc) │ │ │ + -- used 0.293973s (cpu); 0.293974s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc) │ │ │ + -- used 0.605047s (cpu); 0.523239s (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 0.021082s (cpu); 0.021086s (thread); 0s (gc) │ │ │ + -- used 0.0230604s (cpu); 0.0230636s (thread); 0s (gc) │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc) │ │ │ + -- used 0.007758s (cpu); 0.00776329s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of S │ │ │ │ │ │ i13 : │ │ ├── ./usr/share/doc/Macaulay2/Saturation/html/_quotient_lp..._cm__Strategy_eq_gt..._rp.html │ │ │ @@ -129,23 +129,23 @@ │ │ │ │ │ │ o6 : Ideal of S │ │ │
    │ │ │ 
    i7 : time quotient(I^3, J^2, Strategy => Iterate);
│ │ │ - -- used 0.412639s (cpu); 0.315292s (thread); 0s (gc)
│ │ │ + -- used 0.293973s (cpu); 0.293974s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of S
    │ │ │
    │ │ │ 
    i8 : time quotient(I^3, J^2, Strategy => Quotient);
│ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc)
│ │ │ + -- used 0.605047s (cpu); 0.523239s (thread); 0s (gc)
│ │ │  
│ │ │  o8 : Ideal of S
    │ │ │
    │ │ │
    │ │ │

    Strategy => Quotient is faster in other cases:

    │ │ │ @@ -162,23 +162,23 @@ │ │ │ │ │ │ o10 : Ideal of S │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : time quotient(I^5, I^3, Strategy => Iterate);
│ │ │ - -- used 0.021082s (cpu); 0.021086s (thread); 0s (gc)
│ │ │ + -- used 0.0230604s (cpu); 0.0230636s (thread); 0s (gc)
│ │ │  
│ │ │  o11 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time quotient(I^5, I^3, Strategy => Quotient);
│ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc)
│ │ │ + -- used 0.007758s (cpu); 0.00776329s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of S
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -52,32 +52,32 @@ │ │ │ │ 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 0.412639s (cpu); 0.315292s (thread); 0s (gc) │ │ │ │ + -- used 0.293973s (cpu); 0.293974s (thread); 0s (gc) │ │ │ │ │ │ │ │ o7 : Ideal of S │ │ │ │ i8 : time quotient(I^3, J^2, Strategy => Quotient); │ │ │ │ - -- used 0.660447s (cpu); 0.585082s (thread); 0s (gc) │ │ │ │ + -- used 0.605047s (cpu); 0.523239s (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 0.021082s (cpu); 0.021086s (thread); 0s (gc) │ │ │ │ + -- used 0.0230604s (cpu); 0.0230636s (thread); 0s (gc) │ │ │ │ │ │ │ │ o11 : Ideal of S │ │ │ │ i12 : time quotient(I^5, I^3, Strategy => Quotient); │ │ │ │ - -- used 0.00785722s (cpu); 0.00785869s (thread); 0s (gc) │ │ │ │ + -- used 0.007758s (cpu); 0.00776329s (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/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ aW5jbHVzaW9uKC4uLixTdWJEaW1lbnNpb249Pi4uLik= │ │ │ #:len=260 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg1Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbaW5jbHVzaW9uLFN1YkRpbWVuc2lvbl0sImluY2x1 │ │ ├── ./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 0.00475218s (cpu); 0.00474919s (thread); 0s (gc) │ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc) │ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc) │ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc) │ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc) │ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc) │ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc) │ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc) │ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc) │ │ │ + -- used 0.00594132s (cpu); 0.00593976s (thread); 0s (gc) │ │ │ + -- used 0.00618061s (cpu); 0.00618559s (thread); 0s (gc) │ │ │ + -- used 0.00955573s (cpu); 0.00955987s (thread); 0s (gc) │ │ │ + -- used 0.0156952s (cpu); 0.0156997s (thread); 0s (gc) │ │ │ + -- used 0.0299372s (cpu); 0.0299434s (thread); 0s (gc) │ │ │ + -- used 0.0590319s (cpu); 0.0590403s (thread); 0s (gc) │ │ │ + -- used 0.0974063s (cpu); 0.0974123s (thread); 0s (gc) │ │ │ + -- used 0.140349s (cpu); 0.140355s (thread); 0s (gc) │ │ │ + -- used 0.356161s (cpu); 0.243848s (thread); 0s (gc) │ │ │ │ │ │ o7 = {1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, │ │ │ ------------------------------------------------------------------------ │ │ │ 289139638632755625, 520764738758073845321} │ │ │ │ │ │ o7 : List │ │ ├── ./usr/share/doc/Macaulay2/Schubert2/html/___Lines_spon_sphypersurfaces.html │ │ │ @@ -131,23 +131,23 @@ │ │ │ │ │ │ o6 : FunctionClosure │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : for n from 2 to 10 list time f n
│ │ │ - -- used 0.00475218s (cpu); 0.00474919s (thread); 0s (gc)
│ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc)
│ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc)
│ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc)
│ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc)
│ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc)
│ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc)
│ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc)
│ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc)
│ │ │ + -- used 0.00594132s (cpu); 0.00593976s (thread); 0s (gc)
│ │ │ + -- used 0.00618061s (cpu); 0.00618559s (thread); 0s (gc)
│ │ │ + -- used 0.00955573s (cpu); 0.00955987s (thread); 0s (gc)
│ │ │ + -- used 0.0156952s (cpu); 0.0156997s (thread); 0s (gc)
│ │ │ + -- used 0.0299372s (cpu); 0.0299434s (thread); 0s (gc)
│ │ │ + -- used 0.0590319s (cpu); 0.0590403s (thread); 0s (gc)
│ │ │ + -- used 0.0974063s (cpu); 0.0974123s (thread); 0s (gc)
│ │ │ + -- used 0.140349s (cpu); 0.140355s (thread); 0s (gc)
│ │ │ + -- used 0.356161s (cpu); 0.243848s (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 0.00475218s (cpu); 0.00474919s (thread); 0s (gc) │ │ │ │ - -- used 0.00626054s (cpu); 0.00626186s (thread); 0s (gc) │ │ │ │ - -- used 0.0101612s (cpu); 0.010164s (thread); 0s (gc) │ │ │ │ - -- used 0.0184984s (cpu); 0.018504s (thread); 0s (gc) │ │ │ │ - -- used 0.0345749s (cpu); 0.0345811s (thread); 0s (gc) │ │ │ │ - -- used 0.0689669s (cpu); 0.0689727s (thread); 0s (gc) │ │ │ │ - -- used 0.104701s (cpu); 0.104707s (thread); 0s (gc) │ │ │ │ - -- used 0.307541s (cpu); 0.188629s (thread); 0s (gc) │ │ │ │ - -- used 0.386656s (cpu); 0.275611s (thread); 0s (gc) │ │ │ │ + -- used 0.00594132s (cpu); 0.00593976s (thread); 0s (gc) │ │ │ │ + -- used 0.00618061s (cpu); 0.00618559s (thread); 0s (gc) │ │ │ │ + -- used 0.00955573s (cpu); 0.00955987s (thread); 0s (gc) │ │ │ │ + -- used 0.0156952s (cpu); 0.0156997s (thread); 0s (gc) │ │ │ │ + -- used 0.0299372s (cpu); 0.0299434s (thread); 0s (gc) │ │ │ │ + -- used 0.0590319s (cpu); 0.0590403s (thread); 0s (gc) │ │ │ │ + -- used 0.0974063s (cpu); 0.0974123s (thread); 0s (gc) │ │ │ │ + -- used 0.140349s (cpu); 0.140355s (thread); 0s (gc) │ │ │ │ + -- used 0.356161s (cpu); 0.243848s (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/SchurComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=12 │ │ │ c2NodXJDb21wbGV4 │ │ │ #:len=3519 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiU2NodXIgZnVuY3RvcnMgb2YgY2hhaW4g │ │ │ Y29tcGxleGVzIiwgImxpbmVudW0iID0+IDU5NywgSW5wdXRzID0+IHtTUEFOe1RUeyJsYW1iZGEi │ │ ├── ./usr/share/doc/Macaulay2/SchurFunctors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=27 │ │ │ aXNXZXlsU3RhbmRhcmQoV2V5bEZpbGxpbmcp │ │ │ #:len=281 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTcwMywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoaXNXZXlsU3RhbmRhcmQsV2V5bEZpbGxpbmcpLCJp │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ c2NodXJSZXNvbHV0aW9uKFJpbmdFbGVtZW50LExpc3Qp │ │ │ #:len=286 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjU4Miwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoc2NodXJSZXNvbHV0aW9uLFJpbmdFbGVtZW50LExp │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Eor__H.out │ │ │ @@ -33,13 +33,13 @@ │ │ │ i6 : toS fe == toS fh │ │ │ │ │ │ o6 = true │ │ │ │ │ │ i7 : R = symmetricRing(QQ,12); │ │ │ │ │ │ i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false); │ │ │ - -- .00389276s elapsed │ │ │ + -- .0031273s elapsed │ │ │ │ │ │ i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false); │ │ │ - -- .000092754s elapsed │ │ │ + -- .000123495s elapsed │ │ │ │ │ │ i10 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/___Memoize.out │ │ │ @@ -1,23 +1,23 @@ │ │ │ -- -*- M2-comint -*- hash: 9113092561535051477 │ │ │ │ │ │ i1 : R = symmetricRing(QQ, 10); │ │ │ │ │ │ i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ - -- .000475089s elapsed │ │ │ + -- .000376464s elapsed │ │ │ │ │ │ i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ - -- .000014267s elapsed │ │ │ + -- .000016469s elapsed │ │ │ │ │ │ i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ - -- .000441827s elapsed │ │ │ + -- .000393215s elapsed │ │ │ │ │ │ i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ - -- .000013796s elapsed │ │ │ + -- .000016953s elapsed │ │ │ │ │ │ i6 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ - -- .000013405s elapsed │ │ │ + -- .000012719s elapsed │ │ │ │ │ │ i7 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ - -- .000012954s elapsed │ │ │ + -- .000012492s elapsed │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi.out │ │ │ @@ -50,18 +50,18 @@ │ │ │ i11 : toS fe == toS fh │ │ │ │ │ │ o11 = true │ │ │ │ │ │ i12 : R = symmetricRing(QQ,6); │ │ │ │ │ │ i13 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ - -- .000476681s elapsed │ │ │ + -- .000407909s elapsed │ │ │ │ │ │ i14 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ - -- .000014307s elapsed │ │ │ + -- .000015828s elapsed │ │ │ │ │ │ i15 : R = symmetricRing(QQ,5); │ │ │ │ │ │ i16 : S = schurRing R; │ │ │ │ │ │ i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1}) │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/example-output/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.out │ │ │ @@ -3,16 +3,16 @@ │ │ │ i1 : R = symmetricRing(QQ,6); │ │ │ │ │ │ i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi({4,3,2,1},R,Memoize => false) │ │ │ │ │ │ o2 = true │ │ │ │ │ │ i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ - -- .000459589s elapsed │ │ │ + -- .000401146s elapsed │ │ │ │ │ │ i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ - -- .00001618s elapsed │ │ │ + -- .000014991s elapsed │ │ │ │ │ │ i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false); │ │ │ - -- .000383086s elapsed │ │ │ + -- .000338944s elapsed │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Eor__H.html │ │ │ @@ -143,21 +143,21 @@ │ │ │ │ │ │ 
    i7 : R = symmetricRing(QQ,12);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false);
│ │ │ - -- .00389276s elapsed
    │ │ │ + -- .0031273s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false);
│ │ │ - -- .000092754s elapsed
    │ │ │ + -- .000123495s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Functions with optional argument named EorH:

    │ │ │ ├── html2text {} │ │ │ │ @@ -46,17 +46,17 @@ │ │ │ │ o6 = true │ │ │ │ When the conjugate partition is much longer than lambda itself, the "H"-branch │ │ │ │ requires a smaller determinant and runs measurably faster. For example on │ │ │ │ lambda = (10) the conjugate is $(1^{10})$, so "H" only sets up a 1x1 │ │ │ │ determinant: │ │ │ │ i7 : R = symmetricRing(QQ,12); │ │ │ │ i8 : elapsedTime jacobiTrudi({10},R,EorH => "E",Memoize => false); │ │ │ │ - -- .00389276s elapsed │ │ │ │ + -- .0031273s elapsed │ │ │ │ i9 : elapsedTime jacobiTrudi({10},R,EorH => "H",Memoize => false); │ │ │ │ - -- .000092754s elapsed │ │ │ │ + -- .000123495s elapsed │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd EEoorrHH:: ********** │ │ │ │ * jacobiTrudi(...,EorH=>...) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _E_o_r_H is a _s_y_m_b_o_l. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/SchurRings.m2:6153:0. │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/___Memoize.html │ │ │ @@ -66,59 +66,59 @@ │ │ │ │ │ │ 
    i1 : R = symmetricRing(QQ, 10);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000475089s elapsed
    │ │ │ + -- .000376464s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true);
│ │ │ - -- .000014267s elapsed
    │ │ │ + -- .000016469s elapsed │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    │ │ │

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

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000441827s elapsed
    │ │ │ + -- .000393215s elapsed │ │ │
    │ │ │ 
    i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true);
│ │ │ - -- .000013796s elapsed
    │ │ │ + -- .000016953s elapsed │ │ │
    │ │ │
    │ │ │

    │ │ │

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

    │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000013405s elapsed
    │ │ │ + -- .000012719s elapsed │ │ │
    │ │ │ 
    i7 : elapsedTime jacobiTrudi({4,3,2,1}, R);
│ │ │ - -- .000012954s elapsed
    │ │ │ + -- .000012492s elapsed │ │ │
    │ │ │
    │ │ │
    │ │ │

    See also

    │ │ │
      │ │ │ ├── html2text {} │ │ │ │ @@ -7,31 +7,31 @@ │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This is an optional argument for the _j_a_c_o_b_i_T_r_u_d_i function, allowing one to │ │ │ │ store its values in order to speed up computations. When Memoize => true, every │ │ │ │ computed value is cached in a hash table on the symmetric ring, so repeated │ │ │ │ calls on the same partition return the cached value in constant time. │ │ │ │ i1 : R = symmetricRing(QQ, 10); │ │ │ │ i2 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ │ - -- .000475089s elapsed │ │ │ │ + -- .000376464s elapsed │ │ │ │ i3 : elapsedTime jacobiTrudi({4,3,2,1}, R, Memoize => true); │ │ │ │ - -- .000014267s elapsed │ │ │ │ + -- .000016469s elapsed │ │ │ │ The cache is attached to the ring R. After one partition is memoized, │ │ │ │ subsequent calls with a different partition perform the full Jacobi-Trudi │ │ │ │ determinant expansion, then cache it as well: │ │ │ │ i4 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ │ - -- .000441827s elapsed │ │ │ │ + -- .000393215s elapsed │ │ │ │ i5 : elapsedTime jacobiTrudi({5,3,2}, R, Memoize => true); │ │ │ │ - -- .000013796s elapsed │ │ │ │ + -- .000016953s elapsed │ │ │ │ Without Memoize => true, each call recomputes the determinant from scratch; for │ │ │ │ large partitions this can be substantially more expensive than a single cached │ │ │ │ lookup. │ │ │ │ i6 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ │ - -- .000013405s elapsed │ │ │ │ + -- .000012719s elapsed │ │ │ │ i7 : elapsedTime jacobiTrudi({4,3,2,1}, R); │ │ │ │ - -- .000012954s elapsed │ │ │ │ + -- .000012492s elapsed │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi │ │ │ │ function. │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _M_e_m_o_i_z_e is a _s_y_m_b_o_l. │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi.html │ │ │ @@ -186,21 +186,21 @@ │ │ │ │ │ │ 
      i12 : R = symmetricRing(QQ,6);
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i13 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000476681s elapsed
      │ │ │ + -- .000407909s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : elapsedTime jacobiTrudi({4,3,2,1},R);
│ │ │ - -- .000014307s elapsed
      │ │ │ + -- .000015828s elapsed │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │

      │ │ │

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

      │ │ │

      │ │ │ ├── html2text {} │ │ │ │ @@ -67,17 +67,17 @@ │ │ │ │ │ │ │ │ o11 = true │ │ │ │ The routine caches intermediate subdeterminants on the ring via _j_a_c_o_b_i_T_r_u_d_i │ │ │ │ _(_._._._,_M_e_m_o_i_z_e_=_>_._._._), so a second call on a large partition returns almost │ │ │ │ instantly: │ │ │ │ i12 : R = symmetricRing(QQ,6); │ │ │ │ i13 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ │ - -- .000476681s elapsed │ │ │ │ + -- .000407909s elapsed │ │ │ │ i14 : elapsedTime jacobiTrudi({4,3,2,1},R); │ │ │ │ - -- .000014307s elapsed │ │ │ │ + -- .000015828s elapsed │ │ │ │ Passing a partition through _t_o_S_y_m_m applied to the corresponding Schur label │ │ │ │ reproduces the Jacobi-Trudi output: │ │ │ │ i15 : R = symmetricRing(QQ,5); │ │ │ │ i16 : S = schurRing R; │ │ │ │ i17 : jacobiTrudi({3,2,1},R) == toSymm(S_{3,2,1}) │ │ │ │ │ │ │ │ o17 = true │ │ ├── ./usr/share/doc/Macaulay2/SchurRings/html/_jacobi__Trudi_lp..._cm__Memoize_eq_gt..._rp.html │ │ │ @@ -85,27 +85,27 @@ │ │ │ │ │ │ o2 = true │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .000459589s elapsed
      │ │ │ + -- .000401146s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true);
│ │ │ - -- .00001618s elapsed
      │ │ │ + -- .000014991s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false);
│ │ │ - -- .000383086s elapsed
      │ │ │ + -- .000338944s elapsed │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │
      │ │ │
      │ │ │

      Functions with optional argument named Memoize:

      │ │ │ ├── html2text {} │ │ │ │ @@ -17,19 +17,19 @@ │ │ │ │ extra memory in the ring. │ │ │ │ i1 : R = symmetricRing(QQ,6); │ │ │ │ i2 : jacobiTrudi({4,3,2,1},R,Memoize => true) == jacobiTrudi( │ │ │ │ {4,3,2,1},R,Memoize => false) │ │ │ │ │ │ │ │ o2 = true │ │ │ │ i3 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ │ - -- .000459589s elapsed │ │ │ │ + -- .000401146s elapsed │ │ │ │ i4 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => true); │ │ │ │ - -- .00001618s elapsed │ │ │ │ + -- .000014991s elapsed │ │ │ │ i5 : elapsedTime jacobiTrudi({5,4,3,2,1},R,Memoize => false); │ │ │ │ - -- .000383086s elapsed │ │ │ │ + -- .000338944s elapsed │ │ │ │ ********** FFuunnccttiioonnss wwiitthh ooppttiioonnaall aarrgguummeenntt nnaammeedd MMeemmooiizzee:: ********** │ │ │ │ * _j_a_c_o_b_i_T_r_u_d_i_(_._._._,_M_e_m_o_i_z_e_=_>_._._._) -- Store values of the jacobiTrudi │ │ │ │ function. │ │ │ │ ********** FFuurrtthheerr iinnffoorrmmaattiioonn ********** │ │ │ │ * Default value: _t_r_u_e │ │ │ │ * Function: _j_a_c_o_b_i_T_r_u_d_i -- Jacobi-Trudi determinant │ │ │ │ * Option key: _M_e_m_o_i_z_e -- Option to record values of the jacobiTrudi │ │ ├── ./usr/share/doc/Macaulay2/SchurVeronese/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ bWFrZUJldHRpVGFsbHk= │ │ │ #:len=1334 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29udmVydHMgYSBoYXNoIHRhYmxlIHJl │ │ │ cHJlc2VudGluZyBhIEJldHRpIHRhYmxlIHRvIGEgQmV0dGkgdGFsbHkiLCAibGluZW51bSIgPT4g │ │ ├── ./usr/share/doc/Macaulay2/SectionRing/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ bVJlZ3VsYXI= │ │ │ #:len=1136 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibVJlZ3VsYXIoRixHKSBjb21wdXRlcyB0 │ │ │ aGUgcmVndWxhcml0eSBvZiBGIHdpdGggcmVzcGVjdCB0byBHIChnbG9iYWxseSBnZW5lcmF0ZWQp │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=26 │ │ │ bWFrZVByb2R1Y3RSaW5nKFJpbmcsTGlzdCk= │ │ │ #:len=277 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gOTQxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWtlUHJvZHVjdFJpbmcsUmluZyxMaXN0KSwibWFr │ │ ├── ./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 4.54047s (cpu); 3.61426s (thread); 0s (gc) │ │ │ + -- used 6.08963s (cpu); 3.23069s (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 54.8103s (cpu); 50.9154s (thread); 0s (gc) │ │ │ + -- used 54.4934s (cpu); 50.7614s (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 0.261889s (cpu); 0.158635s (thread); 0s (gc) │ │ │ + -- used 1.9807s (cpu); 0.400095s (thread); 0s (gc) │ │ │ │ │ │ 2 2 │ │ │ o6 = 2H + 4H H + 2H │ │ │ 1 1 2 2 │ │ │ │ │ │ o6 : A │ │ │ │ │ │ i7 : time segre(X,Y,A) │ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc) │ │ │ + -- used 0.246609s (cpu); 0.119065s (thread); 0s (gc) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o7 = 12H H - 6H H - 6H H + 2H + 4H H + 2H │ │ │ 1 2 1 2 1 2 1 1 2 2 │ │ │ │ │ │ o7 : A │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_is__Component__Contained.html │ │ │ @@ -167,15 +167,15 @@ │ │ │ │ │ │ o10 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i11 : time isComponentContained(X,Y)
│ │ │ - -- used 4.54047s (cpu); 3.61426s (thread); 0s (gc)
│ │ │ + -- used 6.08963s (cpu); 3.23069s (thread); 0s (gc)
│ │ │  
│ │ │  o11 = true
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i12 : print "we could confirm this with the computation:"
│ │ │ @@ -194,15 +194,15 @@
│ │ │  
│ │ │  o13 : Ideal of R
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i14 : time isSubset(saturate(Y,B),saturate(X,B))
│ │ │ - -- used 54.8103s (cpu); 50.9154s (thread); 0s (gc)
│ │ │ + -- used 54.4934s (cpu); 50.7614s (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 4.54047s (cpu); 3.61426s (thread); 0s (gc) │ │ │ │ + -- used 6.08963s (cpu); 3.23069s (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 54.8103s (cpu); 50.9154s (thread); 0s (gc) │ │ │ │ + -- used 54.4934s (cpu); 50.7614s (thread); 0s (gc) │ │ │ │ │ │ │ │ o14 = true │ │ │ │ ********** WWaayyss ttoo uussee iissCCoommppoonneennttCCoonnttaaiinneedd:: ********** │ │ │ │ * isComponentContained(Ideal,Ideal) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _i_s_C_o_m_p_o_n_e_n_t_C_o_n_t_a_i_n_e_d is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n_ _w_i_t_h_ _o_p_t_i_o_n_s. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/SegreClasses/html/_segre__Dim__X.html │ │ │ @@ -123,27 +123,27 @@ │ │ │ │ │ │ o5 : QuotientRing │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i6 : time s = segreDimX(X,Y,A)
│ │ │ - -- used 0.261889s (cpu); 0.158635s (thread); 0s (gc)
│ │ │ + -- used 1.9807s (cpu); 0.400095s (thread); 0s (gc)
│ │ │  
│ │ │         2             2
│ │ │  o6 = 2H  + 4H H  + 2H
│ │ │         1     1 2     2
│ │ │  
│ │ │  o6 : A
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i7 : time segre(X,Y,A)
│ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc)
│ │ │ + -- used 0.246609s (cpu); 0.119065s (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 0.261889s (cpu); 0.158635s (thread); 0s (gc) │ │ │ │ + -- used 1.9807s (cpu); 0.400095s (thread); 0s (gc) │ │ │ │ │ │ │ │ 2 2 │ │ │ │ o6 = 2H + 4H H + 2H │ │ │ │ 1 1 2 2 │ │ │ │ │ │ │ │ o6 : A │ │ │ │ i7 : time segre(X,Y,A) │ │ │ │ - -- used 0.158315s (cpu); 0.114099s (thread); 0s (gc) │ │ │ │ + -- used 0.246609s (cpu); 0.119065s (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/SemidefiniteProgramming/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=28 │ │ │ b3B0aW1pemUoLi4uLFZlcmJvc2l0eT0+Li4uKQ== │ │ │ #:len=300 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzkyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tvcHRpbWl6ZSxWZXJib3NpdHldLCJvcHRpbWl6ZSgu │ │ ├── ./usr/share/doc/Macaulay2/Seminormalization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=22 │ │ │ YmV0dGVyTm9ybWFsaXphdGlvbk1hcA== │ │ │ #:len=1762 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAibm9ybWFsaXplcyBub24gZG9tYWlucyIs │ │ │ ICJsaW5lbnVtIiA9PiA4ODEsIElucHV0cyA9PiB7U1BBTntUVHsiUyJ9LCIsICIsU1BBTnsiYSAi │ │ ├── ./usr/share/doc/Macaulay2/Serialization/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ U2VyaWFsaXphdGlvbg== │ │ │ #:len=554 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicmV2ZXJzaWJsZSBjb252ZXJzaW9uIG9m │ │ │ IGFsbCBNYWNhdWxheTIgb2JqZWN0cyB0byBzdHJpbmdzIiwgRGVzY3JpcHRpb24gPT4gMTooRElW │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=20 │ │ │ YXJYaXYoU3RyaW5nLFN0cmluZyk= │ │ │ #:len=236 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzA5LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhhclhpdixTdHJpbmcsU3RyaW5nKSwiYXJYaXYoU3Ry │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/example-output/_test__Example.out │ │ │ @@ -1,7 +1,7 @@ │ │ │ -- -*- M2-comint -*- hash: 1331702921222 │ │ │ │ │ │ i1 : check SimpleDoc │ │ │ - -- capturing check(0, "SimpleDoc") -- .36758s elapsed │ │ │ - -- capturing check(1, "SimpleDoc") -- .401467s elapsed │ │ │ + -- capturing check(0, "SimpleDoc") -- .179537s elapsed │ │ │ + -- capturing check(1, "SimpleDoc") -- .154542s elapsed │ │ │ │ │ │ i2 : │ │ ├── ./usr/share/doc/Macaulay2/SimpleDoc/html/_test__Example.html │ │ │ @@ -79,16 +79,16 @@ │ │ │
      │ │ │

      The check method executes all package tests defined this way.

      │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i1 : check SimpleDoc
│ │ │ - -- capturing check(0, "SimpleDoc")           -- .36758s elapsed
│ │ │ - -- capturing check(1, "SimpleDoc")           -- .401467s elapsed
      │ │ │ + -- capturing check(0, "SimpleDoc") -- .179537s elapsed │ │ │ + -- capturing check(1, "SimpleDoc") -- .154542s elapsed │ │ │
      │ │ │
      │ │ │
      │ │ │

      See also

      │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -10,16 +10,16 @@ │ │ │ │ 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") -- .36758s elapsed │ │ │ │ - -- capturing check(1, "SimpleDoc") -- .401467s elapsed │ │ │ │ + -- capturing check(0, "SimpleDoc") -- .179537s elapsed │ │ │ │ + -- capturing check(1, "SimpleDoc") -- .154542s 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/SimplicialComplexes/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=18 │ │ │ ZWxlbWVudGFyeUNvbGxhcHNl │ │ │ #:len=365 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDU1Miwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsiZWxlbWVudGFyeUNvbGxhcHNlIiwiZWxlbWVudGFy │ │ ├── ./usr/share/doc/Macaulay2/SimplicialDecomposability/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ c2hlbGxpbmdPcmRlciguLi4sUmFuZG9tPT4uLi4p │ │ │ #:len=311 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODkzLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tzaGVsbGluZ09yZGVyLFJhbmRvbV0sInNoZWxsaW5n │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=44 │ │ │ bm9ybWFsaXplKFNpbXBsaWNpYWxNb2R1bGUsWlosQ2hlY2tTdW09Pi4uLik= │ │ │ #:len=375 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzg0Nywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbKG5vcm1hbGl6ZSxTaW1wbGljaWFsTW9kdWxlLFpa │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/___Simplicial__Modules.out │ │ │ @@ -111,25 +111,25 @@ │ │ │ o7 = R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o7 : SimplicialModule │ │ │ │ │ │ i8 : elapsedTime simplicialModule(K,6) --specify top degree 6 │ │ │ - -- .0671636s elapsed │ │ │ + -- .0659143s elapsed │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ o8 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o8 : SimplicialModule │ │ │ │ │ │ i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true) │ │ │ - -- .195242s elapsed │ │ │ + -- .145094s elapsed │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o9 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_ext__Power.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ o2 = Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ i3 : w3K = elapsedTime prune extPower(3, K) │ │ │ - -- 7.53161s elapsed │ │ │ + -- 4.96256s elapsed │ │ │ │ │ │ 1 18 63 91 60 15 │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 1 2 3 4 5 6 │ │ │ │ │ │ o3 : Complex │ │ │ @@ -93,15 +93,15 @@ │ │ │ 1 2 │ │ │ 2 : Q <--------------- Q : 2 │ │ │ {2} | 0 b | │ │ │ │ │ │ o7 : ComplexMap │ │ │ │ │ │ i8 : f = elapsedTime prune extPower(2, phi) │ │ │ - -- .604017s elapsed │ │ │ + -- .364216s elapsed │ │ │ │ │ │ 3 6 │ │ │ o8 = 1 : Q <------------------------- Q : 1 │ │ │ {1} | a b 0 0 0 0 | │ │ │ {1} | 0 0 0 b 0 0 | │ │ │ {2} | 0 0 0 0 ab b2 | │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_exterior__Inclusion.out │ │ │ @@ -12,15 +12,15 @@ │ │ │ o2 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ - -- .210615s elapsed │ │ │ + -- .193978s elapsed │ │ │ │ │ │ i4 : isWellDefined phi │ │ │ │ │ │ o4 = true │ │ │ │ │ │ i5 : isCommutative phi │ │ │ │ │ │ @@ -160,15 +160,15 @@ │ │ │ o10 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o10 : Complex │ │ │ │ │ │ i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ - -- .21521s elapsed │ │ │ + -- .188879s elapsed │ │ │ │ │ │ i12 : isWellDefined phi │ │ │ │ │ │ o12 = true │ │ │ │ │ │ i13 : isCommutative phi │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_forget__Degeneracy_lp__Simplicial__Module_rp.out │ │ │ @@ -21,15 +21,15 @@ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ i4 : elapsedTime S**S │ │ │ - -- .487245s elapsed │ │ │ + -- .296389s elapsed │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o4 : SimplicialModule │ │ │ @@ -40,15 +40,15 @@ │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ │ │ i6 : elapsedTime fS**fS --faster when degeneracy is ignored │ │ │ - -- .463629s elapsed │ │ │ + -- .253896s elapsed │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ o6 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ o6 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.out │ │ │ @@ -114,35 +114,35 @@ │ │ │ o10 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 9 10 │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ │ │ i11 : elapsedTime prune normalize S10 │ │ │ - -- 4.94576s elapsed │ │ │ + -- 3.20219s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o11 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o11 : Complex │ │ │ │ │ │ i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks │ │ │ - -- 8.29592s elapsed │ │ │ + -- 6.03003s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o12 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o12 : Complex │ │ │ │ │ │ i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER! │ │ │ - -- .0571616s elapsed │ │ │ + -- .0321265s elapsed │ │ │ │ │ │ 10 30 30 10 │ │ │ o13 = R <-- R <-- R <-- R │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o13 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_schur__Map.out │ │ │ @@ -14,25 +14,25 @@ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ i4 : S2S = elapsedTime schurMap({2}, S) │ │ │ - -- .142323s elapsed │ │ │ + -- .114875s elapsed │ │ │ │ │ │ 1 6 21 55 120 │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o4 : SimplicialModule │ │ │ │ │ │ i5 : elapsedTime schurMap({2}, S, Degeneracy => true) │ │ │ - -- .309043s elapsed │ │ │ + -- .253149s elapsed │ │ │ │ │ │ 1 6 21 55 120 │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o5 : SimplicialModule │ │ │ @@ -78,15 +78,15 @@ │ │ │ o10 = Q │ │ │ │ │ │ o10 : PolynomialRing │ │ │ │ │ │ i11 : K = koszulComplex vars Q; │ │ │ │ │ │ i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4) │ │ │ - -- 1.00439s elapsed │ │ │ + -- .760079s elapsed │ │ │ │ │ │ 1 9 36 74 81 │ │ │ o12 = Q <-- Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 4 │ │ │ │ │ │ o12 : Complex │ │ │ @@ -273,15 +273,15 @@ │ │ │ 1 2 │ │ │ 2 : Q <--------------- Q : 2 │ │ │ {2} | 0 b | │ │ │ │ │ │ o30 : ComplexMap │ │ │ │ │ │ i31 : f = elapsedTime prune schurMap({2}, phi) │ │ │ - -- .912527s elapsed │ │ │ + -- .81818s elapsed │ │ │ │ │ │ 1 1 │ │ │ o31 = 0 : Q <--------- Q : 0 │ │ │ | 1 | │ │ │ │ │ │ 5 9 │ │ │ 1 : Q <----------------------------------- Q : 1 │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/example-output/_tensor__L__E__S.out │ │ │ @@ -62,15 +62,15 @@ │ │ │ o7 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ o7 : Complex │ │ │ │ │ │ i8 : hL = elapsedTime prune tensorLES(L,4) │ │ │ - -- 2.28436s elapsed │ │ │ + -- 1.57652s elapsed │ │ │ │ │ │ 25 50 25 │ │ │ o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0 0 b a 0 0 0 0 | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q <-- Q <-- Q │ │ │ | 0 b a | | 0 b a | | 0 0 b a | | 0 0 b a 0 0 0 0 | | 0 0 0 0 b a 0 0 | │ │ │ 0 1 2 3 | 0 0 0 0 b a 0 0 | | ab2 a2b 0 0 0 0 b2 a2 | 9 10 11 12 13 14 │ │ │ 4 5 6 | 0 0 0 0 0 0 b a | │ │ │ 8 │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_ext__Power.html │ │ │ @@ -102,15 +102,15 @@ │ │ │ │ │ │ o2 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ - -- 7.53161s elapsed
│ │ │ + -- 4.96256s elapsed
│ │ │  
│ │ │        1      18      63      91      60      15
│ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │                                               
│ │ │       1      2       3       4       5       6
│ │ │  
│ │ │  o3 : Complex
        │ │ │ @@ -203,15 +203,15 @@ │ │ │ │ │ │ o7 : ComplexMap │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ - -- .604017s elapsed
│ │ │ + -- .364216s elapsed
│ │ │  
│ │ │            3                             6
│ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │                 {1} | a b 0 0 0  0  |
│ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -37,15 +37,15 @@
│ │ │ │        1      2      1
│ │ │ │  o2 = Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2
│ │ │ │  
│ │ │ │  o2 : Complex
│ │ │ │  i3 : w3K = elapsedTime prune extPower(3, K)
│ │ │ │ - -- 7.53161s elapsed
│ │ │ │ + -- 4.96256s elapsed
│ │ │ │  
│ │ │ │        1      18      63      91      60      15
│ │ │ │  o3 = Q  <-- Q   <-- Q   <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │       1      2       3       4       5       6
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │ @@ -115,15 +115,15 @@
│ │ │ │  
│ │ │ │            1                   2
│ │ │ │       2 : Q  <--------------- Q  : 2
│ │ │ │                 {2} | 0 b |
│ │ │ │  
│ │ │ │  o7 : ComplexMap
│ │ │ │  i8 : f = elapsedTime prune extPower(2, phi)
│ │ │ │ - -- .604017s elapsed
│ │ │ │ + -- .364216s elapsed
│ │ │ │  
│ │ │ │            3                             6
│ │ │ │  o8 = 1 : Q  <------------------------- Q  : 1
│ │ │ │                 {1} | a b 0 0 0  0  |
│ │ │ │                 {1} | 0 0 0 b 0  0  |
│ │ │ │                 {2} | 0 0 0 0 ab b2 |
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_exterior__Inclusion.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │  
│ │ │  o2 : Complex
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .210615s elapsed
        │ │ │ + -- .193978s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : isWellDefined phi
│ │ │  
│ │ │  o4 = true
        │ │ │ @@ -278,15 +278,15 @@ │ │ │ │ │ │ o10 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
│ │ │ - -- .21521s elapsed
        │ │ │ + -- .188879s elapsed │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i12 : isWellDefined phi
│ │ │  
│ │ │  o12 = true
        │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ 1 3 3 1 │ │ │ │ o2 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o2 : Complex │ │ │ │ i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ │ - -- .210615s elapsed │ │ │ │ + -- .193978s elapsed │ │ │ │ i4 : isWellDefined phi │ │ │ │ │ │ │ │ o4 = true │ │ │ │ i5 : isCommutative phi │ │ │ │ │ │ │ │ o5 = true │ │ │ │ i6 : prune coker phi │ │ │ │ @@ -176,15 +176,15 @@ │ │ │ │ 1 3 3 1 │ │ │ │ o10 = Q <-- Q <-- Q <-- Q │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o10 : Complex │ │ │ │ i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3 │ │ │ │ - -- .21521s elapsed │ │ │ │ + -- .188879s elapsed │ │ │ │ i12 : isWellDefined phi │ │ │ │ │ │ │ │ o12 = true │ │ │ │ i13 : isCommutative phi │ │ │ │ │ │ │ │ o13 = true │ │ │ │ i14 : for i to 2 list prune HH_i source phi │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_forget__Degeneracy_lp__Simplicial__Module_rp.html │ │ │ @@ -109,15 +109,15 @@ │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : elapsedTime S**S
│ │ │ - -- .487245s elapsed
│ │ │ + -- .296389s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o4 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o4 : SimplicialModule
        │ │ │ @@ -134,15 +134,15 @@ │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i6 : elapsedTime fS**fS --faster when degeneracy is ignored
│ │ │ - -- .463629s elapsed
│ │ │ + -- .253896s elapsed
│ │ │  
│ │ │        1      25      225      1225      4900      15876      44100
│ │ │  o6 = Q  <-- Q   <-- Q    <-- Q     <-- Q     <-- Q      <-- Q     <-- ...
│ │ │                                                               
│ │ │       0      1       2        3         4         5          6
│ │ │  
│ │ │  o6 : SimplicialModule
        │ │ │ ├── html2text {} │ │ │ │ @@ -36,15 +36,15 @@ │ │ │ │ 1 5 15 35 70 126 210 │ │ │ │ o3 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o3 : SimplicialModule │ │ │ │ i4 : elapsedTime S**S │ │ │ │ - -- .487245s elapsed │ │ │ │ + -- .296389s elapsed │ │ │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ │ o4 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o4 : SimplicialModule │ │ │ │ @@ -53,15 +53,15 @@ │ │ │ │ 1 5 15 35 70 126 210 │ │ │ │ o5 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o5 : SimplicialModule │ │ │ │ i6 : elapsedTime fS**fS --faster when degeneracy is ignored │ │ │ │ - -- .463629s elapsed │ │ │ │ + -- .253896s elapsed │ │ │ │ │ │ │ │ 1 25 225 1225 4900 15876 44100 │ │ │ │ o6 = Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- Q <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o6 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_normalize_lp__Simplicial__Module_cm__Z__Z_rp.html │ │ │ @@ -237,28 +237,28 @@ │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i11 : elapsedTime prune normalize S10
│ │ │ - -- 4.94576s elapsed
│ │ │ + -- 3.20219s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o11 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o11 : Complex
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times slower; becomes significant for larger ranks
│ │ │ - -- 8.29592s elapsed
│ │ │ + -- 6.03003s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o12 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o12 : Complex
        │ │ │ @@ -268,15 +268,15 @@ │ │ │
        │ │ │

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

        │ │ │
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
        │ │ │ 
        i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER!
│ │ │ - -- .0571616s elapsed
│ │ │ + -- .0321265s elapsed
│ │ │  
│ │ │         10      30      30      10
│ │ │  o13 = R   <-- R   <-- R   <-- R
│ │ │                                 
│ │ │        0       1       2       3
│ │ │  
│ │ │  o13 : Complex
        │ │ │ ├── html2text {} │ │ │ │ @@ -156,38 +156,38 @@ │ │ │ │ │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 7 8 │ │ │ │ 9 10 │ │ │ │ │ │ │ │ o10 : SimplicialModule │ │ │ │ i11 : elapsedTime prune normalize S10 │ │ │ │ - -- 4.94576s elapsed │ │ │ │ + -- 3.20219s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o11 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o11 : Complex │ │ │ │ i12 : elapsedTime prune normalize(S10, CheckSum => false) --about 3-4 times │ │ │ │ slower; becomes significant for larger ranks │ │ │ │ - -- 8.29592s elapsed │ │ │ │ + -- 6.03003s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o12 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o12 : Complex │ │ │ │ The user may also specify the top homological degree to compute the │ │ │ │ normalization up to. Note that this can help speed up computational time; if │ │ │ │ the user knows the normalization should have a shorter length, then they should │ │ │ │ specify this upper bound in the syntax: │ │ │ │ i13 : elapsedTime prune normalize(S10, 3, CheckSum => false) --MUCH FASTER! │ │ │ │ - -- .0571616s elapsed │ │ │ │ + -- .0321265s elapsed │ │ │ │ │ │ │ │ 10 30 30 10 │ │ │ │ o13 = R <-- R <-- R <-- R │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o13 : Complex │ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_schur__Map.html │ │ │ @@ -110,28 +110,28 @@ │ │ │ │ │ │ o3 : SimplicialModule │ │ │
        │ │ │ 
        i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ - -- .142323s elapsed
│ │ │ + -- .114875s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o4 : SimplicialModule
        │ │ │
        │ │ │ 
        i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ - -- .309043s elapsed
│ │ │ + -- .253149s elapsed
│ │ │  
│ │ │        1      6      21      55      120
│ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │                                      
│ │ │       0      1      2       3       4
│ │ │  
│ │ │  o5 : SimplicialModule
        │ │ │ @@ -208,15 +208,15 @@ │ │ │         		│ │ │ 
        i11 : K = koszulComplex vars Q;
        │ │ │
        │ │ │ 
        i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ - -- 1.00439s elapsed
│ │ │ + -- .760079s elapsed
│ │ │  
│ │ │         1      9      36      74      81
│ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │                                       
│ │ │        0      1      2       3       4
│ │ │  
│ │ │  o12 : Complex
        │ │ │ @@ -470,15 +470,15 @@ │ │ │ │ │ │ o30 : ComplexMap │ │ │
        │ │ │ 
        i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ - -- .912527s elapsed
│ │ │ + -- .81818s elapsed
│ │ │  
│ │ │             1             1
│ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │                  | 1 |
│ │ │  
│ │ │             5                                       9
│ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ │ ├── html2text {}
│ │ │ │ @@ -42,24 +42,24 @@
│ │ │ │        1      3      6      10      15
│ │ │ │  o3 = Q  <-- Q  <-- Q  <-- Q   <-- Q  <-- ...
│ │ │ │  
│ │ │ │       0      1      2      3       4
│ │ │ │  
│ │ │ │  o3 : SimplicialModule
│ │ │ │  i4 : S2S = elapsedTime schurMap({2}, S)
│ │ │ │ - -- .142323s elapsed
│ │ │ │ + -- .114875s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o4 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o4 : SimplicialModule
│ │ │ │  i5 : elapsedTime schurMap({2}, S, Degeneracy => true)
│ │ │ │ - -- .309043s elapsed
│ │ │ │ + -- .253149s elapsed
│ │ │ │  
│ │ │ │        1      6      21      55      120
│ │ │ │  o5 = Q  <-- Q  <-- Q   <-- Q   <-- Q   <-- ...
│ │ │ │  
│ │ │ │       0      1      2       3       4
│ │ │ │  
│ │ │ │  o5 : SimplicialModule
│ │ │ │ @@ -106,15 +106,15 @@
│ │ │ │  i10 : Q = ZZ/101[a..c]
│ │ │ │  
│ │ │ │  o10 = Q
│ │ │ │  
│ │ │ │  o10 : PolynomialRing
│ │ │ │  i11 : K = koszulComplex vars Q;
│ │ │ │  i12 : S2K = elapsedTime prune schurMap({2}, K, TopDegree => 4)
│ │ │ │ - -- 1.00439s elapsed
│ │ │ │ + -- .760079s elapsed
│ │ │ │  
│ │ │ │         1      9      36      74      81
│ │ │ │  o12 = Q  <-- Q  <-- Q   <-- Q   <-- Q
│ │ │ │  
│ │ │ │        0      1      2       3       4
│ │ │ │  
│ │ │ │  o12 : Complex
│ │ │ │ @@ -299,15 +299,15 @@
│ │ │ │  
│ │ │ │             1                   2
│ │ │ │        2 : Q  <--------------- Q  : 2
│ │ │ │                  {2} | 0 b |
│ │ │ │  
│ │ │ │  o30 : ComplexMap
│ │ │ │  i31 : f = elapsedTime prune schurMap({2}, phi)
│ │ │ │ - -- .912527s elapsed
│ │ │ │ + -- .81818s elapsed
│ │ │ │  
│ │ │ │             1             1
│ │ │ │  o31 = 0 : Q  <--------- Q  : 0
│ │ │ │                  | 1 |
│ │ │ │  
│ │ │ │             5                                       9
│ │ │ │        1 : Q  <----------------------------------- Q  : 1
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/_tensor__L__E__S.html
│ │ │ @@ -166,15 +166,15 @@
│ │ │  
│ │ │  o7 : Complex
        │ │ │
        │ │ │ 
        i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ - -- 2.28436s elapsed
│ │ │ + -- 1.57652s elapsed
│ │ │  
│ │ │                                                                                                                                                                                                                                                                                                     25      50      25
│ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b | <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0 | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-- cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ │                                                                      | 0 b a |              | 0 b a |              | 0 0 b a |              | 0 0 b a 0 0 0 0 |              | 0   0   0 0 b a 0  0  |                                                                                                              
│ │ │       0                    1                    2     3                                                                                     | 0 0 0 0 b a 0 0 |              | ab2 a2b 0 0 0 0 b2 a2 |     9                              10                        11                             12      13      14
│ │ │                                                             4                      5                      6                                 | 0 0 0 0 0 0 b a |      
│ │ │                                                                                                                                                                     8
│ │ │ ├── html2text {}
│ │ │ │ @@ -114,15 +114,15 @@
│ │ │ │        1      2      2      1
│ │ │ │  o7 = Q  <-- Q  <-- Q  <-- Q
│ │ │ │  
│ │ │ │       0      1      2      3
│ │ │ │  
│ │ │ │  o7 : Complex
│ │ │ │  i8 : hL = elapsedTime prune tensorLES(L,4)
│ │ │ │ - -- 2.28436s elapsed
│ │ │ │ + -- 1.57652s elapsed
│ │ │ │  
│ │ │ │  
│ │ │ │  25      50      25
│ │ │ │  o8 = cokernel | b a | <-- cokernel | b a | <-- 0 <-- 0 <-- cokernel | a 0 b |
│ │ │ │  <-- cokernel | a 0 b | <-- cokernel | b a 0 0 | <-- cokernel | b a 0 0 0 0 0 0
│ │ │ │  | <-- cokernel | 0   0   b a 0 0 0  0  | <-- cokernel | ab2 a2b b2 a2 | <-
│ │ │ │  - cokernel | a2 b2 ab | <-- cokernel | ab2 a2b b2 a2 | <-- Q   <-- Q   <-- Q
│ │ ├── ./usr/share/doc/Macaulay2/SimplicialModules/html/index.html
│ │ │ @@ -211,28 +211,28 @@
│ │ │  
│ │ │  o7 : SimplicialModule
        │ │ │
        │ │ │ 
        i8 : elapsedTime simplicialModule(K,6) --specify top degree 6
│ │ │ - -- .0671636s elapsed
│ │ │ + -- .0659143s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o8 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o8 : SimplicialModule
        │ │ │
        │ │ │ 
        i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true)
│ │ │ - -- .195242s elapsed
│ │ │ + -- .145094s elapsed
│ │ │  
│ │ │        1      4      10      20      35      56      84
│ │ │  o9 = R  <-- R  <-- R   <-- R   <-- R   <-- R   <-- R  <-- ...
│ │ │                                                      
│ │ │       0      1      2       3       4       5       6
│ │ │  
│ │ │  o9 : SimplicialModule
        │ │ │ ├── html2text {} │ │ │ │ @@ -135,24 +135,24 @@ │ │ │ │ 1 4 10 20 │ │ │ │ o7 = R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 │ │ │ │ │ │ │ │ o7 : SimplicialModule │ │ │ │ i8 : elapsedTime simplicialModule(K,6) --specify top degree 6 │ │ │ │ - -- .0671636s elapsed │ │ │ │ + -- .0659143s elapsed │ │ │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ │ o8 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o8 : SimplicialModule │ │ │ │ i9 : elapsedTime S' = simplicialModule(K,6, Degeneracy => true) │ │ │ │ - -- .195242s elapsed │ │ │ │ + -- .145094s elapsed │ │ │ │ │ │ │ │ 1 4 10 20 35 56 84 │ │ │ │ o9 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- ... │ │ │ │ │ │ │ │ 0 1 2 3 4 5 6 │ │ │ │ │ │ │ │ o9 : SimplicialModule │ │ ├── ./usr/share/doc/Macaulay2/SimplicialPosets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ aXNCb29sZWFu │ │ │ #:len=1246 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiRGV0ZXJtaW5lIGlmIGEgcG9zZXQgaXMg │ │ │ YSBib29sZWFuIGFsZ2VicmEuIiwgImxpbmVudW0iID0+IDMzNywgSW5wdXRzID0+IHtTUEFOe1RU │ │ ├── ./usr/share/doc/Macaulay2/SlackIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=14 │ │ │ dW5pdmVyc2FsSWRlYWw= │ │ │ #:len=2182 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZXMgdGhlIHVuaXZlcnNhbCBy │ │ │ ZWFsaXphdGlvbiBpZGVhbCBvZiBhIG1hdHJvaWQiLCAibGluZW51bSIgPT4gMjE0MiwgSW5wdXRz │ │ ├── ./usr/share/doc/Macaulay2/SpaceCurves/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ aXNTbW9vdGhBQ01CZXR0aQ== │ │ │ #:len=902 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSBCZXR0aSB0 │ │ │ YWJsZSBpcyB0aGF0IG9mIGEgc21vb3RoIEFDTSBjdXJ2ZSIsICJsaW5lbnVtIiA9PiAxMjY5LCBJ │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=47 │ │ │ TXVsdGlkaW1lbnNpb25hbE1hdHJpeCAqIE11bHRpZGltZW5zaW9uYWxNYXRyaXg= │ │ │ #:len=1779 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicHJvZHVjdCBvZiBtdWx0aWRpbWVuc2lv │ │ │ bmFsIG1hdHJpY2VzIiwgImxpbmVudW0iID0+IDE0MjUsIElucHV0cyA9PiB7U1BBTntUVHsiTSJ9 │ │ ├── ./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 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc) │ │ │ + -- used 7.133e-05s (cpu); 6.4259e-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 0.0324525s (cpu); 0.0316018s (thread); 0s (gc) │ │ │ + -- used 0.164129s (cpu); 0.0466131s (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 0.44512s (cpu); 0.234802s (thread); 0s (gc) │ │ │ + -- used 0.472677s (cpu); 0.256092s (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 0.0911372s (cpu); 0.0911356s (thread); 0s (gc) │ │ │ + -- used 0.101762s (cpu); 0.101761s (thread); 0s (gc) │ │ │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula │ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc) │ │ │ + -- used 0.345564s (cpu); 0.29129s (thread); 0s (gc) │ │ │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc) │ │ │ + -- used 0.360652s (cpu); 0.313897s (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 0.0864658s (cpu); 0.0851857s (thread); 0s (gc) │ │ │ + -- used 0.350563s (cpu); 0.149678s (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 0.778868s (cpu); 0.689399s (thread); 0s (gc) │ │ │ + -- used 0.455589s (cpu); 0.455453s (thread); 0s (gc) │ │ │ │ │ │ o4 = 912984499996938980479447727885644530753184525786986940737407301278806287 │ │ │ 9257139493926586400187927813888 │ │ │ │ │ │ i5 : │ │ ├── ./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 3.23258s (cpu); 2.80646s (thread); 0s (gc) │ │ │ + -- used 2.46172s (cpu); 2.16308s (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 0.477656s (cpu); 0.408514s (thread); 0s (gc) │ │ │ + -- used 0.674574s (cpu); 0.538203s (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 0.00964993s (cpu); 0.00964869s (thread); 0s (gc) │ │ │ + -- used 0.00956928s (cpu); 0.00956909s (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 0.0310848s (cpu); 0.0302486s (thread); 0s (gc) │ │ │ + -- used 0.220082s (cpu); 0.0638006s (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 0.00326741s (cpu); 0.00326931s (thread); 0s (gc) │ │ │ + -- used 0.00394602s (cpu); 0.00394586s (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 0.277694s (cpu); 0.202525s (thread); 0s (gc) │ │ │ + -- used 0.362597s (cpu); 0.210553s (thread); 0s (gc) │ │ │ │ │ │ i13 : quotientRemainder(UnmixedRes,MixedRes) │ │ │ │ │ │ 2 2 2 2 2 2 │ │ │ o13 = (b c - b b c c + b b c + b c c - 2b b c c - b b c c + b c , 0) │ │ │ 5 2 4 5 2 4 2 5 4 4 2 5 2 5 2 5 2 4 4 5 2 5 │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_degree__Determinant.html │ │ │ @@ -81,15 +81,15 @@ │ │ │ │ │ │ o1 : List │ │ │
        │ │ │ 
        i2 : time degreeDeterminant n
│ │ │ - -- used 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc)
│ │ │ + -- used 7.133e-05s (cpu); 6.4259e-05s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 6
        │ │ │
        │ │ │ 
        i3 : M = genericMultidimensionalMatrix n;
│ │ │ @@ -103,15 +103,15 @@
│ │ │  o3 : 3-dimensional matrix of shape 2 x 3 x 2 over ZZ[a     ..a     ]
│ │ │                                                        0,0,0   1,2,1
        │ │ │
        │ │ │ 
        i4 : time degree determinant M
│ │ │ - -- used 0.0324525s (cpu); 0.0316018s (thread); 0s (gc)
│ │ │ + -- used 0.164129s (cpu); 0.0466131s (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 9.3134e-05s (cpu); 8.2705e-05s (thread); 0s (gc) │ │ │ │ + -- used 7.133e-05s (cpu); 6.4259e-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 0.0324525s (cpu); 0.0316018s (thread); 0s (gc) │ │ │ │ + -- used 0.164129s (cpu); 0.0466131s (thread); 0s (gc) │ │ │ │ │ │ │ │ o4 = {6} │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _d_e_t_e_r_m_i_n_a_n_t_(_M_u_l_t_i_d_i_m_e_n_s_i_o_n_a_l_M_a_t_r_i_x_) -- hyperdeterminant of a │ │ │ │ multidimensional matrix │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Discriminant.html │ │ │ @@ -85,15 +85,15 @@ │ │ │ │ │ │ 
        i1 : (d,n) := (2,3);
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time Disc = denseDiscriminant(d,n)
│ │ │ - -- used 0.44512s (cpu); 0.234802s (thread); 0s (gc)
│ │ │ + -- used 0.472677s (cpu); 0.256092s (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 0.44512s (cpu); 0.234802s (thread); 0s (gc) │ │ │ │ + -- used 0.472677s (cpu); 0.256092s (thread); 0s (gc) │ │ │ │ │ │ │ │ o2 = Disc │ │ │ │ │ │ │ │ o2 : SparseDiscriminant (sparse discriminant associated to | 0 0 0 0 0 0 1 1 1 │ │ │ │ 2 |) │ │ │ │ | 0 0 0 1 1 2 0 0 1 │ │ │ │ 0 | │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_dense__Resultant.html │ │ │ @@ -95,27 +95,27 @@ │ │ │ │ │ │ o1 : Sequence │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time denseResultant(f0,f1,f2); -- using Poisson formula
│ │ │ - -- used 0.0911372s (cpu); 0.0911356s (thread); 0s (gc)
        │ │ │ + -- used 0.101762s (cpu); 0.101761s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay formula
│ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc)
        │ │ │ + -- used 0.345564s (cpu); 0.29129s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant
│ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc)
        │ │ │ + -- used 0.360652s (cpu); 0.313897s (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 0.0911372s (cpu); 0.0911356s (thread); 0s (gc) │ │ │ │ + -- used 0.101762s (cpu); 0.101761s (thread); 0s (gc) │ │ │ │ i3 : time denseResultant(f0,f1,f2,Algorithm=>"Macaulay"); -- using Macaulay │ │ │ │ formula │ │ │ │ - -- used 0.298973s (cpu); 0.23937s (thread); 0s (gc) │ │ │ │ + -- used 0.345564s (cpu); 0.29129s (thread); 0s (gc) │ │ │ │ i4 : time (denseResultant(1,2,2)) (f0,f1,f2); -- using sparseResultant │ │ │ │ - -- used 0.382249s (cpu); 0.323056s (thread); 0s (gc) │ │ │ │ + -- used 0.360652s (cpu); 0.313897s (thread); 0s (gc) │ │ │ │ i5 : assert(o2 == o3 and o3 == o4) │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _s_p_a_r_s_e_R_e_s_u_l_t_a_n_t -- sparse resultant (A-resultant) │ │ │ │ * _a_f_f_i_n_e_R_e_s_u_l_t_a_n_t -- affine resultant │ │ │ │ * _d_e_n_s_e_D_i_s_c_r_i_m_i_n_a_n_t -- dense discriminant (classical discriminant) │ │ │ │ * _e_x_p_o_n_e_n_t_s_M_a_t_r_i_x -- exponents in one or more polynomials │ │ │ │ * _g_e_n_e_r_i_c_L_a_u_r_e_n_t_P_o_l_y_n_o_m_i_a_l_s -- generic (Laurent) polynomials │ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_determinant_lp__Multidimensional__Matrix_rp.html │ │ │ @@ -89,15 +89,15 @@ │ │ │ │ │ │ o1 : 4-dimensional matrix of shape 2 x 2 x 2 x 2 over ZZ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i2 : time det M
│ │ │ - -- used 0.0864658s (cpu); 0.0851857s (thread); 0s (gc)
│ │ │ + -- used 0.350563s (cpu); 0.149678s (thread); 0s (gc)
│ │ │  
│ │ │  o2 = 9698337990421512192
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i3 : M = randomMultidimensionalMatrix(2,2,2,2,5)
│ │ │ @@ -114,15 +114,15 @@
│ │ │  
│ │ │  o3 : 5-dimensional matrix of shape 2 x 2 x 2 x 2 x 5 over ZZ
        │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
        i4 : time det M
│ │ │ - -- used 0.778868s (cpu); 0.689399s (thread); 0s (gc)
│ │ │ + -- used 0.455589s (cpu); 0.455453s (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 0.0864658s (cpu); 0.0851857s (thread); 0s (gc) │ │ │ │ + -- used 0.350563s (cpu); 0.149678s (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 0.778868s (cpu); 0.689399s (thread); 0s (gc) │ │ │ │ + -- used 0.455589s (cpu); 0.455453s (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/_sparse__Discriminant.html │ │ │ @@ -95,15 +95,15 @@ │ │ │ o1 : ZZ[a ..a ][x ..x , y ..y , z ..z ] │ │ │ 0,0,0 1,2,1 0 1 0 2 0 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
      i2 : time sparseDiscriminant f
│ │ │ - -- used 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ + -- used 2.46172s (cpu); 2.16308s (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 3.23258s (cpu); 2.80646s (thread); 0s (gc)
│ │ │ │ + -- used 2.46172s (cpu); 2.16308s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                                                     2
│ │ │ │  o2 = a     a     a     a     a     a      - a     a     a     a     a      -
│ │ │ │        0,1,1 0,2,0 0,2,1 1,0,0 1,0,1 1,1,0    0,1,0 0,2,1 1,0,0 1,0,1 1,1,0
│ │ │ │       ------------------------------------------------------------------------
│ │ │ │              2     2                                2
│ │ │ │       a     a     a     a      + a     a     a     a     a      -
│ │ ├── ./usr/share/doc/Macaulay2/SparseResultants/html/_sparse__Resultant.html
│ │ │ @@ -79,15 +79,15 @@
│ │ │          
      Description
      
│ │ │          
      Alternatively, one can apply the method directly to the list of Laurent polynomials $f_0,\ldots,f_n$. In this case, the matrices $A_0,\ldots,A_n$ are automatically determined by exponentsMatrix. If you want require that $A_0=\cdots=A_n$, then use the option Unmixed=>true (this could be faster). Below we consider some examples.
      
│ │ │          
      In the first example, we calculate the sparse (mixed) resultant associated to the three sets of monomials $(1,x y,x^2 y,x),(y,x^2 y^2,x^2 y,x),(1,y,x y,x)$. Then we evaluate it at the three polynomials $f = c_{(1,1)}+c_{(1,2)} x y+c_{(1,3)} x^2 y+c_{(1,4)} x, g = c_{(2,1)} y+c_{(2,2)} x^2 y^2+c_{(2,3)} x^2 y+c_{(2,4)} x, h = c_{(3,1)}+c_{(3,2)} y+c_{(3,3)} x y+c_{(3,4)} x$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -109,15 +109,15 @@
│ │ │  
│ │ │  o3 : Sequence
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │          
      
│ │ │                
      i1 : time Res = sparseResultant(matrix{{0,1,1,2},{0,0,1,1}},matrix{{0,1,2,2},{1,0,1,2}},matrix{{0,0,1,1},{0,1,0,1}})
│ │ │ - -- used 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ + -- used 0.674574s (cpu); 0.538203s (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 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ + -- used 0.00956928s (cpu); 0.00956909s (thread); 0s (gc)
│ │ │  
│ │ │          2                       4      2   2               4    
│ │ │  o4 = - c   c   c   c   c   c   c    + c   c   c   c   c   c    +
│ │ │          1,2 1,3 1,4 2,1 2,2 2,3 3,1    1,2 1,3 2,1 2,2 2,4 3,1  
│ │ │       ------------------------------------------------------------------------
│ │ │        3       2       3               2                   3        
│ │ │       c   c   c   c   c   c    - 2c   c   c   c   c   c   c   c    +
│ │ │ @@ -830,15 +830,15 @@
│ │ │            
      
│ │ │          
      In the second example, we calculate the sparse unmixed resultant associated to the set of monomials $(1,x,y,xy)$. Then we evaluate it at the three polynomials $f = a_0 + a_1 x + a_2 y + a_3 x y, g = b_0 + b_1 x + b_2 y + b_3 x y, h = c_0 + c_1 x + c_2 y + c_3 x y$. Moreover, we perform all the computation over $\mathbb{Z}/3331$.
      
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ ├── html2text {}
│ │ │ │ @@ -68,15 +68,15 @@
│ │ │ │  
│ │ │ │  o4 = | 0 1 |
│ │ │ │       | 2 3 |
│ │ │ │       | 4 |
│ │ │ │  
│ │ │ │  o4 : YoungTableau
│ │ │ │  i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ │ + -- used 0.00144178s (cpu); 0.00143975s (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 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ │ + -- used 0.00130489s (cpu); 0.00130481s (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 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ │ + -- used 0.00197823s (cpu); 0.00198046s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 = (- x  + x )(- x  + x )(- x  + x )(- x  + x )((x  + x  + x )(x )(x ) + (x )
│ │ │ │  (x )(x ))
│ │ │ │           0    2     0    4     2    4     1    3    0    2    4   3   1      4
│ │ │ │  2   0
│ │ │ │  
│ │ │ │  o7 : Expression of class Product
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_representation__Multiplicity.html
│ │ │ @@ -131,15 +131,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -116,15 +116,15 @@
│ │ │  
│ │ │  o4 : ProjectiveVariety, a point in PP^5
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │ @@ -126,15 +126,15 @@
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -22,15 +22,15 @@
│ │ │ │  2*x_2*x_4-x_3*x_4-
│ │ │ │  2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-
│ │ │ │  3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and
│ │ │ │  sectional genus 0
│ │ │ │  i3 : time toGrass X
│ │ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ │ + -- used 4.14566s (cpu); 2.40827s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces
│ │ │ │  of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass_lp__Embedded__Projective__Variety_rp.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │ ├── html2text {}
│ │ │ │ @@ -25,15 +25,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, 5-dimensional subvariety of PP^8
│ │ │ │  i3 : time toGrass X
│ │ │ │  warning: clearing value of symbol x to allow access to subscripted variables
│ │ │ │  based on it
│ │ │ │         : debug with expression   debug 9868   or with command line option   --
│ │ │ │  debug 9868
│ │ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ │ + -- used 4.51342s (cpu); 2.61054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │ │  
│ │ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_unirational__Parametrization.html
│ │ │ @@ -87,15 +87,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
      
│ │ │                
      i6 : time Res = sparseResultant(matrix{{0,0,1,1},{0,1,0,1}},CoefficientRing=>ZZ/3331);
│ │ │ - -- used 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ + -- used 0.220082s (cpu); 0.0638006s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : SparseResultant (sparse unmixed resultant associated to | 0 0 1 1 | over ZZ/3331)
│ │ │                                                               | 0 1 0 1 |
      
│ │ │              
      
│ │ │ @@ -854,15 +854,15 @@
│ │ │  
│ │ │  o8 : Sequence
│ │ │              
      
│ │ │                
      i9 : time Res(f,g,h)
│ │ │ - -- used 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ + -- used 0.00394602s (cpu); 0.00394586s (thread); 0s (gc)
│ │ │  
│ │ │        2     2            2            2        2 2    2          
│ │ │  o9 = a b b c  - a a b b c  - a a b b c  + a a b c  - a b b c c  -
│ │ │        3 1 2 0    2 3 1 3 0    1 3 2 3 0    1 2 3 0    3 0 2 0 1  
│ │ │       ------------------------------------------------------------------------
│ │ │                           2                       2                         
│ │ │       a a b b c c  + a a b c c  + a a b b c c  + a b b c c  - a a b b c c  +
│ │ │ @@ -943,15 +943,15 @@
│ │ │  
│ │ │  o11 : Sequence
      
│ │ │              
      
│ │ │                
      i12 : time (MixedRes,UnmixedRes) = (sparseResultant(f,g,h),sparseResultant(f,g,h,Unmixed=>true));
│ │ │ - -- used 0.277694s (cpu); 0.202525s (thread); 0s (gc)
      
│ │ │ + -- used 0.362597s (cpu); 0.210553s (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 0.477656s (cpu); 0.408514s (thread); 0s (gc)
│ │ │ │ + -- used 0.674574s (cpu); 0.538203s (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 0.00964993s (cpu); 0.00964869s (thread); 0s (gc)
│ │ │ │ + -- used 0.00956928s (cpu); 0.00956909s (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 0.0310848s (cpu); 0.0302486s (thread); 0s (gc)
│ │ │ │ + -- used 0.220082s (cpu); 0.0638006s (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 0.00326741s (cpu); 0.00326931s (thread); 0s (gc)
│ │ │ │ + -- used 0.00394602s (cpu); 0.00394586s (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 0.277694s (cpu); 0.202525s (thread); 0s (gc)
│ │ │ │ + -- used 0.362597s (cpu); 0.210553s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=38
│ │ │  c2NodXJQb2x5bm9taWFsKC4uLixBc0V4cHJlc3Npb249Pi4uLik=
│ │ │  #:len=287
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzMwNSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2NodXJQb2x5bm9taWFsLEFzRXhwcmVzc2lvbl0s
│ │ ├── ./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 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ + -- used 0.00144178s (cpu); 0.00143975s (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 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ + -- used 0.00130489s (cpu); 0.00130481s (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 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ + -- used 0.00197823s (cpu); 0.00198046s (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 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ + -- used 0.398034s (cpu); 0.302173s (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 0.815181s (cpu); 0.539564s (thread); 0s (gc)
│ │ │ + -- used 0.835955s (cpu); 0.575786s (thread); 0s (gc)
│ │ │  
│ │ │  i4 : seco#(new Partition from {2,2,2})
│ │ │  
│ │ │                                                        2 2 2       4 2   2     2   2 2     2 2     2   4   2   2   2     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2 2       2 2   2     1   2 2     1 2 2       2 2 2       1 2 2       1 2   2     2   2 2     1 2   2     2 2   2     1   2 2     1   2 2     2 2 2       4 2 2       2 2 2       2 2 2       4 2 2       2 2 2       1 2   2     1   2 2     2 2   2     1 2   2     2   2 2     1   2 2     1 2   2     2 2   2     1 2   2     1   2 2     2   2 2     1   2 2     1 2     2   2   2   2   1     2 2   2 2     2   1   2   2   1 2     2   1 2     2   1   2   2   2   2   2   1     2 2   2     2 2   1     2 2   1 2     2   1   2   2   2 2     2   1 2     2   2   2   2   1   2   2   1 2     2   2 2     2   1 2     2   1   2   2   2   2   2   1   2   2   2     2 2   4     2 2   2     2 2   2     2 2   4     2 2   2     2 2
│ │ │  o4 = HashTable{{0, 1, 2, 3, 4, 5} => HashTable{0 => - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  + -x x x x  - -x x x x  - -x x x x  + -x x x x  - -x x x x }                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        }
│ │ │                                                        3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 4   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 5   3 1 2 3 5   3 1 2 3 5   3 1 2 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 4 5   3 1 3 4 5   3 2 3 4 5   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 3 6   3 1 2 3 6   3 1 2 3 6   3 1 2 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 4 6   3 1 3 4 6   3 2 3 4 6   3 1 2 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6   3 1 2 5 6   3 1 3 5 6   3 2 3 5 6   3 1 4 5 6   3 2 4 5 6   3 3 4 5 6
│ │ │                                                      2 3 2 2     4 2 3 2     2 2 2 3     4 3 2   2   2 2 3   2   2 3   2 2   2   3 2 2   2 2   3 2   4   2 3 2   2 2 2   3   4 2   2 3   2   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   1 3 2 2     2 2 3 2     1 2 2 3     2 3 2 2     1 2 3 2     1 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 2 3     2 2 2 3     1 2 2 3     1 3 2 2     1 2 3 2     2 3 2 2     1 3 2 2     2 2 3 2     1 2 3 2     1 3 2 2     2 3 2 2     1 3 2 2     1 2 3 2     2 2 3 2     1 2 3 2     2 2 2 3     4 2 2 3     2 2 2 3     2 2 2 3     4 2 2 3     2 2 2 3     2 3 2   2   1 2 3   2   1 3   2 2   1   3 2 2   1 2   3 2   2   2 3 2   1 3 2   2   2 2 3   2   1 3 2   2   2 3 2   2   1 2 3   2   1 2 3   2   1 3   2 2   1   3 2 2   2 3   2 2   1 3   2 2   2   3 2 2   1   3 2 2   2 2   3 2   1   2 3 2   1 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1 3 2   2   1 2 3   2   2 3 2   2   1 3 2   2   2 2 3   2   1 2 3   2   1 3 2   2   2 3 2   2   1 3 2   2   1 2 3   2   2 2 3   2   1 2 3   2   2 3   2 2   2   3 2 2   4 3   2 2   2 3   2 2   4   3 2 2   2   3 2 2   2 3   2 2   4 3   2 2   2 3   2 2   2   3 2 2   4   3 2 2   2   3 2 2   1 2   3 2   1   2 3 2   2 2   3 2   1 2   3 2   2   2 3 2   1   2 3 2   1 2   3 2   2 2   3 2   1 2   3 2   1   2 3 2   2   2 3 2   1   2 3 2   1 2 2   3   2 2   2 3   1   2 2 3   1 2 2   3   2 2 2   3   1 2 2   3   1 2   2 3   2   2 2 3   1 2   2 3   2 2   2 3   1   2 2 3   1   2 2 3   2 2 2   3   4 2 2   3   2 2 2   3   2 2 2   3   4 2 2   3   2 2 2   3   1 2   2 3   1   2 2 3   2 2   2 3   1 2   2 3   2   2 2 3   1   2 2 3   1 2   2 3   2 2   2 3   1 2   2 3   1   2 2 3   2   2 2 3   1   2 2 3
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_higher__Specht__Polynomial_lp__Young__Tableau_cm__Young__Tableau_cm__Polynomial__Ring_rp.html
│ │ │ @@ -130,15 +130,15 @@
│ │ │  
│ │ │  o4 : YoungTableau
      
│ │ │              
      
│ │ │                
      i5 : time higherSpechtPolynomial(S,T,R)
│ │ │ - -- used 0.00150749s (cpu); 0.00150181s (thread); 0s (gc)
│ │ │ + -- used 0.00144178s (cpu); 0.00143975s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o5 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -154,15 +154,15 @@
│ │ │  
│ │ │  o5 : R
      
│ │ │              
      
│ │ │                
      i6 : time higherSpechtPolynomial(S,T,R, Robust => false)
│ │ │ - -- used 0.0012075s (cpu); 0.00120561s (thread); 0s (gc)
│ │ │ + -- used 0.00130489s (cpu); 0.00130481s (thread); 0s (gc)
│ │ │  
│ │ │        3 2          2 3      3     2        3 2    3   2      2   3    
│ │ │  o6 = x x x x  - x x x x  - x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │        0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 3    0 1 2 4    0 1 2 4  
│ │ │       ------------------------------------------------------------------------
│ │ │        3 2        3 2        2 3        2 3        3   2        3 2    
│ │ │       x x x x  - x x x x  + x x x x  + x x x x  + x x x x  - x x x x  -
│ │ │ @@ -178,15 +178,15 @@
│ │ │  
│ │ │  o6 : R
      
│ │ │              
      
│ │ │                
      i7 : time higherSpechtPolynomial(S,T,R, Robust => false, AsExpression => true)
│ │ │ - -- used 0.00201494s (cpu); 0.00201272s (thread); 0s (gc)
│ │ │ + -- used 0.00197823s (cpu); 0.00198046s (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 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ + -- used 0.398034s (cpu); 0.302173s (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 0.425094s (cpu); 0.317058s (thread); 0s (gc)
│ │ │ │ + -- used 0.398034s (cpu); 0.302173s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = HashTable{Partition{1, 1, 1, 1, 1, 1} => 1}
│ │ │ │                 Partition{2, 1, 1, 1, 1} => 0
│ │ │ │                 Partition{2, 2, 1, 1} => 1
│ │ │ │                 Partition{2, 2, 2} => 1
│ │ │ │                 Partition{3, 1, 1, 1} => 0
│ │ │ │                 Partition{3, 2, 1} => 0
│ │ ├── ./usr/share/doc/Macaulay2/SpechtModule/html/_secondary__Invariants_lp__List_cm__Polynomial__Ring_rp.html
│ │ │ @@ -114,15 +114,15 @@
│ │ │  (Partition{3, 3}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{3, 2, 1}, Ambient_Dimension, 16, Rank, 0)
│ │ │  (Partition{3, 1, 1, 1}, Ambient_Dimension, 10, Rank, 0)
│ │ │  (Partition{2, 2, 2}, Ambient_Dimension, 5, Rank, 1)
│ │ │  (Partition{2, 2, 1, 1}, Ambient_Dimension, 9, Rank, 1)
│ │ │  (Partition{2, 1, 1, 1, 1}, Ambient_Dimension, 5, Rank, 0)
│ │ │  (Partition{1, 1, 1, 1, 1, 1}, Ambient_Dimension, 1, Rank, 1)
│ │ │ - -- used 0.815181s (cpu); 0.539564s (thread); 0s (gc)
      
│ │ │ + -- used 0.835955s (cpu); 0.575786s (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 0.815181s (cpu); 0.539564s (thread); 0s (gc)
│ │ │ │ + -- used 0.835955s (cpu); 0.575786s (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/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=34
│ │ │  ZGV0ZWN0Q29uZ3J1ZW5jZSguLi4sVmVyYm9zZT0+Li4uKQ==
│ │ │  #:len=296
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTUsIHN5bWJvbCBEb2N1bWVudFRhZyA9
│ │ │  PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7W2RldGVjdENvbmdydWVuY2UsVmVyYm9zZV0sImRldGVj
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__Castelnuovo__Surface.out
│ │ │ @@ -45,15 +45,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │       of discriminant 31 = det| 8 1 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -42,15 +42,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 0 seconds (cpu: 1 second)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -48,15 +48,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 5 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │  
│ │ │  i4 : describe X
│ │ │  
│ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │       containing a surface of degree 2 and sectional genus 0
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -8,28 +8,28 @@
│ │ │  i2 : describe X
│ │ │  
│ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ + -- used 2.77326s (cpu); 1.67129s (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 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ + -- used 0.388553s (cpu); 0.273321s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │  
│ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ + -- used 16.4469s (cpu); 6.55054s (thread); 0s (gc)
│ │ │  number lines contained in the image of the quadratic map and passing through a general point: 7
│ │ │  number 1-secant lines = 6
│ │ │  number 3-secant conics = 1
│ │ │  
│ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │  
│ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ @@ -29,15 +29,15 @@
│ │ │  i5 : p := point Y -- random point on Y
│ │ │  
│ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -12725, 1]
│ │ │  
│ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │  
│ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ + -- used 0.633925s (cpu); 0.361693s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │  
│ │ │  i7 : S = surface X;
│ │ │  
│ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, curve in PP^5
│ │ │  
│ │ │  i3 : X = random({{2},{2},{2}},S);
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ + -- used 0.368245s (cpu); 0.205643s (thread); 0s (gc)
│ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3 hypersurfaces of degree 2
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 32
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ │  dim GG(2,9) = 21
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Cubic__Fourfold_rp.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold V;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ + -- used 0.594558s (cpu); 0.353461s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.out
│ │ │ @@ -11,15 +11,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │  
│ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
│ │ │  
│ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ + -- used 3.10937s (cpu); 2.23547s (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 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ + -- used 0.823676s (cpu); 0.521774s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │       source variety: PP^4
│ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees 1^2 2^5 
│ │ │       dominance: true
│ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_polarized__K3surface.out
│ │ │ @@ -72,15 +72,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 11 seconds (cpu: 15 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -108,16 +108,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 4 seconds)
│ │ │ +-- total time (K3 surface + polarization): 14 seconds (cpu: 20 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -205,15 +205,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 5 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -226,15 +226,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 5 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_to__Grass.out
│ │ │ @@ -3,15 +3,15 @@
│ │ │  i1 : x := gens ring PP_(ZZ/33331)^8;
│ │ │  
│ │ │  i2 : X = gushelMukaiFourfold(ideal(x_6-x_7, x_5, x_3-x_4, x_1, x_0-x_4, x_2*x_7-x_4*x_8), ideal(x_4*x_6-x_3*x_7+x_1*x_8, x_4*x_5-x_2*x_7+x_0*x_8, x_3*x_5-x_2*x_6+x_0*x_8+x_1*x_8-x_5*x_8, x_1*x_5-x_0*x_6+x_0*x_7+x_1*x_7-x_5*x_7, x_1*x_2-x_0*x_3+x_0*x_4+x_1*x_4-x_2*x_7+x_0*x_8, x_0^2+x_0*x_1+x_1^2+x_0*x_2+2*x_0*x_3+x_1*x_3+x_2*x_3+x_3^2-x_0*x_4-x_1*x_4-2*x_2*x_4-x_3*x_4-2*x_4^2+x_0*x_5+x_2*x_5+x_5^2+2*x_0*x_6+x_1*x_6+2*x_2*x_6+x_3*x_6+x_5*x_6+x_6^2-3*x_4*x_7+2*x_5*x_7-x_7^2+x_1*x_8+x_3*x_8-3*x_4*x_8+2*x_5*x_8+x_6*x_8-x_7*x_8));
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
│ │ │  
│ │ │  i3 : time toGrass X
│ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ + -- used 4.14566s (cpu); 2.40827s (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 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ + -- used 4.51342s (cpu); 2.61054s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/example-output/_unirational__Parametrization.out
│ │ │ @@ -5,15 +5,15 @@
│ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │  
│ │ │  i3 : X = cubicFourfold S;
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
│ │ │  
│ │ │  i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ + -- used 0.901878s (cpu); 0.519995s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │  
│ │ │  i5 : degreeSequence f
│ │ │  
│ │ │  o5 = {[10]}
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__Castelnuovo__Surface.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │    -- top 1, degrees: 1^1 2^3 3^3 
│ │ │    -- top 2, degrees: 2^4 3^3 
│ │ │    -- top 3, degrees: 2^3 3^4 
│ │ │    -- top 4, degrees: 2^3 3^3 4^1 
│ │ │    -- top 5, degrees: 2^3 3^3 5^1 
│ │ │    -- top 6, degrees: 2^3 3^3 6^1 
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -80,15 +80,15 @@
│ │ │ │    -- top 1, degrees: 1^1 2^3 3^3
│ │ │ │    -- top 2, degrees: 2^4 3^3
│ │ │ │    -- top 3, degrees: 2^3 3^4
│ │ │ │    -- top 4, degrees: 2^3 3^3 4^1
│ │ │ │    -- top 5, degrees: 2^3 3^3 5^1
│ │ │ │    -- top 6, degrees: 2^3 3^3 6^1
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ associated Castelnuovo successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated Castelnuovo successfully completed in 1 second (cpu: 2 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, Castelnuovo surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Complete intersection of 3 quadrics in PP^7
│ │ │ │       of discriminant 31 = det| 8 1 |
│ │ │ │                               | 1 4 |
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -144,15 +144,15 @@
│ │ │  -- computing the top components of (U ∩ U')\{exceptional lines} via interpolation
│ │ │    -- top 1, degrees: 1^4 2^1 
│ │ │    -- top 2, degrees: 1^3 2^2 
│ │ │    -- top 3, degrees: 1^3 2^1 3^1 
│ │ │    -- top 4, degrees: 1^3 2^1 4^1 
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ + ✦ associated K3 successfully completed in 0 seconds (cpu: 1 second)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -79,15 +79,15 @@
│ │ │ │  interpolation
│ │ │ │    -- top 1, degrees: 1^4 2^1
│ │ │ │    -- top 2, degrees: 1^3 2^2
│ │ │ │    -- top 3, degrees: 1^3 2^1 3^1
│ │ │ │    -- top 4, degrees: 1^3 2^1 4^1
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │ - ✦ associated K3 successfully completed in 2 seconds (cpu: 1 second)
│ │ │ │ + ✦ associated K3 successfully completed in 0 seconds (cpu: 1 second)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special cubic fourfold of discriminant 14
│ │ │ │       containing a rational surface of degree 4 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degree 2
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_associated__K3surface_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -150,15 +150,15 @@
│ │ │    -- top 3, degrees: 1^1 2^4 3^2 
│ │ │    -- top 4, degrees: 1^1 2^4 4^2 
│ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │  -- computing normalization of the surface image
│ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 5 seconds)
│ │ │  
│ │ │  o3 : ProjectiveVariety, K3 surface associated to X
      
│ │ │              
      
│ │ │                
      i4 : describe X
│ │ │ ├── html2text {}
│ │ │ │ @@ -84,15 +84,15 @@
│ │ │ │    -- top 3, degrees: 1^1 2^4 3^2
│ │ │ │    -- top 4, degrees: 1^1 2^4 4^2
│ │ │ │  -- exceptional curves computed: obtained 2 line(s)
│ │ │ │  -- computing the map f from U to the minimal K3 surface
│ │ │ │  -- computing the image of f via 'F4' algorithm...
│ │ │ │  -- note: invariant mismatch for standard K3 surface
│ │ │ │  -- computing normalization of the surface image
│ │ │ │ - ✦ associated K3 successfully completed in 6 seconds (cpu: 6 seconds)
│ │ │ │ + ✦ associated K3 successfully completed in 4 seconds (cpu: 5 seconds)
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, K3 surface associated to X
│ │ │ │  i4 : describe X
│ │ │ │  
│ │ │ │  o4 = Special Gushel-Mukai fourfold of discriminant 10(')
│ │ │ │       containing a surface of degree 2 and sectional genus 0
│ │ │ │       cut out by 6 hypersurfaces of degrees 1^5 2^1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Cubic__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -96,15 +96,15 @@
│ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │       cut out by 13 hypersurfaces of degree 3
      
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ + -- used 2.77326s (cpu); 1.67129s (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 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ + -- used 0.388553s (cpu); 0.273321s (thread); 0s (gc)
│ │ │  
│ │ │  o5 : ProjectiveVariety, curve in PP^5
      
│ │ │              
      
│ │ │                
      i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree(C * surface X) == 5 and isSubset(p, C))
      
│ │ │ ├── html2text {}
│ │ │ │ @@ -29,28 +29,28 @@
│ │ │ │  sectional genus 0
│ │ │ │  i2 : describe X
│ │ │ │  
│ │ │ │  o2 = Special cubic fourfold of discriminant 26
│ │ │ │       containing a 3-nodal surface of degree 7 and sectional genus 0
│ │ │ │       cut out by 13 hypersurfaces of degree 3
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 3.79687s (cpu); 2.14089s (thread); 0s (gc)
│ │ │ │ + -- used 2.77326s (cpu); 1.67129s (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 0.325742s (cpu); 0.325011s (thread); 0s (gc)
│ │ │ │ + -- used 0.388553s (cpu); 0.273321s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, curve in PP^5
│ │ │ │  i6 : assert(dim C == 1 and degree C == 2 and dim(C * surface X) == 0 and degree
│ │ │ │  (C * surface X) == 5 and isSubset(p, C))
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _d_e_t_e_c_t_C_o_n_g_r_u_e_n_c_e_(_G_u_s_h_e_l_M_u_k_a_i_F_o_u_r_f_o_l_d_,_Z_Z_) -- detect and return a
│ │ │ │        congruence of (2e-1)-secant curves of degree e inside a del Pezzo
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_detect__Congruence_lp__Gushel__Mukai__Fourfold_cm__Z__Z_rp.html
│ │ │ @@ -99,15 +99,15 @@
│ │ │       Type: ordinary
│ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │              
      
│ │ │                
      i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ + -- used 16.4469s (cpu); 6.55054s (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 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ + -- used 0.633925s (cpu); 0.361693s (thread); 0s (gc)
│ │ │  
│ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
      
│ │ │              
      
│ │ │                
      i7 : S = surface X;
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  o2 = Special Gushel-Mukai fourfold of discriminant 20
│ │ │ │       containing a surface of degree 9 and sectional genus 2
│ │ │ │       cut out by 19 hypersurfaces of degree 2
│ │ │ │       and with class in G(1,4) given by 6*s_(3,1)+3*s_(2,2)
│ │ │ │       Type: ordinary
│ │ │ │       (case 17 of Table 1 in arXiv:2002.07026)
│ │ │ │  i3 : time f = detectCongruence(X,Verbose=>true);
│ │ │ │ - -- used 15.0623s (cpu); 7.94045s (thread); 0s (gc)
│ │ │ │ + -- used 16.4469s (cpu); 6.55054s (thread); 0s (gc)
│ │ │ │  number lines contained in the image of the quadratic map and passing through a
│ │ │ │  general point: 7
│ │ │ │  number 1-secant lines = 6
│ │ │ │  number 3-secant conics = 1
│ │ │ │  
│ │ │ │  o3 : Congruence of 3-secant conics to surface in a fivefold in PP^8
│ │ │ │  i4 : Y = ambientFivefold X; -- del Pezzo fivefold containing X
│ │ │ │ @@ -52,15 +52,15 @@
│ │ │ │  i5 : p := point Y -- random point on Y
│ │ │ │  
│ │ │ │  o5 = point of coordinates [14360, -1933, -494, -6471, -10457, -2246, -11879, -
│ │ │ │  12725, 1]
│ │ │ │  
│ │ │ │  o5 : ProjectiveVariety, a point in PP^8
│ │ │ │  i6 : time C = f p; -- 3-secant conic to the surface
│ │ │ │ - -- used 0.828054s (cpu); 0.576108s (thread); 0s (gc)
│ │ │ │ + -- used 0.633925s (cpu); 0.361693s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 : ProjectiveVariety, curve in PP^8 (subvariety of codimension 4 in Y)
│ │ │ │  i7 : S = surface X;
│ │ │ │  
│ │ │ │  o7 : ProjectiveVariety, surface in PP^8 (subvariety of codimension 3 in Y)
│ │ │ │  i8 : assert(dim C == 1 and degree C == 2 and dim(C*S) == 0 and degree(C*S) == 3
│ │ │ │  and isSubset(p,C) and isSubset(C,Y))
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, surface in PP^5
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(S,X,Verbose=>true)
│ │ │ - -- used 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ + -- used 0.368245s (cpu); 0.205643s (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 0.427896s (cpu); 0.253136s (thread); 0s (gc)
│ │ │ │ + -- used 0.368245s (cpu); 0.205643s (thread); 0s (gc)
│ │ │ │  S: rational normal curve of degree 5 in PP^5
│ │ │ │  X: smooth surface of degree 8 and sectional genus 5 in PP^5 cut out by 3
│ │ │ │  hypersurfaces of degree 2
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 32
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,P^5}(2)) = 10 = h^0(O_(P^5)(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,P^5}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Cubic__Fourfold_rp.html
│ │ │ @@ -94,15 +94,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ + -- used 0.594558s (cpu); 0.353461s (thread); 0s (gc)
│ │ │  S: Veronese surface in PP^5
│ │ │  X: smooth cubic hypersurface in PP^5
│ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │  h^0(N_{S,P^5}) = 27
│ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ │ ├── html2text {}
│ │ │ │ @@ -33,15 +33,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold V;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 0.7594s (cpu); 0.447905s (thread); 0s (gc)
│ │ │ │ + -- used 0.594558s (cpu); 0.353461s (thread); 0s (gc)
│ │ │ │  S: Veronese surface in PP^5
│ │ │ │  X: smooth cubic hypersurface in PP^5
│ │ │ │  (assumption: h^1(N_{S,P^5}) = 0)
│ │ │ │  h^0(N_{S,P^5}) = 27
│ │ │ │  h^1(O_S(3)) = 0, and h^0(I_{S,P^5}(3)) = 28 = h^0(O_(P^5)(3)) - \chi(O_S(3));
│ │ │ │  in particular, h^0(I_{S,P^5}(3)) is minimal
│ │ │ │  h^0(N_{S,P^5}) + 27 = 54
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parameter__Count_lp__Gushel__Mukai__Fourfold_rp.html
│ │ │ @@ -103,15 +103,15 @@
│ │ │  
│ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i4 : time parameterCount(X,Verbose=>true)
│ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ + -- used 3.10937s (cpu); 2.23547s (thread); 0s (gc)
│ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │  X: GM fourfold containing S
│ │ │  Y: del Pezzo fivefold containing X
│ │ │  h^1(N_{S,Y}) = 0
│ │ │  h^0(N_{S,Y}) = 11
│ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ │ ├── html2text {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^9 (subvariety of codimension 4 in G)
│ │ │ │  i3 : X = gushelMukaiFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, GM fourfold containing a surface of degree 3 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time parameterCount(X,Verbose=>true)
│ │ │ │ - -- used 4.53348s (cpu); 2.63125s (thread); 0s (gc)
│ │ │ │ + -- used 3.10937s (cpu); 2.23547s (thread); 0s (gc)
│ │ │ │  S: cubic surface in PP^8 cut out by 7 hypersurfaces of degrees (1,1,1,1,2,2,2)
│ │ │ │  X: GM fourfold containing S
│ │ │ │  Y: del Pezzo fivefold containing X
│ │ │ │  h^1(N_{S,Y}) = 0
│ │ │ │  h^0(N_{S,Y}) = 11
│ │ │ │  h^1(O_S(2)) = 0, and h^0(I_{S,Y}(2)) = 28 = h^0(O_Y(2)) - \chi(O_S(2));
│ │ │ │  in particular, h^0(I_{S,Y}(2)) is minimal
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_parametrize__Fano__Fourfold.html
│ │ │ @@ -93,15 +93,15 @@
│ │ │  o3 = 4-dimensional subvariety of PP^9 cut out by 7 hypersurfaces of degrees
│ │ │       1^2 2^5
      
│ │ │              
      
│ │ │                
      i4 : time parametrizeFanoFourfold X
│ │ │ - -- used 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ + -- used 0.823676s (cpu); 0.521774s (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 0.850872s (cpu); 0.559168s (thread); 0s (gc)
│ │ │ │ + -- used 0.823676s (cpu); 0.521774s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 = multi-rational map consisting of one single rational map
│ │ │ │       source variety: PP^4
│ │ │ │       target variety: 4-dimensional subvariety of PP^9 cut out by 7
│ │ │ │  hypersurfaces of degrees 1^2 2^5
│ │ │ │       dominance: true
│ │ │ │       degree: 1
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_polarized__K3surface.html
│ │ │ @@ -176,15 +176,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- surface was already irreducible
│ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 11 seconds (cpu: 15 seconds)
│ │ │  
│ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -215,16 +215,16 @@
│ │ │  -- computing p2^*(H_PP^2)
│ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │    -- computing image on K3 surface...
│ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8 
│ │ │  -- constructing lattice polarization...
│ │ │  -- verifying self-intersection of the curve...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 4 seconds)
│ │ │ +-- total time (K3 surface + polarization): 14 seconds (cpu: 20 seconds)
│ │ │  
│ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15 hypersurfaces of degree 2
│ │ │       No exceptional curves
│ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │       Lattice intersection matrix on Ũ: | 14 7 |
│ │ │ @@ -327,15 +327,15 @@
│ │ │  -- analyzing the base locus of the inverse map...
│ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │  -- projecting to PP^3 for surface decomposition
│ │ │    -- removing 1 components of degrees {3}
│ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ + ✦ underlying K3 successfully completed in 5 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice polarization: not yet computed; use 'polarize' or 'polarizedK3surface'
│ │ │ @@ -351,15 +351,15 @@
│ │ │  -- available strategies: "SpecialCurve", "MapFromW", "MapFromU", "MapFromW-Virtual", "MapFromU-Virtual"
│ │ │  -- special curves already detected on U
│ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1 
│ │ │  -- constructing lattice polarization...
│ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ +-- total time (K3 surface + polarization): 5 seconds (cpu: 8 seconds)
│ │ │  
│ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │        Mirror fourfold: PP^4
│ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        No exceptional curves
│ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1 
│ │ │        Lattice intersection matrix on Ũ: | 6 5  |
│ │ │ ├── html2text {}
│ │ │ │ @@ -119,15 +119,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- surface was already irreducible
│ │ │ │  -- result: surface in PP^8 cut out by 15 hypersurfaces of degree 2
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 21 seconds (cpu: 23 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 11 seconds (cpu: 15 seconds)
│ │ │ │  
│ │ │ │  o5 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -160,16 +160,16 @@
│ │ │ │    -- obtained the curve on U: curve in PP^8 cut out by 11 hypersurfaces of
│ │ │ │  degrees 1^3 2^8
│ │ │ │    -- computing image on K3 surface...
│ │ │ │    -- image curve: curve in PP^8 cut out by 11 hypersurfaces of degrees 1^3 2^8
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- verifying self-intersection of the curve...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (8, 7, 2)
│ │ │ │ - ✦ polarization successfully completed in 4 seconds (cpu: 3 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 25 seconds (cpu: 26 seconds)
│ │ │ │ + ✦ polarization successfully completed in 3 seconds (cpu: 4 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 14 seconds (cpu: 20 seconds)
│ │ │ │  
│ │ │ │  o6 = Fourfold: X, cubic fourfold in C_8
│ │ │ │       Mirror fourfold: ≋ ℙ² × ℙ² ⊂ ℙ⁸
│ │ │ │       Surface U of degree 14, sectional genus 8, χ(O_U) = 2, cut out by 15
│ │ │ │  hypersurfaces of degree 2
│ │ │ │       No exceptional curves
│ │ │ │       Minimal K3 surface Ũ: degree 14 and sectional genus 8 in PP^8 cut out by
│ │ │ │ @@ -267,15 +267,15 @@
│ │ │ │  -- analyzing the base locus of the inverse map...
│ │ │ │  -- surface found in base locus; equidimensionality already known, skipping...
│ │ │ │  -- projecting to PP^3 for surface decomposition
│ │ │ │    -- removing 1 components of degrees {3}
│ │ │ │  -- result: surface in PP^4 cut out by 2 hypersurfaces of degrees 2^1 3^1
│ │ │ │  -- U ∩ U' contains no (exceptional) curves
│ │ │ │  -- U is already in the target space; defining f as the identity map
│ │ │ │ - ✦ underlying K3 successfully completed in 7 seconds (cpu: 7 seconds)
│ │ │ │ + ✦ underlying K3 successfully completed in 5 seconds (cpu: 8 seconds)
│ │ │ │  
│ │ │ │  o10 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ │ │ @@ -292,15 +292,15 @@
│ │ │ │  Virtual", "MapFromU-Virtual"
│ │ │ │  -- special curves already detected on U
│ │ │ │    -- pushing forward curve to K3 (1/1)...
│ │ │ │    -- image curve: curve in PP^4 cut out by 5 hypersurfaces of degrees 2^4 3^1
│ │ │ │  -- constructing lattice polarization...
│ │ │ │  -- constructing lattice polarized K3 with (g, d, C^2) = (4, 5, -2)
│ │ │ │   ✦ polarization successfully completed in 0 seconds (cpu: 0 seconds)
│ │ │ │ --- total time (K3 surface + polarization): 7 seconds (cpu: 7 seconds)
│ │ │ │ +-- total time (K3 surface + polarization): 5 seconds (cpu: 8 seconds)
│ │ │ │  
│ │ │ │  o11 = Fourfold: X, cubic fourfold in C_20 ∩ C_8
│ │ │ │        Mirror fourfold: PP^4
│ │ │ │        Surface U of degree 6, sectional genus 4, χ(O_U) = 2, cut out by 2
│ │ │ │  hypersurfaces of degrees 2^1 3^1
│ │ │ │        No exceptional curves
│ │ │ │        Minimal K3 surface Ũ: degree 6 and sectional genus 4 in PP^4 cut out by
│ │ ├── ./usr/share/doc/Macaulay2/SpecialFanoFourfolds/html/_to__Grass.html
│ │ │ @@ -84,15 +84,15 @@
│ │ │  
│ │ │  o2 : ProjectiveVariety, GM fourfold containing a surface of degree 2 and sectional genus 0
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │ - -- used 4.10188s (cpu); 2.6702s (thread); 0s (gc)
│ │ │ + -- used 4.14566s (cpu); 2.40827s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 4-dimensional subvariety of PP^8 cut out by 6 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i3 : time toGrass X
│ │ │  warning: clearing value of symbol x to allow access to subscripted variables based on it
│ │ │         : debug with expression   debug 9868   or with command line option   --debug 9868
│ │ │ - -- used 4.22961s (cpu); 2.84603s (thread); 0s (gc)
│ │ │ + -- used 4.51342s (cpu); 2.61054s (thread); 0s (gc)
│ │ │  
│ │ │  o3 = multi-rational map consisting of one single rational map
│ │ │       source variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
│ │ │       target variety: GG(1,4) ⊂ PP^9
│ │ │  
│ │ │  o3 : MultirationalMap (rational map from X to GG(1,4))
      
│ │ │              
      
│ │ │                
      i4 : time f = unirationalParametrization X;
│ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ + -- used 0.901878s (cpu); 0.519995s (thread); 0s (gc)
│ │ │  
│ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
      
│ │ │              
      
│ │ │                
      i5 : degreeSequence f
│ │ │ ├── html2text {}
│ │ │ │ @@ -18,15 +18,15 @@
│ │ │ │  
│ │ │ │  o2 : ProjectiveVariety, surface in PP^5
│ │ │ │  i3 : X = cubicFourfold S;
│ │ │ │  
│ │ │ │  o3 : ProjectiveVariety, cubic fourfold containing a surface of degree 4 and
│ │ │ │  sectional genus 0
│ │ │ │  i4 : time f = unirationalParametrization X;
│ │ │ │ - -- used 0.856935s (cpu); 0.482912s (thread); 0s (gc)
│ │ │ │ + -- used 0.901878s (cpu); 0.519995s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o4 : MultirationalMap (rational map from PP^4 to X)
│ │ │ │  i5 : degreeSequence f
│ │ │ │  
│ │ │ │  o5 = {[10]}
│ │ │ │  
│ │ │ │  o5 : List
│ │ ├── ./usr/share/doc/Macaulay2/SpectralSequences/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  U3BlY3RyYWxTZXF1ZW5jZSBeIEluZmluaXRlTnVtYmVy
│ │ │  #:len=967
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGluZmluaXR5IHBhZ2Ugb2YgYSBz
│ │ │  cGVjdHJhbCBzZXF1ZW5jZSIsICJsaW5lbnVtIiA9PiAzNDExLCBJbnB1dHMgPT4ge1NQQU57VFR7
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=19
│ │ │  aXNMb29wbGVzcyhEaWdyYXBoKQ==
│ │ │  #:len=247
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODU0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhpc0xvb3BsZXNzLERpZ3JhcGgpLCJpc0xvb3BsZXNz
│ │ ├── ./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/example-output/_to__String_lp__Mixed__Graph_rp.out
│ │ │ @@ -11,12 +11,12 @@
│ │ │                  Graph => Graph{1 => {3}}
│ │ │                                 3 => {1}
│ │ │  
│ │ │  o1 : MixedGraph
│ │ │  
│ │ │  i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ -     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3},
│ │ │ -     {{1, 2}, {2, 3}})}
│ │ │ +o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ +     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ +     3}, {4, 2}})}
│ │ │  
│ │ │  i3 :
│ │ ├── ./usr/share/doc/Macaulay2/StatGraphs/html/_graph_lp__Mixed__Graph_rp.html
│ │ │ @@ -121,15 +121,15 @@
│ │ │  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/StatGraphs/html/_to__String_lp__Mixed__Graph_rp.html
│ │ │ @@ -93,17 +93,17 @@
│ │ │  o1 : MixedGraph
      
│ │ │              
      
│ │ │                
      i2 : toString G
│ │ │  
│ │ │ -o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph =>
│ │ │ -     bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3},
│ │ │ -     {{1, 2}, {2, 3}})}
      
│ │ │ +o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}),
│ │ │ +     Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4,
│ │ │ +     3}, {4, 2}})}
│ │ │              
      
│ │ │
      │ │ │
      │ │ │

      See also

      │ │ │
        │ │ │ ├── html2text {} │ │ │ │ @@ -23,17 +23,17 @@ │ │ │ │ 3 => {} │ │ │ │ Graph => Graph{1 => {3}} │ │ │ │ 3 => {1} │ │ │ │ │ │ │ │ o1 : MixedGraph │ │ │ │ i2 : toString G │ │ │ │ │ │ │ │ -o2 = new HashTable from {Graph => graph ({3, 1}, {{1, 3}}), Bigraph => │ │ │ │ - bigraph ({3, 4, 2}, {{4, 3}, {4, 2}}), Digraph => digraph ({1, 2, 3}, │ │ │ │ - {{1, 2}, {2, 3}})} │ │ │ │ +o2 = new HashTable from {Digraph => digraph ({1, 2, 3}, {{1, 2}, {2, 3}}), │ │ │ │ + Graph => graph ({3, 1}, {{1, 3}}), Bigraph => bigraph ({3, 4, 2}, {{4, │ │ │ │ + 3}, {4, 2}})} │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _M_i_x_e_d_G_r_a_p_h -- a graph that has undirected, directed and bidirected edges │ │ │ │ * _n_e_t_(_M_i_x_e_d_G_r_a_p_h_) -- print a mixed graph as a net │ │ │ │ * _S_t_r_i_n_g -- the class of all strings │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _t_o_S_t_r_i_n_g_(_M_i_x_e_d_G_r_a_p_h_) -- print a mixed graph as a string │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/StatePolytope/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=13 │ │ │ aW5pdGlhbElkZWFscw== │ │ │ #:len=950 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2FsbHMgZ2ZhbiBhbmQgcmV0dXJucyB0 │ │ │ aGUgbGlzdCBvZiBpbml0aWFsIGlkZWFscyIsICJsaW5lbnVtIiA9PiAxNDAsIElucHV0cyA9PiB7 │ │ ├── ./usr/share/doc/Macaulay2/StronglyStableIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=34 │ │ │ bWFjYXVsYXlEZWNvbXBvc2l0aW9uKFJpbmdFbGVtZW50KQ== │ │ │ #:len=329 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODUxLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhtYWNhdWxheURlY29tcG9zaXRpb24sUmluZ0VsZW1l │ │ ├── ./usr/share/doc/Macaulay2/Style/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=39 │ │ │ Z2VuZXJhdGVHcmFtbWFyKFN0cmluZyxTdHJpbmcsRnVuY3Rpb24p │ │ │ #:len=282 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgyLCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhnZW5lcmF0ZUdyYW1tYXIsU3RyaW5nLFN0cmluZyxG │ │ ├── ./usr/share/doc/Macaulay2/Style/example-output/_generate__Grammar.out │ │ │ @@ -1,16 +1,16 @@ │ │ │ -- -*- M2-comint -*- hash: 3455701143666534588 │ │ │ │ │ │ i1 : outfile = temporaryFileName() │ │ │ │ │ │ -o1 = /tmp/M2-10829-0/0 │ │ │ +o1 = /tmp/M2-10929-0/0 │ │ │ │ │ │ i2 : template = outfile | ".in" │ │ │ │ │ │ -o2 = /tmp/M2-10829-0/0.in │ │ │ +o2 = /tmp/M2-10929-0/0.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/M2-10829-0/0 │ │ │ + -- generating /tmp/M2-10929-0/0 │ │ │ │ │ │ i12 : get outfile │ │ │ │ │ │ o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually. │ │ │ │ │ │ This is an example file for the generateGrammar method! │ │ │ String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///" │ │ ├── ./usr/share/doc/Macaulay2/Style/html/_generate__Grammar.html │ │ │ @@ -87,22 +87,22 @@ │ │ │

        The function demarkf indicates how the elements of each of the lists will be demarked in the resulting file. The file outfile will then be generated, replacing each of these strings as indicated above.

        │ │ │
      │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -148,15 +148,15 @@ │ │ │ @M2KEYWORDS@ │ │ │ } │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
      │ │ │ 
      i1 : outfile = temporaryFileName()
│ │ │  
│ │ │ -o1 = /tmp/M2-10829-0/0
      │ │ │ +o1 = /tmp/M2-10929-0/0 │ │ │
      │ │ │ 
      i2 : template = outfile | ".in"
│ │ │  
│ │ │ -o2 = /tmp/M2-10829-0/0.in
      │ │ │ +o2 = /tmp/M2-10929-0/0.in │ │ │
      │ │ │ 
      i3 : template << "@M2BANNER@" << endl << endl;
      │ │ │
      │ │ │ 
      i11 : generateGrammar(template, outfile, x -> demark(",\n    ", x))
│ │ │ - -- generating /tmp/M2-10829-0/0
      │ │ │ + -- generating /tmp/M2-10929-0/0 │ │ │
      │ │ │ 
      i12 : get outfile
│ │ │  
│ │ │  o12 = Auto-generated for Macaulay2-1.26.06. 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/M2-10829-0/0
│ │ │ │ +o1 = /tmp/M2-10929-0/0
│ │ │ │  i2 : template = outfile | ".in"
│ │ │ │  
│ │ │ │ -o2 = /tmp/M2-10829-0/0.in
│ │ │ │ +o2 = /tmp/M2-10929-0/0.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/M2-10829-0/0
│ │ │ │ + -- generating /tmp/M2-10929-0/0
│ │ │ │  i12 : get outfile
│ │ │ │  
│ │ │ │  o12 = Auto-generated for Macaulay2-1.26.06. Do not modify this file manually.
│ │ │ │  
│ │ │ │        This is an example file for the generateGrammar method!
│ │ │ │        String regex: "///\\(/?/?[^/]\\|\\(//\\)*////[^/]\\)*\\(//\\)*///"
│ │ │ │        List of keywords: {
│ │ ├── ./usr/share/doc/Macaulay2/SubalgebraBases/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=24
│ │ │  c2FnYmkoLi4uLFN0cmF0ZWd5PT4uLi4p
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU3OSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbc2FnYmksU3RyYXRlZ3ldLCJzYWdiaSguLi4sU3Ry
│ │ ├── ./usr/share/doc/Macaulay2/SumsOfSquares/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=30
│ │ │  bG93ZXJCb3VuZCguLi4sVmVyYm9zaXR5PT4uLi4p
│ │ │  #:len=302
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20ge1tsb3dlckJvdW5kLFZlcmJvc2l0eV0sImxvd2VyQm91
│ │ ├── ./usr/share/doc/Macaulay2/SuperLinearAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=6
│ │ │  cGFyaXR5
│ │ │  #:len=1709
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAicGFyaXR5IG9mIGFuIGVsZW1lbnQgb2Yg
│ │ │  YSBzdXBlciByaW5nLiIsICJsaW5lbnVtIiA9PiA2NDcsIElucHV0cyA9PiB7U1BBTntUVHsiZiJ9
│ │ ├── ./usr/share/doc/Macaulay2/SwitchingFields/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=33
│ │ │  ZmllbGRCYXNlQ2hhbmdlKFJpbmcsR2Fsb2lzRmllbGQp
│ │ │  #:len=300
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhmaWVsZEJhc2VDaGFuZ2UsUmluZyxHYWxvaXNGaWVs
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  bm9QYWNrZWRBbGxTdWJz
│ │ │  #:len=1151
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgYWxsIHN1YnN0aXR1dGlvbnMg
│ │ │  b2YgdmFyaWFibGVzIGJ5IDEgYW5kL29yIDAgZm9yIHdoaWNoIGlkZWFsIGlzIG5vdCBLb25pZy4i
│ │ ├── ./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 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ + -- used 0.453915s (cpu); 0.206273s (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 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ + -- used 0.107704s (cpu); 0.0467543s (thread); 0s (gc)
│ │ │  
│ │ │  o10 : Ideal of QQ[x..z]
│ │ │  
│ │ │  i11 :
│ │ ├── ./usr/share/doc/Macaulay2/SymbolicPowers/html/_symbolic__Power.html
│ │ │ @@ -146,15 +146,15 @@
│ │ │  
│ │ │  o6 = false
      │ │ │
      │ │ │ 
      i7 : time symbolicPower(P,4);
│ │ │ - -- used 0.441308s (cpu); 0.240277s (thread); 0s (gc)
│ │ │ + -- used 0.453915s (cpu); 0.206273s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of QQ[x..z]
      │ │ │
      │ │ │ 
      i8 : Q = ker map(QQ[t],QQ[x,y,z, Degrees => {3,4,5}],{t^3,t^4,t^5})
│ │ │ @@ -171,15 +171,15 @@
│ │ │  
│ │ │  o9 = true
      │ │ │
      │ │ │ 
      i10 : time symbolicPower(Q,4);
│ │ │ - -- used 0.115068s (cpu); 0.0574305s (thread); 0s (gc)
│ │ │ + -- used 0.107704s (cpu); 0.0467543s (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 0.441308s (cpu); 0.240277s (thread); 0s (gc) │ │ │ │ + -- used 0.453915s (cpu); 0.206273s (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 0.115068s (cpu); 0.0574305s (thread); 0s (gc) │ │ │ │ + -- used 0.107704s (cpu); 0.0467543s (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/SymmetricPolynomials/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=32 │ │ │ YnVpbGRTeW1tZXRyaWNHQihQb2x5bm9taWFsUmluZyk= │ │ │ #:len=1047 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiR3JvZWJuZXIgYmFzaXMgb2YgZWxlbWVu │ │ │ dGFyeSBzeW1tZXRyaWMgcG9seW5vbWlhbHMgYWxnZWJyYSIsICJsaW5lbnVtIiA9PiAxNzksIElu │ │ ├── ./usr/share/doc/Macaulay2/TSpreadIdeals/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=31 │ │ │ Y291bnRUTGV4TW9uKC4uLixGaXhlZE1heD0+Li4uKQ== │ │ │ #:len=268 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTQwMSwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY291bnRUTGV4TW9uLEZpeGVkTWF4XSwiY291bnRU │ │ ├── ./usr/share/doc/Macaulay2/Tableaux/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ aXNDb3JuZXI= │ │ │ #:len=1388 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIGlmIGEgYm94IGlzIGEgY29y │ │ │ bmVyIG9mIGEgdGFibGVhdSIsICJsaW5lbnVtIiA9PiA2MTEsIElucHV0cyA9PiB7U1BBTntUVHsi │ │ ├── ./usr/share/doc/Macaulay2/TangentCone/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=11 │ │ │ VGFuZ2VudENvbmU= │ │ │ #:len=312 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGFuZ2VudCBjb25lcyIsIERlc2NyaXB0 │ │ │ aW9uID0+IDE6KCJUaGlzIHBhY2thZ2UgcHJvdmlkZXMgYSBzaW5nbGUgZnVuY3Rpb24gdGhhdCBj │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=52 │ │ │ cHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyguLi4sQ29lZmZpY2llbnRGaWVsZD0+Li4uKQ== │ │ │ #:len=347 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDM3Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbcHJvZHVjdE9mUHJvamVjdGl2ZVNwYWNlcyxDb2Vm │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/example-output/_beilinson__Window.out │ │ │ @@ -10,15 +10,15 @@ │ │ │ o3 = 0 <-- E <-- 0 │ │ │ │ │ │ -1 0 1 │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ i4 : time T=tateExtension W; │ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc) │ │ │ + -- used 0.856799s (cpu); 0.584706s (thread); 0s (gc) │ │ │ │ │ │ i5 : cohomologyMatrix(T,-{3,3},{3,3}) │ │ │ │ │ │ o5 = | 8h 4h 0 4 8 12 16 | │ │ │ | 6h 3h 0 3 6 9 12 | │ │ │ | 4h 2h 0 2 4 6 8 | │ │ │ | 2h h 0 1 2 3 4 | │ │ ├── ./usr/share/doc/Macaulay2/TateOnProducts/html/_beilinson__Window.html │ │ │ @@ -97,15 +97,15 @@ │ │ │ │ │ │ o3 : Complex │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : time T=tateExtension W;
│ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
    │ │ │ + -- used 0.856799s (cpu); 0.584706s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : cohomologyMatrix(T,-{3,3},{3,3})
│ │ │  
│ │ │  o5 = | 8h  4h  0 4  8  12 16 |
│ │ │ ├── html2text {}
│ │ │ │ @@ -23,15 +23,15 @@
│ │ │ │               1
│ │ │ │  o3 = 0  <-- E  <-- 0
│ │ │ │  
│ │ │ │       -1     0      1
│ │ │ │  
│ │ │ │  o3 : Complex
│ │ │ │  i4 : time T=tateExtension W;
│ │ │ │ - -- used 1.08449s (cpu); 0.725679s (thread); 0s (gc)
│ │ │ │ + -- used 0.856799s (cpu); 0.584706s (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/TensorComplexes/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=31
│ │ │  bWlub3JzTWFwKE1hdHJpeCxMYWJlbGVkTW9kdWxlKQ==
│ │ │  #:len=285
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTkzOCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsobWlub3JzTWFwLE1hdHJpeCxMYWJlbGVkTW9kdWxl
│ │ ├── ./usr/share/doc/Macaulay2/TerraciniLoci/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=26
│ │ │  dGVycmFjaW5pTG9jdXMoWlosUmluZ01hcCk=
│ │ │  #:len=278
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXJyYWNpbmlMb2N1cyxaWixSaW5nTWFwKSwidGVy
│ │ ├── ./usr/share/doc/Macaulay2/TestAudit/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  dGVzdEF1ZGl0KFN0cmluZyk=
│ │ │  #:len=238
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNTg4LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0ZXN0QXVkaXQsU3RyaW5nKSwidGVzdEF1ZGl0KFN0
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=17
│ │ │  ZnJvYmVuaXVzUHJlaW1hZ2U=
│ │ │  #:len=934
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZmluZHMgdGhlIGlkZWFsIG9mIGVsZW1l
│ │ │  bnRzIG1hcHBlZCBpbnRvIGEgZ2l2ZW4gaWRlYWwsIHVuZGVyIGFsbCAkcF57LWV9JC1saW5lYXIg
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/example-output/_frobenius__Root.out
│ │ │ @@ -63,20 +63,20 @@
│ │ │  o15 : Ideal of R
│ │ │  
│ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │  
│ │ │  o16 : Ideal of R
│ │ │  
│ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ + -- used 1.02676s (cpu); 0.658283s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
│ │ │  
│ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ + -- used 2.52076s (cpu); 2.07413s (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 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ + -- used 0.00278373s (cpu); 0.00278018s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ + -- used 0.00451069s (cpu); 0.00451626s (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 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ + -- used 0.026571s (cpu); 0.02657s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
│ │ │  
│ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ + -- used 1.60007s (cpu); 1.19375s (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 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ + -- used 0.0650209s (cpu); 0.0650233s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
│ │ │  
│ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ + -- used 0.0678094s (cpu); 0.0678193s (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 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ + -- used 0.673089s (cpu); 0.609574s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
│ │ │  
│ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ + -- used 0.281576s (cpu); 0.213346s (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 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ + -- used 0.134484s (cpu); 0.0747341s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
│ │ │  
│ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ + -- used 0.131576s (cpu); 0.131585s (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 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ + -- used 0.216462s (cpu); 0.159601s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
│ │ │  
│ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ + -- used 0.279976s (cpu); 0.219433s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = ideal (y, x)
│ │ │  
│ │ │  o24 : Ideal of R
│ │ │  
│ │ │  i25 :
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_frobenius__Root.html
│ │ │ @@ -231,23 +231,23 @@
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ + -- used 1.02676s (cpu); 0.658283s (thread); 0s (gc)
│ │ │  
│ │ │  o17 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ + -- used 2.52076s (cpu); 2.07413s (thread); 0s (gc)
│ │ │  
│ │ │  o18 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i19 : J1 == J2
│ │ │ ├── html2text {}
│ │ │ │ @@ -106,19 +106,19 @@
│ │ │ │  i15 : I2 = ideal(x^20*y^100, x + z^100);
│ │ │ │  
│ │ │ │  o15 : Ideal of R
│ │ │ │  i16 : I3 = ideal(x^50*y^50*z^50);
│ │ │ │  
│ │ │ │  o16 : Ideal of R
│ │ │ │  i17 : time J1 = frobeniusRoot(1, {8, 10, 12}, {I1, I2, I3});
│ │ │ │ - -- used 1.06996s (cpu); 0.723329s (thread); 0s (gc)
│ │ │ │ + -- used 1.02676s (cpu); 0.658283s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o17 : Ideal of R
│ │ │ │  i18 : time J2 = frobeniusRoot(1, I1^8*I2^10*I3^12);
│ │ │ │ - -- used 2.70336s (cpu); 2.192s (thread); 0s (gc)
│ │ │ │ + -- used 2.52076s (cpu); 2.07413s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o18 : Ideal of R
│ │ │ │  i19 : J1 == J2
│ │ │ │  
│ │ │ │  o19 = true
│ │ │ │  For legacy reasons, the last ideal in the list can be specified separately,
│ │ │ │  using frobeniusRoot(e, \{a_1,\ldots,a_n\}, \{I_1,\ldots,I_n\}, I). The last
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__Cohen__Macaulay.html
│ │ │ @@ -101,23 +101,23 @@
│ │ │              
│ │ │                
    i4 : R = S/(ker g);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : time isCohenMacaulay(R)
│ │ │ - -- used 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ + -- used 0.00278373s (cpu); 0.00278018s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ + -- used 0.00451069s (cpu); 0.00451626s (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 0.00348147s (cpu); 0.00347735s (thread); 0s (gc)
│ │ │ │ + -- used 0.00278373s (cpu); 0.00278018s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  i6 : time isCohenMacaulay(R, AtOrigin => true)
│ │ │ │ - -- used 0.00548896s (cpu); 0.00549078s (thread); 0s (gc)
│ │ │ │ + -- used 0.00451069s (cpu); 0.00451626s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o6 = true
│ │ │ │  i7 : R = QQ[x,y,u,v]/(x*u, x*v, y*u, y*v);
│ │ │ │  i8 : isCohenMacaulay(R)
│ │ │ │  
│ │ │ │  o8 = false
│ │ │ │  The function isCohenMacaulay considers $R$ as a quotient of a polynomial ring,
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Injective.html
│ │ │ @@ -219,23 +219,23 @@
│ │ │  
│ │ │  o19 : QuotientRing
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i20 : time isFInjective(R)
│ │ │ - -- used 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ + -- used 0.026571s (cpu); 0.02657s (thread); 0s (gc)
│ │ │  
│ │ │  o20 = true
    
│ │ │              
    
│ │ │                
    i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ + -- used 1.60007s (cpu); 1.19375s (thread); 0s (gc)
│ │ │  
│ │ │  o21 = true
    
│ │ │              
    
│ │ │          
    
│ │ │            
    If the option AtOrigin (default value false) is set to true, isFInjective will only check $F$-injectivity at the origin. Otherwise, it will check $F$-injectivity globally. Note that checking $F$-injectivity at the origin can be slower than checking it globally. Consider the following example of a non-$F$-injective ring.
    
│ │ │ @@ -245,23 +245,23 @@
│ │ │              
│ │ │                
    i22 : R = ZZ/7[x,y,z]/((x-1)^5 + (y+1)^5 + z^5);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time isFInjective(R)
│ │ │ - -- used 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ + -- used 0.0650209s (cpu); 0.0650233s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time isFInjective(R, AtOrigin => true)
│ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ + -- used 0.0678094s (cpu); 0.0678193s (thread); 0s (gc)
│ │ │  
│ │ │  o24 = true
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    If the option AssumeCM (default value false) is set to true, then isFInjective only checks the Frobenius action on top cohomology (which is typically much faster). Note that it can give an incorrect answer if the non-injective Frobenius occurs in a lower degree.  Consider the example of the cone over a supersingular elliptic curve times $\mathbb{P}^1$.
    
│ │ │ @@ -288,23 +288,23 @@
│ │ │              
│ │ │                
    i28 : R = S/(ker f);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : time isFInjective(R)
│ │ │ - -- used 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ + -- used 0.673089s (cpu); 0.609574s (thread); 0s (gc)
│ │ │  
│ │ │  o29 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : time isFInjective(R, AssumeCM => true)
│ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ + -- used 0.281576s (cpu); 0.213346s (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 0.0276842s (cpu); 0.027688s (thread); 0s (gc)
│ │ │ │ + -- used 0.026571s (cpu); 0.02657s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o20 = true
│ │ │ │  i21 : time isFInjective(R, CanonicalStrategy => null)
│ │ │ │ - -- used 2.0387s (cpu); 1.62786s (thread); 0s (gc)
│ │ │ │ + -- used 1.60007s (cpu); 1.19375s (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 0.109724s (cpu); 0.109728s (thread); 0s (gc)
│ │ │ │ + -- used 0.0650209s (cpu); 0.0650233s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = false
│ │ │ │  i24 : time isFInjective(R, AtOrigin => true)
│ │ │ │ - -- used 0.109777s (cpu); 0.109789s (thread); 0s (gc)
│ │ │ │ + -- used 0.0678094s (cpu); 0.0678193s (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 0.802488s (cpu); 0.65363s (thread); 0s (gc)
│ │ │ │ + -- used 0.673089s (cpu); 0.609574s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o29 = false
│ │ │ │  i30 : time isFInjective(R, AssumeCM => true)
│ │ │ │ - -- used 0.1693s (cpu); 0.169306s (thread); 0s (gc)
│ │ │ │ + -- used 0.281576s (cpu); 0.213346s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o30 = true
│ │ │ │  If the option AssumedReduced is set to true (its default behavior), then the
│ │ │ │  bottom local cohomology is avoided (this means the Frobenius action on the top
│ │ │ │  potentially nonzero Ext is not computed).
│ │ │ │  If the option AssumeNormal (default value false) is set to true, then the
│ │ │ │  bottom two local cohomology modules (or, rather, their duals) need not be
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_is__F__Regular.html
│ │ │ @@ -278,23 +278,23 @@
│ │ │                
    i26 : debugLevel = 1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i27 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 1)
│ │ │  isFRegular: This ring does not appear to be F-regular.  Increasing DepthOfSearch will let the function search more deeply.
│ │ │ - -- used 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ + -- used 0.134484s (cpu); 0.0747341s (thread); 0s (gc)
│ │ │  
│ │ │  o27 = false
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ + -- used 0.131576s (cpu); 0.131585s (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 0.170186s (cpu); 0.0965721s (thread); 0s (gc)
│ │ │ │ + -- used 0.134484s (cpu); 0.0747341s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o27 = false
│ │ │ │  i28 : time isFRegular(S/I, QGorensteinIndex => infinity, DepthOfSearch => 2)
│ │ │ │ - -- used 0.142596s (cpu); 0.142603s (thread); 0s (gc)
│ │ │ │ + -- used 0.131576s (cpu); 0.131585s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o28 = true
│ │ │ │  i29 : debugLevel = 0;
│ │ │ │  ********** SSeeee aallssoo **********
│ │ │ │      * _t_e_s_t_I_d_e_a_l -- compute a test ideal in a Q-Gorenstein ring
│ │ │ │      * _i_s_F_R_a_t_i_o_n_a_l -- whether a ring is F-rational
│ │ │ │  ********** WWaayyss ttoo uussee iissFFRReegguullaarr:: **********
│ │ ├── ./usr/share/doc/Macaulay2/TestIdeals/html/_test__Ideal.html
│ │ │ @@ -260,25 +260,25 @@
│ │ │          
    
│ │ │            
    It is often more efficient to pass a list, as opposed to finding a common denominator and passing a single element, since testIdeal can do things in a more intelligent way for such a list.
    
│ │ │          
    
│ │ │          
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i23 : time testIdeal({3/4, 2/3, 3/5}, L)
│ │ │ - -- used 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ + -- used 0.216462s (cpu); 0.159601s (thread); 0s (gc)
│ │ │  
│ │ │  o23 = ideal (y, x)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
    
│ │ │                
    i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ + -- used 0.279976s (cpu); 0.219433s (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 0.332881s (cpu); 0.257332s (thread); 0s (gc)
│ │ │ │ + -- used 0.216462s (cpu); 0.159601s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 = ideal (y, x)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time testIdeal(1/60, x^45*y^40*(x + y)^36)
│ │ │ │ - -- used 0.45624s (cpu); 0.377858s (thread); 0s (gc)
│ │ │ │ + -- used 0.279976s (cpu); 0.219433s (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/Text/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  bmV3IFRPSCBmcm9tIFRoaW5n
│ │ │  #:len=219
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODQwLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhOZXdGcm9tTWV0aG9kLFRPSCxUaGluZyksIm5ldyBU
│ │ ├── ./usr/share/doc/Macaulay2/ThinSincereQuivers/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=28
│ │ │  Y2hhaW5RdWl2ZXIoLi4uLEhlaWdodD0+Li4uKQ==
│ │ │  #:len=283
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzI5MSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbY2hhaW5RdWl2ZXIsSGVpZ2h0XSwiY2hhaW5RdWl2
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=9
│ │ │  dGdiKExpc3Qp
│ │ │  #:len=213
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDY1LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyh0Z2IsTGlzdCksInRnYihMaXN0KSIsIlRocmVhZGVk
│ │ ├── ./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, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ +                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_matrix_lp__Lineage__Table_rp.out
│ │ │ @@ -4,15 +4,14 @@
│ │ │  
│ │ │  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
│ │ │ -                  ((0, 1), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/example-output/_reduce.out
│ │ │ @@ -2,20 +2,16 @@
│ │ │  
│ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │  
│ │ │  i2 : allowableThreads= 2;
│ │ │  
│ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │                                     2
│ │ │ -                  ((1, 2), 0) => -c
│ │ │ +o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │ @@ -24,18 +20,16 @@
│ │ │                          2
│ │ │                    2 => b
│ │ │  
│ │ │  o3 : LineageTable
│ │ │  
│ │ │  i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o4 = LineageTable{((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/_tgb.out
│ │ │ @@ -6,34 +6,36 @@
│ │ │  
│ │ │  o2 : Ideal of R
│ │ │  
│ │ │  i3 : allowableThreads  = 4;
│ │ │  
│ │ │  i4 : H = tgb I
│ │ │  
│ │ │ -                                   4 4      2 6
│ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ -                                    2 6      2 5
│ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ -                                           4 4     2 4
│ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ -                                    3 4      2 4
│ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ -                                    3 4      2 5
│ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ -                                    2 5
│ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ -                                   2 4
│ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ +                                               2 5
│ │ │ +o4 = LineageTable{(((0, 1), 2), (0, 2)) => -22y z   }
│ │ │ +                                   4 4      3 6
│ │ │ +                  ((0, 1), 2) => 9y z  - 27y z
│ │ │ +                                    2 6
│ │ │ +                  ((0, 2), 3) => -2y z
│ │ │ +                                         5       2 4
│ │ │ +                  ((0, 3), (0, 1)) => 46y z + 40y z
│ │ │ +                                               2 4
│ │ │ +                  ((2, 3), ((0, 1), 2)) => -47y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      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
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/___Minimal.html
│ │ │ @@ -78,16 +78,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb( ideal "abc+c2,ab2-b3c+ac,b2", Minimal=>true)
│ │ │  
│ │ │ -o3 = LineageTable{((0, 2), 0) => null}
│ │ │ -                  ((0, 2), 1) => null
│ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ +                  ((0, 1), 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 @@
│ │ │ │  Gröbner basis is minimized. Lineages of non-minimal Gröbner 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, 2), 0) => null}
│ │ │ │ -                  ((0, 2), 1) => null
│ │ │ │ +o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ +                  ((0, 1), 0) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_matrix_lp__Lineage__Table_rp.html
│ │ │ @@ -93,15 +93,14 @@
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = reduce tgb( ideal "abc+c2,ab2-b3c+ac,b2")
│ │ │  
│ │ │  o3 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │                    ((0, 1), 0) => null
│ │ │ -                  ((0, 1), 1) => null
│ │ │                                    2
│ │ │                    ((1, 2), 0) => c
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │ ├── html2text {}
│ │ │ │ @@ -21,15 +21,14 @@
│ │ │ │  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
│ │ │ │ -                  ((0, 1), 1) => null
│ │ │ │                                    2
│ │ │ │                    ((1, 2), 0) => c
│ │ │ │                    (0, 1) => null
│ │ │ │                    (0, 2) => null
│ │ │ │                    (1, 2) => a*c
│ │ │ │                    0 => null
│ │ │ │                    1 => null
│ │ ├── ./usr/share/doc/Macaulay2/ThreadedGB/html/_reduce.html
│ │ │ @@ -87,20 +87,16 @@
│ │ │                
    i2 : allowableThreads= 2;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │  
│ │ │ -                                        3
│ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ -                                     2
│ │ │ -                  ((0, 1), 0) => -a*c
│ │ │                                     2
│ │ │ -                  ((1, 2), 0) => -c
│ │ │ +o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │                               2
│ │ │                    (0, 1) => a c
│ │ │                                 2
│ │ │                    (0, 2) => b*c
│ │ │                    (1, 2) => -a*c
│ │ │                                  2
│ │ │                    0 => a*b*c + c
│ │ │ @@ -112,18 +108,16 @@
│ │ │  o3 : LineageTable
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : reduce T
│ │ │  
│ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ -                  ((0, 1), 0) => null
│ │ │                                    2
│ │ │ -                  ((1, 2), 0) => c
│ │ │ +o4 = LineageTable{((1, 2), 0) => c }
│ │ │                    (0, 1) => null
│ │ │                    (0, 2) => null
│ │ │                    (1, 2) => a*c
│ │ │                    0 => null
│ │ │                    1 => null
│ │ │                          2
│ │ │                    2 => b
│ │ │ ├── html2text {}
│ │ │ │ @@ -20,20 +20,16 @@
│ │ │ │  remainder on the division by the remaining values H.
│ │ │ │  If values H constitute a Gröbner basis of the ideal they generate, this method
│ │ │ │  returns a reduced Gröbner basis.
│ │ │ │  i1 : R = ZZ/101[a,b,c];
│ │ │ │  i2 : allowableThreads= 2;
│ │ │ │  i3 : T = tgb ideal "abc+c2,ab2-b3c+ac,b2"
│ │ │ │  
│ │ │ │ -                                        3
│ │ │ │ -o3 = LineageTable{(((0, 1), 0), 0) => -c }
│ │ │ │ -                                     2
│ │ │ │ -                  ((0, 1), 0) => -a*c
│ │ │ │                                     2
│ │ │ │ -                  ((1, 2), 0) => -c
│ │ │ │ +o3 = LineageTable{((1, 2), 0) => -c      }
│ │ │ │                               2
│ │ │ │                    (0, 1) => a c
│ │ │ │                                 2
│ │ │ │                    (0, 2) => b*c
│ │ │ │                    (1, 2) => -a*c
│ │ │ │                                  2
│ │ │ │                    0 => a*b*c + c
│ │ │ │ @@ -41,18 +37,16 @@
│ │ │ │                    1 => - b c + a*b  + a*c
│ │ │ │                          2
│ │ │ │                    2 => b
│ │ │ │  
│ │ │ │  o3 : LineageTable
│ │ │ │  i4 : reduce T
│ │ │ │  
│ │ │ │ -o4 = LineageTable{(((0, 1), 0), 0) => null}
│ │ │ │ -                  ((0, 1), 0) => null
│ │ │ │                                    2
│ │ │ │ -                  ((1, 2), 0) => c
│ │ │ │ +o4 = LineageTable{((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/_tgb.html
│ │ │ @@ -100,34 +100,36 @@
│ │ │                
    i3 : allowableThreads  = 4;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : H = tgb I
│ │ │  
│ │ │ -                                   4 4      2 6
│ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ -                                    2 6      2 5
│ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ -                                           4 4     2 4
│ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ -                                    3 4      2 4
│ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ -                                    3 4      2 5
│ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ -                                    2 5
│ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ -                                   2 4
│ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ +                                               2 5
│ │ │ +o4 = LineageTable{(((0, 1), 2), (0, 2)) => -22y z   }
│ │ │ +                                   4 4      3 6
│ │ │ +                  ((0, 1), 2) => 9y z  - 27y z
│ │ │ +                                    2 6
│ │ │ +                  ((0, 2), 3) => -2y z
│ │ │ +                                         5       2 4
│ │ │ +                  ((0, 3), (0, 1)) => 46y z + 40y z
│ │ │ +                                               2 4
│ │ │ +                  ((2, 3), ((0, 1), 2)) => -47y z
│ │ │                                   5 2      3 4
│ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ -                                 3 5     2 4
│ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ -                               5       3 4
│ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ +                              5 3     2 4
│ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ +                              5 2      5
│ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ +                                 4 5      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
│ │ │                    2 => 5x*y z  + 9x*z
│ │ │                             3         3
│ │ │ ├── html2text {}
│ │ │ │ @@ -26,34 +26,36 @@
│ │ │ │  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
│ │ │ │  
│ │ │ │ -                                   4 4      2 6
│ │ │ │ -o4 = LineageTable{((0, 1), 2) => 9y z  + 23y z        }
│ │ │ │ -                                    2 6      2 5
│ │ │ │ -                  ((0, 1), 3) => 23y z  + 48y z
│ │ │ │ -                                           4 4     2 4
│ │ │ │ -                  ((0, 2), (0, 1)) => - 13y z  + 9y z
│ │ │ │ -                                    3 4      2 4
│ │ │ │ -                  ((0, 2), 1) => 18y z  - 33y z
│ │ │ │ -                                    3 4      2 5
│ │ │ │ -                  ((0, 2), 2) => 37y z  - 45y z
│ │ │ │ -                                    2 5
│ │ │ │ -                  ((0, 2), 3) => 16y z
│ │ │ │ -                                   2 4
│ │ │ │ -                  ((0, 3), 2) => 5y z
│ │ │ │ +                                               2 5
│ │ │ │ +o4 = LineageTable{(((0, 1), 2), (0, 2)) => -22y z   }
│ │ │ │ +                                   4 4      3 6
│ │ │ │ +                  ((0, 1), 2) => 9y z  - 27y z
│ │ │ │ +                                    2 6
│ │ │ │ +                  ((0, 2), 3) => -2y z
│ │ │ │ +                                         5       2 4
│ │ │ │ +                  ((0, 3), (0, 1)) => 46y z + 40y z
│ │ │ │ +                                               2 4
│ │ │ │ +                  ((2, 3), ((0, 1), 2)) => -47y z
│ │ │ │                                   5 2      3 4
│ │ │ │                    (0, 1) => - 25y z  - 19y z
│ │ │ │ -                                 3 5     2 4
│ │ │ │ -                  (0, 2) => - 24y z  + 9y z
│ │ │ │ -                               5       3 4
│ │ │ │ -                  (0, 3) => 28y z - 24y z
│ │ │ │ +                              5 3     2 4
│ │ │ │ +                  (0, 2) => 5y z  + 9y z
│ │ │ │ +                              5 2      5
│ │ │ │ +                  (0, 3) => 5y z  + 28y z
│ │ │ │ +                                 4 5      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
│ │ │ │                    2 => 5x*y z  + 9x*z
│ │ │ │                             3         3
│ │ ├── ./usr/share/doc/Macaulay2/Topcom/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=15
│ │ │  Y2hpcm90b3BlU3RyaW5n
│ │ │  #:len=253
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNzUzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJjaGlyb3RvcGVTdHJpbmciLCJjaGlyb3RvcGVTdHJp
│ │ ├── ./usr/share/doc/Macaulay2/TorAlgebra/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=12
│ │ │  aXNHb3JlbnN0ZWlu
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│ │ │  bnN0ZWluIiwgImxpbmVudW0iID0+IDEyNDMsIElucHV0cyA9PiB7U1BBTntUVHsiUiJ9LCIsICIs
│ │ ├── ./usr/share/doc/Macaulay2/ToricHigherDirectImages/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
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│ │ │  #:len=23
│ │ │  VG9yaWNIaWdoZXJEaXJlY3RJbWFnZXM=
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│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0YXRpb25zIGludm9sdmluZyBw
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│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=5
│ │ │  ZWREZWc=
│ │ │  #:len=2501
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIChnZW5lcmljKSBF
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│ │ ├── ./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 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ + -- used 1.02663s (cpu); 0.676922s (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 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ + -- used 4.14724s (cpu); 2.59274s (thread); 0s (gc)
│ │ │  
│ │ │  o5 = 252
│ │ │  
│ │ │  o5 : QQ
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/ToricInvariants/html/_ed__Deg.html
│ │ │ @@ -136,15 +136,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ + -- used 1.02663s (cpu); 0.676922s (thread); 0s (gc)
│ │ │  
│ │ │  o4 = 252
│ │ │  
│ │ │  o4 : QQ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │ @@ -155,15 +155,15 @@
│ │ │  The dual variety has degree = 45, and codimension = 1
│ │ │  Chern-Mather Volumes: (V_0,..,V_(d-1)) = {20, 23, 31, 28}
│ │ │  Polar Degrees: {45, 98, 81, 28}
│ │ │  ED Degree = 252
│ │ │  
│ │ │                         5      4      3      2
│ │ │  Chern-Mather Class: 20h  + 23h  + 31h  + 28h
│ │ │ - -- used 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ + -- used 4.14724s (cpu); 2.59274s (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 1.33817s (cpu); 0.918817s (thread); 0s (gc)
│ │ │ │ + -- used 1.02663s (cpu); 0.676922s (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 4.78144s (cpu); 3.03944s (thread); 0s (gc)
│ │ │ │ + -- used 4.14724s (cpu); 2.59274s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o5 = 252
│ │ │ │  
│ │ │ │  o5 : QQ
│ │ │ │  ********** WWaayyss ttoo uussee eeddDDeegg:: **********
│ │ │ │      * edDeg(Matrix)
│ │ │ │  ********** FFoorr tthhee pprrooggrraammmmeerr **********
│ │ ├── ./usr/share/doc/Macaulay2/ToricTopology/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=25
│ │ │  YmV0dGkoTm9ybWFsVG9yaWNWYXJpZXR5KQ==
│ │ │  #:len=272
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNDc0LCBzeW1ib2wgRG9jdW1lbnRUYWcg
│ │ │  PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhiZXR0aSxOb3JtYWxUb3JpY1ZhcmlldHkpLCJiZXR0
│ │ ├── ./usr/share/doc/Macaulay2/ToricVectorBundles/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  ZHVhbChUb3JpY1ZlY3RvckJ1bmRsZSk=
│ │ │  #:len=1504
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiIHRoZSBkdWFsIGJ1bmRsZSBvZiBhIHRv
│ │ │  cmljIHZlY3RvciBidW5kbGUiLCAibGluZW51bSIgPT4gMjgwOCwgSW5wdXRzID0+IHtTUEFOe1RU
│ │ ├── ./usr/share/doc/Macaulay2/TriangularSets/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=14
│ │ │  Y2hlY2tJbnRlcmZhY2U=
│ │ │  #:len=792
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAid2hldGhlciB0aGUgTWFwbGUgaW50ZXJm
│ │ │  YWNlIGlzIHdvcmtpbmcgKGZvciBkZXZlbG9wZXJzKSIsICJsaW5lbnVtIiA9PiAzMzUsIElucHV0
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=50
│ │ │  cmVndWxhclRyaWFuZ3VsYXRpb25XZWlnaHRzKC4uLixEZWdyZWVNYXRyaXg9Pi4uLik=
│ │ │  #:len=347
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTgzNCwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbcmVndWxhclRyaWFuZ3VsYXRpb25XZWlnaHRzLERl
│ │ ├── ./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);
│ │ │ - -- .116686s elapsed
│ │ │ + -- .0894708s 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;
│ │ │ - -- 2.08042s elapsed
│ │ │ + -- .998968s elapsed
│ │ │  
│ │ │  i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
│ │ │  
│ │ │  i10 :
│ │ ├── ./usr/share/doc/Macaulay2/Triangulations/html/index.html
│ │ │ @@ -168,15 +168,15 @@
│ │ │                4       10
│ │ │  o2 : Matrix ZZ  <-- ZZ
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : elapsedTime Ts = allTriangulations(A, Fine => true);
│ │ │ - -- .116686s elapsed
    
│ │ │ + -- .0894708s 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,
│ │ │ @@ -216,15 +216,15 @@
│ │ │  
│ │ │  o7 : Triangulation
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : elapsedTime Ts2 = generateTriangulations T;
│ │ │ - -- 2.08042s elapsed
    
│ │ │ + -- .998968s elapsed
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i9 : #Ts2 == #Ts
│ │ │  
│ │ │  o9 = true
    │ │ │ ├── html2text {} │ │ │ │ @@ -88,15 +88,15 @@ │ │ │ │ | 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); │ │ │ │ - -- .116686s elapsed │ │ │ │ + -- .0894708s 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, │ │ │ │ @@ -120,15 +120,15 @@ │ │ │ │ 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; │ │ │ │ - -- 2.08042s elapsed │ │ │ │ + -- .998968s elapsed │ │ │ │ i9 : #Ts2 == #Ts │ │ │ │ │ │ │ │ o9 = true │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _P_o_l_y_h_e_d_r_a -- convex polyhedra │ │ │ │ * _T_o_p_c_o_m -- interface to the topcom software package which in particular │ │ │ │ computes triangulations │ │ ├── ./usr/share/doc/Macaulay2/Triplets/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=7 │ │ │ cm90Rm9ydw== │ │ │ #:len=229 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gODM2LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyJyb3RGb3J3Iiwicm90Rm9ydyIsIlRyaXBsZXRzIn0s │ │ ├── ./usr/share/doc/Macaulay2/Tropical/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ aXNCYWxhbmNlZA== │ │ │ #:len=1097 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY2hlY2tzIHdoZXRoZXIgYSB0cm9waWNh │ │ │ bCBjeWNsZSBpcyBiYWxhbmNlZCIsICJsaW5lbnVtIiA9PiA5NTEsIElucHV0cyA9PiB7U1BBTntU │ │ ├── ./usr/share/doc/Macaulay2/TropicalToric/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=43 │ │ │ Y2xhc3NGcm9tVHJvcGljYWwoTm9ybWFsVG9yaWNWYXJpZXR5LElkZWFsKQ== │ │ │ #:len=335 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTM1LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhjbGFzc0Zyb21Ucm9waWNhbCxOb3JtYWxUb3JpY1Zh │ │ ├── ./usr/share/doc/Macaulay2/Truncations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=30 │ │ │ dHJ1bmNhdGUoSW5maW5pdGVOdW1iZXIsVGhpbmcp │ │ │ #:len=222 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjAsICJ1bmRvY3VtZW50ZWQiID0+IHRy │ │ │ dWUsIHN5bWJvbCBEb2N1bWVudFRhZyA9PiBuZXcgRG9jdW1lbnRUYWcgZnJvbSB7KHRydW5jYXRl │ │ ├── ./usr/share/doc/Macaulay2/Units/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=5 │ │ │ VW5pdHM= │ │ │ #:len=298 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5pdHMgY29udmVyc2lvbiBhbmQgcGh5 │ │ │ c2ljYWwgY29uc3RhbnRzIiwgRGVzY3JpcHRpb24gPT4gMTooRElWe1BBUkF7VEVYeyJUaGlzIHBh │ │ ├── ./usr/share/doc/Macaulay2/VNumber/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ c3RhYmxlTWF4 │ │ │ #:len=1353 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiY29tcHV0ZSB0aGUgc2V0IG9mIHN0YWJs │ │ │ ZSBwcmltZXMgb2YgYSBtb25vbWlhbCBpZGVhbCB0aGF0IGFyZSBtYXhpbWFsIHdpdGggcmVzcGVj │ │ ├── ./usr/share/doc/Macaulay2/Valuations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=9 │ │ │ VmFsdWF0aW9u │ │ │ #:len=1226 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiVGhlIHR5cGUgb2YgYWxsIHZhbHVhdGlv │ │ │ bnMiLCAibGluZW51bSIgPT4gNzk4LCBTZWVBbHNvID0+IERJVntIRUFERVIyeyJTZWUgYWxzbyJ9 │ │ ├── ./usr/share/doc/Macaulay2/Varieties/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=16 │ │ │ PiBJbmZpbml0ZU51bWJlcg== │ │ │ #:len=253 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjk4LCBzeW1ib2wgRG9jdW1lbnRUYWcg │ │ │ PT4gbmV3IERvY3VtZW50VGFnIGZyb20geyhzeW1ib2wgPixJbmZpbml0ZU51bWJlciksIj4gSW5m │ │ ├── ./usr/share/doc/Macaulay2/VectorFields/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=10 │ │ │ Y29tbXV0YXRvcg== │ │ │ #:len=3282 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidGhlIGNvbW11dGF0b3Igb2YgYSBjb2xs │ │ │ ZWN0aW9uIG9mIHZlY3RvciBmaWVsZHMiLCAibGluZW51bSIgPT4gMjE1NSwgSW5wdXRzID0+IHtT │ │ ├── ./usr/share/doc/Macaulay2/VectorGraphics/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ dGV4KFNWRyk= │ │ │ #:len=181 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTg4OCwgInVuZG9jdW1lbnRlZCIgPT4g │ │ │ dHJ1ZSwgc3ltYm9sIERvY3VtZW50VGFnID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsodGV4LFNW │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ bGlmdERlZm9ybWF0aW9uKC4uLixWZXJib3NlPT4uLi4p │ │ │ #:len=285 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjA1MCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbbGlmdERlZm9ybWF0aW9uLFZlcmJvc2VdLCJsaWZ0 │ │ ├── ./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 0.688076s (cpu); 0.53591s (thread); 0s (gc) │ │ │ + -- used 0.6341s (cpu); 0.512558s (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 0.537824s (cpu); 0.386837s (thread); 0s (gc) │ │ │ + -- used 0.51105s (cpu); 0.387935s (thread); 0s (gc) │ │ │ │ │ │ i7 : sum G │ │ │ │ │ │ o7 = | t_1t_16 │ │ │ | 2t_5t_10t_11t_16+t_7t_11^2t_16-2t_6t_10t_16+3t_10^2t_16-t_8t_11t_16+ │ │ │ | -t_5t_10^2t_16-2t_7t_10t_11t_16-3t_2t_11^2t_16+t_8t_10t_16+2t_3t_11t │ │ │ | 2t_5t_10t_16^2+2t_7t_11t_16^2+4t_10t_12t_16+2t_11t_13t_16-t_8t_16^2- │ │ ├── ./usr/share/doc/Macaulay2/VersalDeformations/html/___Smart__Lift.html │ │ │ @@ -76,15 +76,15 @@ │ │ │ │ │ │ │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ @@ -103,15 +103,15 @@ │ │ │ │ │ │
    │ │ │ 
    i3 : time (F,R,G,C)=localHilbertScheme(F0);
│ │ │ - -- used 0.688076s (cpu); 0.53591s (thread); 0s (gc)
    │ │ │ + -- used 0.6341s (cpu); 0.512558s (thread); 0s (gc) │ │ │
    │ │ │ 
    i4 : T=ring first G;
    │ │ │
    │ │ │

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

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -26,15 +26,15 @@ │ │ │ │ o2 : List │ │ │ │ i3 : range1=makeRange(L,{4,6}) │ │ │ │ │ │ │ │ o3 = {4, 6, 8, 12} │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1) │ │ │ │ - -- .850657s elapsed │ │ │ │ + -- .394264s elapsed │ │ │ │ │ │ │ │ 3 2 4 8 2 2 2 6 3 2 │ │ │ │ o4 = (true, ideal (x - x x - x z - x z , x x - x - x z , x - x x + │ │ │ │ 0 1 3 0 0 0 1 3 0 1 0 3 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 4 6 8 │ │ │ │ x x z - x x z + x z ), {0}) │ │ │ │ @@ -46,23 +46,23 @@ │ │ │ │ i5 : range2=drop(makeRange(L,{1}),9) │ │ │ │ │ │ │ │ o5 = {10, 11, 12, 13, 14, 15} │ │ │ │ │ │ │ │ o5 : List │ │ │ │ i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1) │ │ │ │ time to decompose J1 : │ │ │ │ - -- .000769529s elapsed │ │ │ │ + -- .000809625s elapsed │ │ │ │ │ │ │ │ component number = 0 │ │ │ │ deformation weights = {{10}} │ │ │ │ semigroup = {4, 5, 7} │ │ │ │ smoothing components numbers = {0} │ │ │ │ - -- .00146584s elapsed │ │ │ │ + -- .000935996s elapsed │ │ │ │ flat = true │ │ │ │ - -- .720102s elapsed │ │ │ │ + -- .370054s elapsed │ │ │ │ │ │ │ │ 3 2 2 10 3 2 10 │ │ │ │ o6 = (true, ideal (x - x x , x x - x - x z , x - x x - x z ), {0}) │ │ │ │ 0 1 3 0 1 3 0 1 0 3 1 │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_prune__Family.html │ │ │ @@ -132,15 +132,15 @@ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : time (F,R,G,C)=localHilbertScheme(F0,SmartLift=>false);
│ │ │ - -- used 0.537824s (cpu); 0.386837s (thread); 0s (gc)
    │ │ │ + -- used 0.51105s (cpu); 0.387935s (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 0.688076s (cpu); 0.53591s (thread); 0s (gc)
│ │ │ │ + -- used 0.6341s (cpu); 0.512558s (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 0.537824s (cpu); 0.386837s (thread); 0s (gc)
│ │ │ │ + -- used 0.51105s (cpu); 0.387935s (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/VirtualResolutions/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=42
│ │ │  bXVsdGlncmFkZWRSZWd1bGFyaXR5KC4uLixMb3dlckxpbWl0PT4uLi4p
│ │ │  #:len=334
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gNjAzLCBzeW1ib2wgRG9jdW1lbnRUYWcg
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│ │ ├── ./usr/share/doc/Macaulay2/Visualize/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=29
│ │ │  dmlzdWFsaXplKEdyYXBoLFZlcmJvc2U9Pi4uLik=
│ │ │  #:len=1329
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYW4gdG8gdmlldyBjb21tdW5pY2F0aW9u
│ │ │  IGJldHdlZW4gTWFjYXVsYXkyIHNlcnZlciBhbmQgYnJvd3NlciAiLCAibGluZW51bSIgPT4gMjEw
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=35
│ │ │  Z2V0UmFuZ2VPZk9uZVBhcmFtZXRlckZhbWlseShJZGVhbCk=
│ │ │  #:len=349
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjMwMSwgc3ltYm9sIERvY3VtZW50VGFn
│ │ │  ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2V0UmFuZ2VPZk9uZVBhcmFtZXRlckZhbWlseSxJ
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_append__Family.out
│ │ │ @@ -21,15 +21,15 @@
│ │ │  i5 : setRandomSeed("always successful");
│ │ │   -- setting random seed to 11500776554390917551191162798934277
│ │ │  
│ │ │  i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │  number of components = 1, codimension of components = {0}
│ │ │  semigroup = {5, 7, 9}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .817637s elapsed
│ │ │ + -- .530796s elapsed
│ │ │  
│ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z 
│ │ │                      2    0 4    2      0            0    2 4    4      0 2  
│ │ │       ------------------------------------------------------------------------
│ │ │              9    3 10        11        13    2 15      16      18      20  
│ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ @@ -43,15 +43,15 @@
│ │ │       x z   - x z   - z  ))
│ │ │        0       0
│ │ │  
│ │ │  o6 : Sequence
│ │ │  
│ │ │  i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │  semigroup = {5, 7, 9}
│ │ │ - -- .0189065s elapsed
│ │ │ + -- .0165426s elapsed
│ │ │  
│ │ │  o7 = true
│ │ │  
│ │ │  i8 : assert(flatten drop(degrees ring J,-1)==L)
│ │ │  
│ │ │  i9 : "appendFamily(L,J,X,Xdbm)";
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_get__Smoothing__Family__With__Versal__Deformation.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  Calculating first order relations
│ │ │  Calculating standard expressions for obstructions
│ │ │  Starting lifting
│ │ │  Order 2
│ │ │  Order 3
│ │ │  Order 4
│ │ │  Solution is polynomial
│ │ │ - -- .253446s elapsed
│ │ │ + -- .131429s elapsed
│ │ │  
│ │ │                      3    2      2   4        5    3 6    2 7        9      10
│ │ │  o2 = (true, ideal (x  - x x  - x x z  - x x z  - x z  - x z  + x x z  - x z  
│ │ │                      2    0 1    0 2      0 1      0      2      0 2      1   
│ │ │       ------------------------------------------------------------------------
│ │ │          2 11      14       16     21   3      2        4    2   5        6  
│ │ │       - x z   + x z   - 3x z   - 2z  , x x  - x  - x x z  - x x z  - x x z  -
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_make__Range.out
│ │ │ @@ -15,15 +15,15 @@
│ │ │  i3 : range1=makeRange(L,{4,6})
│ │ │  
│ │ │  o3 = {4, 6, 8, 12}
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ - -- .850657s elapsed
│ │ │ + -- .394264s elapsed
│ │ │  
│ │ │                      3           2 4      8     2    2    2 6   3    2    
│ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │                      0    1 3    0      0     0 1    3    0     1    0 3  
│ │ │       ------------------------------------------------------------------------
│ │ │            4        6      8
│ │ │       x x z  - x x z  + x z ), {0})
│ │ │ @@ -35,23 +35,23 @@
│ │ │  
│ │ │  o5 = {10, 11, 12, 13, 14, 15}
│ │ │  
│ │ │  o5 : List
│ │ │  
│ │ │  i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │  time to decompose J1 : 
│ │ │ - -- .000769529s elapsed
│ │ │ + -- .000809625s elapsed
│ │ │  
│ │ │  component number = 0
│ │ │  deformation weights = {{10}}
│ │ │  semigroup = {4, 5, 7}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .00146584s elapsed
│ │ │ + -- .000935996s elapsed
│ │ │  flat = true
│ │ │ - -- .720102s elapsed
│ │ │ + -- .370054s elapsed
│ │ │  
│ │ │                      3            2    2      10   3    2        10
│ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │  
│ │ │  o6 : Sequence
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_prune__Family.out
│ │ │ @@ -23,15 +23,15 @@
│ │ │  i6 : as=apply(numgens I,i->a=drop(support unfolding_{i},#L));
│ │ │  
│ │ │  i7 : rL=apply(#as,i->select(as_i,m->(degree m)_0> b));
│ │ │  
│ │ │  i8 : restrictionList=apply(flatten rL,m->sub(m,A));
│ │ │  
│ │ │  i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ - -- .325212s elapsed
│ │ │ + -- .175756s elapsed
│ │ │  
│ │ │  i10 : leadTerm J
│ │ │  
│ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │                {1, 4} {2, 6}   {2, 4} {1, 4}   {2, 3} {1, 4}   {2, 4} {1, 3} 
│ │ │        -----------------------------------------------------------------------
│ │ │        a      a      , a      a      )
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/example-output/_smoothness__With__Reductions.out
│ │ │ @@ -27,12 +27,12 @@
│ │ │          0      3       1       0      0    1 3    0 3      3
│ │ │  
│ │ │  o4 : Ideal of R
│ │ │  
│ │ │  i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │  semigroup = {5, 6, 8}
│ │ │  dim and degree singF = (0, 4)
│ │ │ - -- .0188212s elapsed
│ │ │ + -- .0186353s elapsed
│ │ │  
│ │ │  o5 = true
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_append__Family.html
│ │ │ @@ -113,15 +113,15 @@
│ │ │
    │ │ │ 
    i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1)
│ │ │  number of components = 1, codimension of components = {0}
│ │ │  semigroup = {5, 7, 9}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .817637s elapsed
│ │ │ + -- .530796s elapsed
│ │ │  
│ │ │                      2             7      9     14   5      2    2 7    2   8
│ │ │  o6 = (true, ideal (x  - x x  - x z  - x z  - 2z  , x  - x x  - x z  - x x z 
│ │ │                      2    0 4    2      0            0    2 4    4      0 2  
│ │ │       ------------------------------------------------------------------------
│ │ │              9    3 10        11        13    2 15      16      18      20  
│ │ │       - x x z  - x z   - x x z   - x x z   - x z   - x z   - x z   - x z   -
│ │ │ @@ -138,15 +138,15 @@
│ │ │  o6 : Sequence
    │ │ │
    │ │ │ 
    i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1)
│ │ │  semigroup = {5, 7, 9}
│ │ │ - -- .0189065s elapsed
│ │ │ + -- .0165426s elapsed
│ │ │  
│ │ │  o7 = true
    │ │ │
    │ │ │ 
    i8 : assert(flatten drop(degrees ring J,-1)==L)
    │ │ │ ├── html2text {} │ │ │ │ @@ -30,15 +30,15 @@ │ │ │ │ o4 = fam8.dbm │ │ │ │ i5 : setRandomSeed("always successful"); │ │ │ │ -- setting random seed to 11500776554390917551191162798934277 │ │ │ │ i6 : elapsedTime (smooth,J)=getSmoothingFamily(L,6,Verbose=>1) │ │ │ │ number of components = 1, codimension of components = {0} │ │ │ │ semigroup = {5, 7, 9} │ │ │ │ smoothing components numbers = {0} │ │ │ │ - -- .817637s elapsed │ │ │ │ + -- .530796s elapsed │ │ │ │ │ │ │ │ 2 7 9 14 5 2 2 7 2 8 │ │ │ │ o6 = (true, ideal (x - x x - x z - x z - 2z , x - x x - x z - x x z │ │ │ │ 2 0 4 2 0 0 2 4 4 0 2 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 9 3 10 11 13 2 15 16 18 20 │ │ │ │ - x x z - x z - x x z - x x z - x z - x z - x z - x z - │ │ │ │ @@ -51,15 +51,15 @@ │ │ │ │ 2 17 22 27 │ │ │ │ x z - x z - z )) │ │ │ │ 0 0 │ │ │ │ │ │ │ │ o6 : Sequence │ │ │ │ i7 : elapsedTime smoothnessWithReductions(J,Verbose=>1) │ │ │ │ semigroup = {5, 7, 9} │ │ │ │ - -- .0189065s elapsed │ │ │ │ + -- .0165426s elapsed │ │ │ │ │ │ │ │ o7 = true │ │ │ │ i8 : assert(flatten drop(degrees ring J,-1)==L) │ │ │ │ i9 : "appendFamily(L,J,X,Xdbm)"; │ │ │ │ Reading and writing to the disk does not work in the documentation. Hence we │ │ │ │ give the command in quotes. │ │ │ │ ********** SSeeee aallssoo ********** │ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_get__Smoothing__Family__With__Versal__Deformation.html │ │ │ @@ -99,15 +99,15 @@ │ │ │ Calculating first order relations │ │ │ Calculating standard expressions for obstructions │ │ │ Starting lifting │ │ │ Order 2 │ │ │ Order 3 │ │ │ Order 4 │ │ │ Solution is polynomial │ │ │ - -- .253446s elapsed │ │ │ + -- .131429s elapsed │ │ │ │ │ │ 3 2 2 4 5 3 6 2 7 9 10 │ │ │ o2 = (true, ideal (x - x x - x x z - x x z - x z - x z + x x z - x z │ │ │ 2 0 1 0 2 0 1 0 2 0 2 1 │ │ │ ------------------------------------------------------------------------ │ │ │ 2 11 14 16 21 3 2 4 2 5 6 │ │ │ - x z + x z - 3x z - 2z , x x - x - x x z - x x z - x x z - │ │ │ ├── html2text {} │ │ │ │ @@ -32,15 +32,15 @@ │ │ │ │ Calculating first order relations │ │ │ │ Calculating standard expressions for obstructions │ │ │ │ Starting lifting │ │ │ │ Order 2 │ │ │ │ Order 3 │ │ │ │ Order 4 │ │ │ │ Solution is polynomial │ │ │ │ - -- .253446s elapsed │ │ │ │ + -- .131429s elapsed │ │ │ │ │ │ │ │ 3 2 2 4 5 3 6 2 7 9 10 │ │ │ │ o2 = (true, ideal (x - x x - x x z - x x z - x z - x z + x x z - x z │ │ │ │ 2 0 1 0 2 0 1 0 2 0 2 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ 2 11 14 16 21 3 2 4 2 5 6 │ │ │ │ - x z + x z - 3x z - 2z , x x - x - x x z - x x z - x x z - │ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_make__Range.html │ │ │ @@ -106,15 +106,15 @@ │ │ │ │ │ │ o3 : List │ │ │
    │ │ │ 
    i4 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range1)
│ │ │ - -- .850657s elapsed
│ │ │ + -- .394264s elapsed
│ │ │  
│ │ │                      3           2 4      8     2    2    2 6   3    2    
│ │ │  o4 = (true, ideal (x  - x x  - x z  - x z , x x  - x  - x z , x  - x x  +
│ │ │                      0    1 3    0      0     0 1    3    0     1    0 3  
│ │ │       ------------------------------------------------------------------------
│ │ │            4        6      8
│ │ │       x x z  - x x z  + x z ), {0})
│ │ │ @@ -137,23 +137,23 @@
│ │ │  o5 : List
    │ │ │
    │ │ │ 
    i6 : elapsedTime (smooth,fib, comps)=getSmoothingFamily(L,range2,Verbose=>1)
│ │ │  time to decompose J1 : 
│ │ │ - -- .000769529s elapsed
│ │ │ + -- .000809625s elapsed
│ │ │  
│ │ │  component number = 0
│ │ │  deformation weights = {{10}}
│ │ │  semigroup = {4, 5, 7}
│ │ │   smoothing components numbers = {0}
│ │ │ - -- .00146584s elapsed
│ │ │ + -- .000935996s elapsed
│ │ │  flat = true
│ │ │ - -- .720102s elapsed
│ │ │ + -- .370054s elapsed
│ │ │  
│ │ │                      3            2    2      10   3    2        10
│ │ │  o6 = (true, ideal (x  - x x , x x  - x  - x z  , x  - x x  - x z  ), {0})
│ │ │                      0    1 3   0 1    3    0      1    0 3    1
│ │ │  
│ │ │  o6 : Sequence
    │ │ │     		│ │ │ 
    i8 : restrictionList=apply(flatten rL,m->sub(m,A));
    │ │ │
    │ │ │ 
    i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ - -- .325212s elapsed
    │ │ │ + -- .175756s elapsed │ │ │
    │ │ │ 
    i10 : leadTerm J
│ │ │  
│ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │ ├── html2text {}
│ │ │ │ @@ -34,15 +34,15 @@
│ │ │ │                10007  0   1   3   4
│ │ │ │  i4 : (A,unfolding)=makeUnfolding I;
│ │ │ │  i5 : b = genus L-1;
│ │ │ │  i6 : as=apply(numgens I,i->a=drop(support unfolding_{i},#L));
│ │ │ │  i7 : rL=apply(#as,i->select(as_i,m->(degree m)_0> b));
│ │ │ │  i8 : restrictionList=apply(flatten rL,m->sub(m,A));
│ │ │ │  i9 : elapsedTime (J,family)=getFlatFamily(I,A,unfolding,restrictionList);
│ │ │ │ - -- .325212s elapsed
│ │ │ │ + -- .175756s elapsed
│ │ │ │  i10 : leadTerm J
│ │ │ │  
│ │ │ │  o10 = ideal (a      a      , a      a      , a      a      , a      a      ,
│ │ │ │                {1, 4} {2, 6}   {2, 4} {1, 4}   {2, 3} {1, 4}   {2, 4} {1, 3}
│ │ │ │        -----------------------------------------------------------------------
│ │ │ │        a      a      , a      a      )
│ │ │ │         {2, 3} {1, 3}   {1, 2} {2, 3}
│ │ ├── ./usr/share/doc/Macaulay2/WeierstrassSemigroups/html/_smoothness__With__Reductions.html
│ │ │ @@ -122,15 +122,15 @@
│ │ │
    │ │ │ 
    i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2)
│ │ │  semigroup = {5, 6, 8}
│ │ │  dim and degree singF = (0, 4)
│ │ │ - -- .0188212s elapsed
│ │ │ + -- .0186353s elapsed
│ │ │  
│ │ │  o5 = true
    │ │ │
    │ │ │
    │ │ │

    The intermediate output dim and degree singF = (0, 4) says that after computing some minors of the jacobian matrix, we detect that the curve is smooth away from the zero dimensional scheme defined by singF of degree 4.

    │ │ │ ├── html2text {} │ │ │ │ @@ -49,15 +49,15 @@ │ │ │ │ - x z - x z - x z + x z , x - x x + x x z + x z - z ) │ │ │ │ 0 3 1 0 0 1 3 0 3 3 │ │ │ │ │ │ │ │ o4 : Ideal of R │ │ │ │ i5 : elapsedTime smoothnessWithReductions(J,Verbose=>2) │ │ │ │ semigroup = {5, 6, 8} │ │ │ │ dim and degree singF = (0, 4) │ │ │ │ - -- .0188212s elapsed │ │ │ │ + -- .0186353s elapsed │ │ │ │ │ │ │ │ o5 = true │ │ │ │ The intermediate output dim and degree singF = (0, 4) says that after computing │ │ │ │ some minors of the jacobian matrix, we detect that the curve is smooth away │ │ │ │ from the zero dimensional scheme defined by singF of degree 4. │ │ │ │ The function checkSmoothness takes longer, some times much longer. │ │ │ │ ********** WWaayyss ttoo uussee ssmmooootthhnneessssWWiitthhRReedduuccttiioonnss:: ********** │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=19 │ │ │ Zmxvb3IoUldlaWxEaXZpc29yKQ== │ │ │ #:len=275 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMjU1Niwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZmxvb3IsUldlaWxEaXZpc29yKSwiZmxvb3IoUldl │ │ ├── ./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 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ 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*Div(z) + 4/3*Div(y) + 2/3*Div(x) │ │ │ +o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y) │ │ │ │ │ │ o4 : QWeilDivisor on R │ │ │ │ │ │ i5 : 0.6*D │ │ │ │ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x) │ │ │ +o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y) │ │ │ │ │ │ o5 : RWeilDivisor on R │ │ │ │ │ │ i6 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Basic__Divisor_sp_pl_sp__Basic__Divisor.out │ │ │ @@ -64,30 +64,30 @@ │ │ │ │ │ │ o12 : RWeilDivisor on R │ │ │ │ │ │ i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z); │ │ │ │ │ │ i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)}) │ │ │ │ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z) │ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z) │ │ │ │ │ │ o14 : WeilDivisor on R │ │ │ │ │ │ i15 : -D │ │ │ │ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z) │ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z) │ │ │ │ │ │ o15 : WeilDivisor on R │ │ │ │ │ │ i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z) │ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z) │ │ │ │ │ │ o16 : WeilDivisor on R │ │ │ │ │ │ i17 : -E │ │ │ │ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z) │ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z) │ │ │ │ │ │ o17 : WeilDivisor on R │ │ │ │ │ │ i18 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/___Number_sp_st_sp__Basic__Divisor.out │ │ │ @@ -10,27 +10,27 @@ │ │ │ │ │ │ o3 = 1/2*Div(x) + -5/3*Div(y) │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ │ │ i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, CoefficientType=>RR) │ │ │ │ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2) │ │ │ +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) + 8*Div(y) + 16*Div(x) │ │ │ +o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y) │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ i6 : (-2/3)*D │ │ │ │ │ │ -o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x) │ │ │ +o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y) │ │ │ │ │ │ o6 : QWeilDivisor on R │ │ │ │ │ │ i7 : 0.0*D │ │ │ │ │ │ o7 = 0, the zero divisor │ │ │ │ │ │ @@ -46,18 +46,18 @@ │ │ │ │ │ │ o9 = 2.35667*Div(y) + -.707*Div(x) │ │ │ │ │ │ o9 : RWeilDivisor on R │ │ │ │ │ │ i10 : 6*F │ │ │ │ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2) │ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x) │ │ │ │ │ │ o10 : RWeilDivisor on R │ │ │ │ │ │ i11 : (-3/2)*F │ │ │ │ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2) │ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x) │ │ │ │ │ │ o11 : RWeilDivisor on R │ │ │ │ │ │ i12 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_apply__To__Coefficients.out │ │ │ @@ -1,17 +1,17 @@ │ │ │ -- -*- M2-comint -*- hash: 14937934652040812889 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D = divisor(x*y^2/z) │ │ │ │ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y) │ │ │ +o2 = -Div(z) + 2*Div(y) + Div(x) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : applyToCoefficients(D, u->5*u) │ │ │ │ │ │ -o3 = 5*Div(x) + 10*Div(y) + -5*Div(z) │ │ │ +o3 = 10*Div(y) + -5*Div(z) + 5*Div(x) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : │ │ ├── ./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/_clean__Support.out │ │ │ @@ -1,17 +1,17 @@ │ │ │ -- -*- M2-comint -*- hash: 2803818461661759459 │ │ │ │ │ │ i1 : R = QQ[x,y,z]; │ │ │ │ │ │ i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ -o2 = Div(x) + 0*Div(y) + -2*Div(z) │ │ │ +o2 = 0*Div(y) + -2*Div(z) + Div(x) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : cleanSupport(D) │ │ │ │ │ │ -o3 = Div(x) + -2*Div(z) │ │ │ +o3 = -2*Div(z) + Div(x) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_divisor.out │ │ │ @@ -24,15 +24,15 @@ │ │ │ │ │ │ o5 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z) │ │ │ │ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y) │ │ │ +o6 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ o6 : WeilDivisor on R │ │ │ │ │ │ i7 : R = QQ[x,y,z]/ideal(x^2-y*z); │ │ │ │ │ │ i8 : D = divisor({2}, {ideal(x,y)}) │ │ ├── ./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 0.0706325s (cpu); 0.0706342s (thread); 0s (gc) │ │ │ + -- used 0.0430289s (cpu); 0.0430278s (thread); 0s (gc) │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc) │ │ │ + -- used 0.381629s (cpu); 0.381637s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc) │ │ │ + -- used 0.511364s (cpu); 0.438785s (thread); 0s (gc) │ │ │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc) │ │ │ + -- used 0.0027159s (cpu); 0.00272092s (thread); 0s (gc) │ │ │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc) │ │ │ + -- used 0.00248992s (cpu); 0.00250652s (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 0.0637156s (cpu); 0.0637222s (thread); 0s (gc) │ │ │ + -- used 0.0728028s (cpu); 0.0728017s (thread); 0s (gc) │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc) │ │ │ + -- used 0.00631491s (cpu); 0.00632084s (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/_get__Prime__Divisors.out │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 6816075660530398278 │ │ │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ │ │ i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)}) │ │ │ │ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z) │ │ │ +o2 = 0*Div(x^2+z) + -8*Div(x) + 2*Div(y) │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : getPrimeDivisors( D ) │ │ │ │ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)} │ │ │ +o3 = {Div(x^2+z), Div(x), Div(y)} │ │ │ │ │ │ o3 : List │ │ │ │ │ │ i4 : getPrimeDivisors( cleanSupport D ) │ │ │ │ │ │ o4 = {Div(x), Div(y)} │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_is__Cartier.out │ │ │ @@ -24,15 +24,15 @@ │ │ │ │ │ │ o6 = false │ │ │ │ │ │ i7 : R = QQ[x, y, z]; │ │ │ │ │ │ i8 : D = divisor({1, 2}, {ideal(x), ideal(y)}) │ │ │ │ │ │ -o8 = 2*Div(y) + Div(x) │ │ │ +o8 = Div(x) + 2*Div(y) │ │ │ │ │ │ o8 : WeilDivisor on R │ │ │ │ │ │ i9 : isCartier( D ) │ │ │ │ │ │ o9 = true │ │ │ │ │ │ @@ -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 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z) │ │ │ +o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*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 │ │ │ @@ -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 │ │ │ @@ -1,20 +1,20 @@ │ │ │ -- -*- M2-comint -*- hash: 13920959388108803216 │ │ │ │ │ │ i1 : R = QQ[x, y, z] / ideal(x * y - z^2); │ │ │ │ │ │ i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ) │ │ │ │ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z) │ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z) │ │ │ │ │ │ o2 : QWeilDivisor on R │ │ │ │ │ │ i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ) │ │ │ │ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z) │ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z) │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ │ │ i4 : isQLinearEquivalent(10, D, E) │ │ │ │ │ │ o4 = true │ │ │ │ │ │ @@ -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 │ │ │ @@ -12,15 +12,15 @@ │ │ │ │ │ │ o3 = false │ │ │ │ │ │ i4 : R = QQ[x, y]; │ │ │ │ │ │ i5 : D = divisor(x*y*(x+y)) │ │ │ │ │ │ -o5 = Div(x+y) + Div(y) + Div(x) │ │ │ +o5 = Div(y) + Div(x) + Div(x+y) │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ i6 : isSNC( D ) │ │ │ │ │ │ o6 = 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(y) + Div(x) + Div(x+y) │ │ │ │ │ │ o17 : WeilDivisor on R │ │ │ │ │ │ i18 : isSNC( D, IsGraded => true) │ │ │ │ │ │ o18 = false │ │ ├── ./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) + 3*Div(b) + Div(a+1) │ │ │ +o11 = Div(a+1) + Div(a) + 3*Div(b) │ │ │ │ │ │ o11 : WeilDivisor on S │ │ │ │ │ │ i12 : f^* D │ │ │ │ │ │ -o12 = Div(a) + 3*Div(b) + Div(a+1) │ │ │ +o12 = Div(a+1) + Div(a) + 3*Div(b) │ │ │ │ │ │ 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 0.338433s (cpu); 0.253168s (thread); 0s (gc) │ │ │ + -- used 0.243393s (cpu); 0.17634s (thread); 0s (gc) │ │ │ │ │ │ o23 : Ideal of R │ │ │ │ │ │ i24 : time reflexify(J*R^1); │ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc) │ │ │ + -- used 0.390243s (cpu); 0.3239s (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 0.168829s (cpu); 0.168835s (thread); 0s (gc) │ │ │ + -- used 0.0909353s (cpu); 0.0909381s (thread); 0s (gc) │ │ │ │ │ │ 2 2 9 9 11 │ │ │ o29 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ o29 : Ideal of R │ │ │ │ │ │ i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy ) │ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc) │ │ │ + -- used 5.34768s (cpu); 4.02232s (thread); 0s (gc) │ │ │ │ │ │ 2 2 9 9 11 │ │ │ o30 = ideal (x + 5x*y + 3y , x*z - 4y*z , z + x - 4y) │ │ │ │ │ │ o30 : Ideal of R │ │ │ │ │ │ i31 : J1 == J2 │ │ │ │ │ │ o31 = true │ │ │ │ │ │ i32 : time reflexify( M, Strategy=>IdealStrategy ); │ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc) │ │ │ + -- used 5.4074s (cpu); 4.07129s (thread); 0s (gc) │ │ │ │ │ │ i33 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc) │ │ │ + -- used 0.495501s (cpu); 0.332526s (thread); 0s (gc) │ │ │ │ │ │ i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v); │ │ │ │ │ │ i35 : I = ideal(x,u); │ │ │ │ │ │ o35 : Ideal of R │ │ │ │ │ │ i36 : J = I^20; │ │ │ │ │ │ o36 : Ideal of R │ │ │ │ │ │ i37 : M = I^20*R^1; │ │ │ │ │ │ i38 : time reflexify( J, Strategy=>IdealStrategy ) │ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc) │ │ │ + -- used 0.391858s (cpu); 0.232942s (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 0.0135561s (cpu); 0.0135572s (thread); 0s (gc) │ │ │ + -- used 0.0136439s (cpu); 0.0136467s (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 0.0397151s (cpu); 0.0397234s (thread); 0s (gc) │ │ │ + -- used 0.0390337s (cpu); 0.0390383s (thread); 0s (gc) │ │ │ │ │ │ i41 : time reflexify( M, Strategy=>ModuleStrategy ); │ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc) │ │ │ + -- used 0.006433s (cpu); 0.00643849s (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 0.0292441s (cpu); 0.0292465s (thread); 0s (gc) │ │ │ + -- used 0.0300898s (cpu); 0.0300054s (thread); 0s (gc) │ │ │ │ │ │ o7 : Ideal of R │ │ │ │ │ │ i8 : I20 = I^20; │ │ │ │ │ │ o8 : Ideal of R │ │ │ │ │ │ i9 : time J20b = reflexify(I20); │ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc) │ │ │ + -- used 0.134946s (cpu); 0.134946s (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 0.0299195s (cpu); 0.0299242s (thread); 0s (gc) │ │ │ + -- used 0.0319431s (cpu); 0.0319472s (thread); 0s (gc) │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ │ │ i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy); │ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc) │ │ │ + -- used 0.0600871s (cpu); 0.0600926s (thread); 0s (gc) │ │ │ │ │ │ o14 : Ideal of R │ │ │ │ │ │ i15 : J1 == J2 │ │ │ │ │ │ o15 = true │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/example-output/_trim_lp__Basic__Divisor_rp.out │ │ │ @@ -4,21 +4,21 @@ │ │ │ │ │ │ i2 : D = divisor({1,0,-2}, {ideal(x, z), ideal(x-z,y-z), ideal(y+z, z)}); │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ │ │ i3 : cleanSupport(D) │ │ │ │ │ │ -o3 = -2*Div(y+z, z) + Div(x, z) │ │ │ +o3 = Div(x, z) + -2*Div(y+z, z) │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ │ │ i4 : trim(D) │ │ │ │ │ │ -o4 = -2*Div(z, y) + Div(z, x) │ │ │ +o4 = Div(z, x) + -2*Div(z, y) │ │ │ │ │ │ o4 : WeilDivisor on R │ │ │ │ │ │ i5 : D == trim(D) │ │ │ │ │ │ o5 = true │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Basic__Divisor.html │ │ │ @@ -77,15 +77,15 @@ │ │ │ 
    i1 : R = QQ[x,y,z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor(x*y^2*z^3)
│ │ │  
│ │ │ -o2 = 2*Div(y) + 3*Div(z) + Div(x)
│ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : H = new HashTable from D
│ │ │ @@ -99,24 +99,24 @@
│ │ │  o3 : HashTable
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : (2/3)*D
│ │ │  
│ │ │ -o4 = 2*Div(z) + 4/3*Div(y) + 2/3*Div(x)
│ │ │ +o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y)
│ │ │  
│ │ │  o4 : QWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : 0.6*D
│ │ │  
│ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x)
│ │ │ +o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y)
│ │ │  
│ │ │  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 = 2*Div(y) + 3*Div(z) + Div(x) │ │ │ │ +o2 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ │ │ 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*Div(z) + 4/3*Div(y) + 2/3*Div(x) │ │ │ │ +o4 = 2/3*Div(x) + 2*Div(z) + 4/3*Div(y) │ │ │ │ │ │ │ │ o4 : QWeilDivisor on R │ │ │ │ i5 : 0.6*D │ │ │ │ │ │ │ │ -o5 = 1.8*Div(z) + 1.2*Div(y) + .6*Div(x) │ │ │ │ +o5 = .6*Div(x) + 1.8*Div(z) + 1.2*Div(y) │ │ │ │ │ │ │ │ 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/___Basic__Divisor_sp_pl_sp__Basic__Divisor.html │ │ │ @@ -195,42 +195,42 @@ │ │ │ 
    i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z);
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)})
│ │ │  
│ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z)
│ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z)
│ │ │  
│ │ │  o14 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : -D
│ │ │  
│ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z)
│ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z)
│ │ │  
│ │ │  o15 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)})
│ │ │  
│ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z)
│ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z)
│ │ │  
│ │ │  o16 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i17 : -E
│ │ │  
│ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z)
│ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -70,30 +70,30 @@ │ │ │ │ o12 = -Div(y) + 2.75*Div(x) │ │ │ │ │ │ │ │ o12 : RWeilDivisor on R │ │ │ │ Finally, we can negate a divisor. │ │ │ │ i13 : R = ZZ/3[x,y,z]/ideal(x^2-y*z); │ │ │ │ i14 : D = divisor({3, 0, -1}, {ideal(x,z), ideal(y,z), ideal(x-y, x-z)}) │ │ │ │ │ │ │ │ -o14 = 3*Div(x, z) + 0*Div(y, z) + -Div(x-y, x-z) │ │ │ │ +o14 = 0*Div(y, z) + -Div(x-y, x-z) + 3*Div(x, z) │ │ │ │ │ │ │ │ o14 : WeilDivisor on R │ │ │ │ i15 : -D │ │ │ │ │ │ │ │ -o15 = -3*Div(x, z) + Div(x-y, x-z) │ │ │ │ +o15 = Div(x-y, x-z) + -3*Div(x, z) │ │ │ │ │ │ │ │ o15 : WeilDivisor on R │ │ │ │ i16 : E = divisor({3/2, -2/3}, {ideal(x, z), ideal(y, z)}) │ │ │ │ │ │ │ │ -o16 = 3/2*Div(x, z) + -2/3*Div(y, z) │ │ │ │ +o16 = -2/3*Div(y, z) + 3/2*Div(x, z) │ │ │ │ │ │ │ │ o16 : WeilDivisor on R │ │ │ │ i17 : -E │ │ │ │ │ │ │ │ -o17 = -3/2*Div(x, z) + 2/3*Div(y, z) │ │ │ │ +o17 = 2/3*Div(y, z) + -3/2*Div(x, z) │ │ │ │ │ │ │ │ o17 : WeilDivisor on R │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _B_a_s_i_c_D_i_v_i_s_o_r_ _+_ _B_a_s_i_c_D_i_v_i_s_o_r -- add or subtract two divisors, or negate a │ │ │ │ divisor │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/___Number_sp_st_sp__Basic__Divisor.html │ │ │ @@ -99,33 +99,33 @@ │ │ │ o3 : QWeilDivisor on R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, CoefficientType=>RR)
│ │ │  
│ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2)
│ │ │ +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) + 8*Div(y) + 16*Div(x)
│ │ │ +o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : (-2/3)*D
│ │ │  
│ │ │ -o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x)
│ │ │ +o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y)
│ │ │  
│ │ │  o6 : QWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i7 : 0.0*D
│ │ │ @@ -153,24 +153,24 @@
│ │ │  o9 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i10 : 6*F
│ │ │  
│ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2)
│ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x)
│ │ │  
│ │ │  o10 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i11 : (-3/2)*F
│ │ │  
│ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2)
│ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x)
│ │ │  
│ │ │  o11 : RWeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -22,25 +22,25 @@ │ │ │ │ │ │ │ │ o3 = 1/2*Div(x) + -5/3*Div(y) │ │ │ │ │ │ │ │ o3 : QWeilDivisor on R │ │ │ │ i4 : F = divisor({1.5, 0, -3.2}, {ideal(x), ideal(y), ideal(x^2-y^3)}, │ │ │ │ CoefficientType=>RR) │ │ │ │ │ │ │ │ -o4 = 1.5*Div(x) + 0*Div(y) + -3.2*Div(-y^3+x^2) │ │ │ │ +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) + 8*Div(y) + 16*Div(x) │ │ │ │ +o5 = -8*Div(x+y) + 16*Div(x) + 8*Div(y) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ i6 : (-2/3)*D │ │ │ │ │ │ │ │ -o6 = 2/3*Div(x+y) + -2/3*Div(y) + -4/3*Div(x) │ │ │ │ +o6 = 2/3*Div(x+y) + -4/3*Div(x) + -2/3*Div(y) │ │ │ │ │ │ │ │ o6 : QWeilDivisor on R │ │ │ │ i7 : 0.0*D │ │ │ │ │ │ │ │ o7 = 0, the zero divisor │ │ │ │ │ │ │ │ o7 : RWeilDivisor on R │ │ │ │ @@ -52,20 +52,20 @@ │ │ │ │ i9 : (-1.414)*E │ │ │ │ │ │ │ │ o9 = 2.35667*Div(y) + -.707*Div(x) │ │ │ │ │ │ │ │ o9 : RWeilDivisor on R │ │ │ │ i10 : 6*F │ │ │ │ │ │ │ │ -o10 = 9*Div(x) + -19.2*Div(-y^3+x^2) │ │ │ │ +o10 = -19.2*Div(-y^3+x^2) + 9*Div(x) │ │ │ │ │ │ │ │ o10 : RWeilDivisor on R │ │ │ │ i11 : (-3/2)*F │ │ │ │ │ │ │ │ -o11 = -2.25*Div(x) + 4.8*Div(-y^3+x^2) │ │ │ │ +o11 = 4.8*Div(-y^3+x^2) + -2.25*Div(x) │ │ │ │ │ │ │ │ o11 : RWeilDivisor on R │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _N_u_m_b_e_r_ _*_ _B_a_s_i_c_D_i_v_i_s_o_r -- multiply a divisor by a number │ │ │ │ * QQ * RWeilDivisor │ │ │ │ * QQ * WeilDivisor │ │ │ │ * RR * QWeilDivisor │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_apply__To__Coefficients.html │ │ │ @@ -87,24 +87,24 @@ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor(x*y^2/z)
│ │ │  
│ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y)
│ │ │ +o2 = -Div(z) + 2*Div(y) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : applyToCoefficients(D, u->5*u)
│ │ │  
│ │ │ -o3 = 5*Div(x) + 10*Div(y) + -5*Div(z)
│ │ │ +o3 = 10*Div(y) + -5*Div(z) + 5*Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -25,20 +25,20 @@ │ │ │ │ the output D is the same as the class of the input D1 (WeilDivisor, │ │ │ │ QWeilDivisor, RWeilDivisor, BasicDivisor). If Safe is set to true (the default │ │ │ │ is false), then the function will check to make sure the output is a valid │ │ │ │ divisor. │ │ │ │ i1 : R = QQ[x, y, z]; │ │ │ │ i2 : D = divisor(x*y^2/z) │ │ │ │ │ │ │ │ -o2 = Div(x) + -Div(z) + 2*Div(y) │ │ │ │ +o2 = -Div(z) + 2*Div(y) + Div(x) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : applyToCoefficients(D, u->5*u) │ │ │ │ │ │ │ │ -o3 = 5*Div(x) + 10*Div(y) + -5*Div(z) │ │ │ │ +o3 = 10*Div(y) + -5*Div(z) + 5*Div(x) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ ********** SSeeee aallssoo ********** │ │ │ │ * _f_l_o_o_r_(_R_W_e_i_l_D_i_v_i_s_o_r_) -- produce a WeilDivisor whose coefficients are │ │ │ │ ceilings or floors of the divisor │ │ │ │ * _c_e_i_l_i_n_g_(_R_W_e_i_l_D_i_v_i_s_o_r_) -- produce a WeilDivisor whose coefficients are │ │ │ │ ceilings or floors of the divisor │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_ceiling_lp__R__Weil__Divisor_rp.html │ │ │ @@ -83,24 +83,24 @@ │ │ │ 
    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 )
│ │ │ @@ -110,15 +110,15 @@
│ │ │  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/_clean__Support.html
│ │ │ @@ -81,24 +81,24 @@
│ │ │                
    i1 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)})
│ │ │  
│ │ │ -o2 = Div(x) + 0*Div(y) + -2*Div(z)
│ │ │ +o2 = 0*Div(y) + -2*Div(z) + Div(x)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = Div(x) + -2*Div(z)
│ │ │ +o3 = -2*Div(z) + Div(x)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -13,20 +13,20 @@ │ │ │ │ o an instance of the type _B_a_s_i_c_D_i_v_i_s_o_r, │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ This function returns a divisor where all entries with coefficient zero are │ │ │ │ removed. │ │ │ │ i1 : R = QQ[x,y,z]; │ │ │ │ i2 : D = divisor({1,0,-2}, {ideal(x), ideal(y), ideal(z)}) │ │ │ │ │ │ │ │ -o2 = Div(x) + 0*Div(y) + -2*Div(z) │ │ │ │ +o2 = 0*Div(y) + -2*Div(z) + Div(x) │ │ │ │ │ │ │ │ o2 : WeilDivisor on R │ │ │ │ i3 : cleanSupport(D) │ │ │ │ │ │ │ │ -o3 = Div(x) + -2*Div(z) │ │ │ │ +o3 = -2*Div(z) + Div(x) │ │ │ │ │ │ │ │ o3 : WeilDivisor on R │ │ │ │ ********** WWaayyss ttoo uussee cclleeaannSSuuppppoorrtt:: ********** │ │ │ │ * cleanSupport(BasicDivisor) │ │ │ │ ********** FFoorr tthhee pprrooggrraammmmeerr ********** │ │ │ │ The object _c_l_e_a_n_S_u_p_p_o_r_t is a _m_e_t_h_o_d_ _f_u_n_c_t_i_o_n. │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_divisor.html │ │ │ @@ -136,15 +136,15 @@ │ │ │ o5 : WeilDivisor on R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z)
│ │ │  
│ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y)
│ │ │ +o6 = Div(x) + 2*Div(y) + 3*Div(z)
│ │ │  
│ │ │  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.

    │ │ │ ├── html2text {} │ │ │ │ @@ -63,15 +63,15 @@ │ │ │ │ i5 : G = divisor({{1, ideal(x)}, {2, ideal(y)}, {3, ideal(z)}}) │ │ │ │ │ │ │ │ o5 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ │ │ o5 : WeilDivisor on R │ │ │ │ i6 : H = divisor(x) + 2*divisor(y) + 3*divisor(z) │ │ │ │ │ │ │ │ -o6 = Div(x) + 3*Div(z) + 2*Div(y) │ │ │ │ +o6 = Div(x) + 2*Div(y) + 3*Div(z) │ │ │ │ │ │ │ │ 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)}) │ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_dualize.html │ │ │ @@ -168,43 +168,43 @@ │ │ │ │ │ │ 
    i11 : M = J*R^1;
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i12 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0706325s (cpu); 0.0706342s (thread); 0s (gc)
│ │ │ + -- used 0.0430289s (cpu); 0.0430278s (thread); 0s (gc)
│ │ │  
│ │ │  o12 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i13 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc)
│ │ │ + -- used 0.381629s (cpu); 0.381637s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i14 : time dualize(M, Strategy=>IdealStrategy);
│ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc)
    │ │ │ + -- used 0.511364s (cpu); 0.438785s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i15 : time dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc)
    │ │ │ + -- used 0.0027159s (cpu); 0.00272092s (thread); 0s (gc) │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc)
│ │ │ + -- used 0.00248992s (cpu); 0.00250652s (thread); 0s (gc)
│ │ │  
│ │ │  o16 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

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

    │ │ │ @@ -228,23 +228,23 @@ │ │ │ │ │ │ o19 : Ideal of R │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i20 : time dualize(J, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0637156s (cpu); 0.0637222s (thread); 0s (gc)
│ │ │ + -- used 0.0728028s (cpu); 0.0728017s (thread); 0s (gc)
│ │ │  
│ │ │  o20 : Ideal of R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i21 : time dualize(J, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc)
│ │ │ + -- used 0.00631491s (cpu); 0.00632084s (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 0.0706325s (cpu); 0.0706342s (thread); 0s (gc) │ │ │ │ + -- used 0.0430289s (cpu); 0.0430278s (thread); 0s (gc) │ │ │ │ │ │ │ │ o12 : Ideal of R │ │ │ │ i13 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.491025s (cpu); 0.490974s (thread); 0s (gc) │ │ │ │ + -- used 0.381629s (cpu); 0.381637s (thread); 0s (gc) │ │ │ │ │ │ │ │ o13 : Ideal of R │ │ │ │ i14 : time dualize(M, Strategy=>IdealStrategy); │ │ │ │ - -- used 0.528431s (cpu); 0.440429s (thread); 0s (gc) │ │ │ │ + -- used 0.511364s (cpu); 0.438785s (thread); 0s (gc) │ │ │ │ i15 : time dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00269572s (cpu); 0.00269697s (thread); 0s (gc) │ │ │ │ + -- used 0.0027159s (cpu); 0.00272092s (thread); 0s (gc) │ │ │ │ i16 : time embedAsIdeal dualize(M, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00217526s (cpu); 0.00217625s (thread); 0s (gc) │ │ │ │ + -- used 0.00248992s (cpu); 0.00250652s (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 0.0637156s (cpu); 0.0637222s (thread); 0s (gc) │ │ │ │ + -- used 0.0728028s (cpu); 0.0728017s (thread); 0s (gc) │ │ │ │ │ │ │ │ o20 : Ideal of R │ │ │ │ i21 : time dualize(J, Strategy=>ModuleStrategy); │ │ │ │ - -- used 0.00595766s (cpu); 0.00595829s (thread); 0s (gc) │ │ │ │ + -- used 0.00631491s (cpu); 0.00632084s (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/_get__Prime__Divisors.html │ │ │ @@ -81,24 +81,24 @@ │ │ │ 
    i1 : R = QQ[x, y, z];
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │  
│ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ +o2 = 0*Div(x^2+z) + -8*Div(x) + 2*Div(y)
│ │ │  
│ │ │  o2 : WeilDivisor on R
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : getPrimeDivisors( D )
│ │ │  
│ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ +o3 = {Div(x^2+z), Div(x), Div(y)}
│ │ │  
│ │ │  o3 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : getPrimeDivisors( cleanSupport D )
│ │ │ ├── html2text {}
│ │ │ │ @@ -14,20 +14,20 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  This function returns the list of prime divisors of a given divisor. The prime
│ │ │ │  divisors are all of the class WeilDivisor. If you do not call cleanSupport, you
│ │ │ │  may obtain divisors with zero coefficients.
│ │ │ │  i1 : R = QQ[x, y, z];
│ │ │ │  i2 : D = divisor({-8, 2, 0}, {ideal(x), ideal(y), ideal(x^2+z)})
│ │ │ │  
│ │ │ │ -o2 = -8*Div(x) + 2*Div(y) + 0*Div(x^2+z)
│ │ │ │ +o2 = 0*Div(x^2+z) + -8*Div(x) + 2*Div(y)
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : getPrimeDivisors( D )
│ │ │ │  
│ │ │ │ -o3 = {Div(x), Div(y), Div(x^2+z)}
│ │ │ │ +o3 = {Div(x^2+z), Div(x), Div(y)}
│ │ │ │  
│ │ │ │  o3 : List
│ │ │ │  i4 : getPrimeDivisors( cleanSupport D )
│ │ │ │  
│ │ │ │  o4 = {Div(x), Div(y)}
│ │ │ │  
│ │ │ │  o4 : List
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_is__Cartier.html
│ │ │ @@ -137,15 +137,15 @@
│ │ │                
    i7 : R = QQ[x, y, z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
│ │ │  
│ │ │ -o8 = 2*Div(y) + Div(x)
│ │ │ +o8 = Div(x) + 2*Div(y)
│ │ │  
│ │ │  o8 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : isCartier( D )
│ │ │ @@ -186,15 +186,15 @@
│ │ │                
    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 {}
│ │ │ │ @@ -35,15 +35,15 @@
│ │ │ │  i6 : isCartier( D )
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  Of course the next divisor is Cartier.
│ │ │ │  i7 : R = QQ[x, y, z];
│ │ │ │  i8 : D = divisor({1, 2}, {ideal(x), ideal(y)})
│ │ │ │  
│ │ │ │ -o8 = 2*Div(y) + Div(x)
│ │ │ │ +o8 = Div(x) + 2*Div(y)
│ │ │ │  
│ │ │ │  o8 : WeilDivisor on R
│ │ │ │  i9 : isCartier( D )
│ │ │ │  
│ │ │ │  o9 = true
│ │ │ │  If the option IsGraded is set to true (it is false by default), this will check
│ │ │ │  as if D is a divisor on the $Proj$ of the ambient graded ring.
│ │ │ │ @@ -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
│ │ │ @@ -82,15 +82,15 @@
│ │ │                
    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 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │ +o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*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 = Div(x*y-z^2) + 2*Div(-x^2+y*z) + 3*Div(-y^2+x*z)
│ │ │ │ +o2 = 3*Div(-y^2+x*z) + Div(x*y-z^2) + 2*Div(-x^2+y*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
│ │ │ @@ -86,24 +86,24 @@
│ │ │                
    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)
│ │ │ @@ -121,24 +121,24 @@
│ │ │                
    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
│ │ │ @@ -169,24 +169,24 @@
│ │ │                
    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 {}
│ │ │ │ @@ -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
│ │ │ @@ -87,24 +87,24 @@
│ │ │                
    i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │  
│ │ │  o2 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │  
│ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │  
│ │ │  o3 : QWeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : isQLinearEquivalent(10, D, E)
│ │ │ @@ -160,24 +160,24 @@
│ │ │                
    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 {}
│ │ │ │ @@ -19,20 +19,20 @@
│ │ │ │  ********** DDeessccrriippttiioonn **********
│ │ │ │  Given two rational divisors, this method returns true if they linearly
│ │ │ │  equivalent after clearing denominators or if some further multiple up to n
│ │ │ │  makes them linearly equivalent. Otherwise it returns false.
│ │ │ │  i1 : R = QQ[x, y, z] / ideal(x * y - z^2);
│ │ │ │  i2 : D = divisor({1/2, 3/4}, {ideal(x, z), ideal(y, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o2 = 1/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │ +o2 = 3/4*Div(y, z) + 1/2*Div(x, z)
│ │ │ │  
│ │ │ │  o2 : QWeilDivisor on R
│ │ │ │  i3 : E = divisor({3/4, 5/2}, {ideal(y, z), ideal(x, z)}, CoefficientType => QQ)
│ │ │ │  
│ │ │ │ -o3 = 5/2*Div(x, z) + 3/4*Div(y, z)
│ │ │ │ +o3 = 3/4*Div(y, z) + 5/2*Div(x, z)
│ │ │ │  
│ │ │ │  o3 : QWeilDivisor on R
│ │ │ │  i4 : isQLinearEquivalent(10, D, E)
│ │ │ │  
│ │ │ │  o4 = true
│ │ │ │  In the above ring, every pair of divisors is Q-linearly equivalent because the
│ │ │ │  Weil divisor class group is isomorphic to Z/2. However, if we don't set n high
│ │ │ │ @@ -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
│ │ │ @@ -108,15 +108,15 @@
│ │ │                
    i4 : R = QQ[x, y];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o5 = Div(x+y) + Div(y) + Div(x)
│ │ │ +o5 = Div(y) + Div(x) + Div(x+y)
│ │ │  
│ │ │  o5 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i6 : isSNC( D )
│ │ │ @@ -157,15 +157,15 @@
│ │ │                
    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 )
│ │ │ @@ -203,15 +203,15 @@
│ │ │                
    i16 : R = QQ[x,y,z];
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i17 : D = divisor(x*y*(x+y))
│ │ │  
│ │ │ -o17 = Div(x) + Div(x+y) + Div(y)
│ │ │ +o17 = Div(y) + Div(x) + Div(x+y)
│ │ │  
│ │ │  o17 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i18 : isSNC( D, IsGraded => true)
│ │ │ ├── html2text {}
│ │ │ │ @@ -24,15 +24,15 @@
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : isSNC( D )
│ │ │ │  
│ │ │ │  o3 = false
│ │ │ │  i4 : R = QQ[x, y];
│ │ │ │  i5 : D = divisor(x*y*(x+y))
│ │ │ │  
│ │ │ │ -o5 = Div(x+y) + Div(y) + Div(x)
│ │ │ │ +o5 = Div(y) + Div(x) + Div(x+y)
│ │ │ │  
│ │ │ │  o5 : WeilDivisor on R
│ │ │ │  i6 : isSNC( D )
│ │ │ │  
│ │ │ │  o6 = false
│ │ │ │  i7 : R = QQ[x, y];
│ │ │ │  i8 : D = divisor(x*y*(x+1))
│ │ │ │ @@ -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(y) + Div(x) + Div(x+y)
│ │ │ │  
│ │ │ │  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/_pullback_lp__Ring__Map_cm__R__Weil__Divisor_rp.html
│ │ │ @@ -99,24 +99,24 @@
│ │ │  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)
│ │ │ @@ -155,24 +155,24 @@
│ │ │  o10 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i11 : D1 = pullback(f, D)
│ │ │  
│ │ │ -o11 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ +o11 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │  
│ │ │  o11 : WeilDivisor on S
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i12 : f^* D
│ │ │  
│ │ │ -o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ +o12 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │  
│ │ │  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) + 3*Div(b) + Div(a+1)
│ │ │ │ +o11 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ │  
│ │ │ │  o11 : WeilDivisor on S
│ │ │ │  i12 : f^* D
│ │ │ │  
│ │ │ │ -o12 = Div(a) + 3*Div(b) + Div(a+1)
│ │ │ │ +o12 = Div(a+1) + Div(a) + 3*Div(b)
│ │ │ │  
│ │ │ │  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
│ │ │ @@ -272,23 +272,23 @@
│ │ │  
│ │ │  o22 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i23 : time reflexify(J);
│ │ │ - -- used 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ + -- used 0.243393s (cpu); 0.17634s (thread); 0s (gc)
│ │ │  
│ │ │  o23 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i24 : time reflexify(J*R^1);
│ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
    
│ │ │ + -- used 0.390243s (cpu); 0.3239s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    Because of this, there are two strategies for computing a reflexification (at least if the module embeds as an ideal).
    
│ │ │          
    
│ │ │          
    
│ │ │ @@ -324,26 +324,26 @@
│ │ │              
│ │ │                
    i28 : M = J*R^1;
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i29 : J1 = time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ + -- used 0.0909353s (cpu); 0.0909381s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o29 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ + -- used 5.34768s (cpu); 4.02232s (thread); 0s (gc)
│ │ │  
│ │ │                2            2     9       9   11
│ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │  
│ │ │  o30 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │ @@ -353,21 +353,21 @@
│ │ │  
│ │ │  o31 = true
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
    
│ │ │ + -- used 5.4074s (cpu); 4.07129s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
    
│ │ │ + -- used 0.495501s (cpu); 0.332526s (thread); 0s (gc)
    
│ │ │              
│ │ │            
│ │ │          
│ │ │          
    
│ │ │            
    However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
    
│ │ │          
    
│ │ │          
│ │ │ @@ -394,15 +394,15 @@
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │            
│ │ │          
    
│ │ │                
    i37 : M = I^20*R^1;
    
│ │ │              
    
│ │ │                
    i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ + -- used 0.391858s (cpu); 0.232942s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o38 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -411,15 +411,15 @@
│ │ │  
│ │ │  o38 : Ideal of R
    
│ │ │              
    
│ │ │                
    i39 : time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ - -- used 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ + -- used 0.0136439s (cpu); 0.0136467s (thread); 0s (gc)
│ │ │  
│ │ │                20     19   2 18   3 17   4 16   5 15   6 14   7 13   8 12 
│ │ │  o39 = ideal (u  , x*u  , x u  , x u  , x u  , x u  , x u  , x u  , x u  ,
│ │ │        -----------------------------------------------------------------------
│ │ │         9 11   10 10   11 9   12 8   13 7   14 6   15 5   16 4   17 3   18 2 
│ │ │        x u  , x  u  , x  u , x  u , x  u , x  u , x  u , x  u , x  u , x  u ,
│ │ │        -----------------------------------------------------------------------
│ │ │ @@ -428,21 +428,21 @@
│ │ │  
│ │ │  o39 : Ideal of R
    
│ │ │              
    
│ │ │                
    i40 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ - -- used 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
    
│ │ │ + -- used 0.0390337s (cpu); 0.0390383s (thread); 0s (gc)
│ │ │              
    
│ │ │                
    i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
    
│ │ │ + -- used 0.006433s (cpu); 0.00643849s (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 0.338433s (cpu); 0.253168s (thread); 0s (gc)
│ │ │ │ + -- used 0.243393s (cpu); 0.17634s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o23 : Ideal of R
│ │ │ │  i24 : time reflexify(J*R^1);
│ │ │ │ - -- used 0.671713s (cpu); 0.510526s (thread); 0s (gc)
│ │ │ │ + -- used 0.390243s (cpu); 0.3239s (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 0.168829s (cpu); 0.168835s (thread); 0s (gc)
│ │ │ │ + -- used 0.0909353s (cpu); 0.0909381s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o29 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o29 : Ideal of R
│ │ │ │  i30 : J2 = time reflexify( J, Strategy=>ModuleStrategy )
│ │ │ │ - -- used 7.92394s (cpu); 5.32977s (thread); 0s (gc)
│ │ │ │ + -- used 5.34768s (cpu); 4.02232s (thread); 0s (gc)
│ │ │ │  
│ │ │ │                2            2     9       9   11
│ │ │ │  o30 = ideal (x  + 5x*y + 3y , x*z  - 4y*z , z   + x - 4y)
│ │ │ │  
│ │ │ │  o30 : Ideal of R
│ │ │ │  i31 : J1 == J2
│ │ │ │  
│ │ │ │  o31 = true
│ │ │ │  i32 : time reflexify( M, Strategy=>IdealStrategy );
│ │ │ │ - -- used 6.96991s (cpu); 5.49609s (thread); 0s (gc)
│ │ │ │ + -- used 5.4074s (cpu); 4.07129s (thread); 0s (gc)
│ │ │ │  i33 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.831741s (cpu); 0.473654s (thread); 0s (gc)
│ │ │ │ + -- used 0.495501s (cpu); 0.332526s (thread); 0s (gc)
│ │ │ │  However, sometimes ModuleStrategy is faster, especially for Monomial ideals.
│ │ │ │  i34 : R = QQ[x,y,u,v]/ideal(x*y-u*v);
│ │ │ │  i35 : I = ideal(x,u);
│ │ │ │  
│ │ │ │  o35 : Ideal of R
│ │ │ │  i36 : J = I^20;
│ │ │ │  
│ │ │ │  o36 : Ideal of R
│ │ │ │  i37 : M = I^20*R^1;
│ │ │ │  i38 : time reflexify( J, Strategy=>IdealStrategy )
│ │ │ │ - -- used 0.461613s (cpu); 0.267865s (thread); 0s (gc)
│ │ │ │ + -- used 0.391858s (cpu); 0.232942s (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 0.0135561s (cpu); 0.0135572s (thread); 0s (gc)
│ │ │ │ + -- used 0.0136439s (cpu); 0.0136467s (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 0.0397151s (cpu); 0.0397234s (thread); 0s (gc)
│ │ │ │ + -- used 0.0390337s (cpu); 0.0390383s (thread); 0s (gc)
│ │ │ │  i41 : time reflexify( M, Strategy=>ModuleStrategy );
│ │ │ │ - -- used 0.00682599s (cpu); 0.00682691s (thread); 0s (gc)
│ │ │ │ + -- used 0.006433s (cpu); 0.00643849s (thread); 0s (gc)
│ │ │ │  For ideals, if KnownDomain is false (default value is true), then the function
│ │ │ │  will check whether it is a domain. If it is a domain (or assumed to be a
│ │ │ │  domain), it will reflexify using a strategy which can speed up computation, if
│ │ │ │  not it will compute using a sometimes slower method which is essentially
│ │ │ │  reflexifying it as a module.
│ │ │ │  Consider the following example showing the importance of making the correct
│ │ │ │  assumption about the ring being a domain.
│ │ ├── ./usr/share/doc/Macaulay2/WeilDivisors/html/_reflexive__Power.html
│ │ │ @@ -129,30 +129,30 @@
│ │ │  
│ │ │  o6 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i7 : time J20a = reflexivePower(20, I);
│ │ │ - -- used 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ + -- used 0.0300898s (cpu); 0.0300054s (thread); 0s (gc)
│ │ │  
│ │ │  o7 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i8 : I20 = I^20;
│ │ │  
│ │ │  o8 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i9 : time J20b = reflexify(I20);
│ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ + -- used 0.134946s (cpu); 0.134946s (thread); 0s (gc)
│ │ │  
│ │ │  o9 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i10 : J20a == J20b
│ │ │ @@ -176,23 +176,23 @@
│ │ │  
│ │ │  o12 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i13 : time J1 = reflexivePower(20, I, Strategy=>IdealStrategy);
│ │ │ - -- used 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ + -- used 0.0319431s (cpu); 0.0319472s (thread); 0s (gc)
│ │ │  
│ │ │  o13 : Ideal of R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ + -- used 0.0600871s (cpu); 0.0600926s (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 0.0292441s (cpu); 0.0292465s (thread); 0s (gc)
│ │ │ │ + -- used 0.0300898s (cpu); 0.0300054s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o7 : Ideal of R
│ │ │ │  i8 : I20 = I^20;
│ │ │ │  
│ │ │ │  o8 : Ideal of R
│ │ │ │  i9 : time J20b = reflexify(I20);
│ │ │ │ - -- used 0.142968s (cpu); 0.14297s (thread); 0s (gc)
│ │ │ │ + -- used 0.134946s (cpu); 0.134946s (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 0.0299195s (cpu); 0.0299242s (thread); 0s (gc)
│ │ │ │ + -- used 0.0319431s (cpu); 0.0319472s (thread); 0s (gc)
│ │ │ │  
│ │ │ │  o13 : Ideal of R
│ │ │ │  i14 : time J2 = reflexivePower(20, I, Strategy=>ModuleStrategy);
│ │ │ │ - -- used 0.164355s (cpu); 0.0959583s (thread); 0s (gc)
│ │ │ │ + -- used 0.0600871s (cpu); 0.0600926s (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/_trim_lp__Basic__Divisor_rp.html
│ │ │ @@ -93,24 +93,24 @@
│ │ │  o2 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i3 : cleanSupport(D)
│ │ │  
│ │ │ -o3 = -2*Div(y+z, z) + Div(x, z)
│ │ │ +o3 = Div(x, z) + -2*Div(y+z, z)
│ │ │  
│ │ │  o3 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : trim(D)
│ │ │  
│ │ │ -o4 = -2*Div(z, y) + Div(z, x)
│ │ │ +o4 = Div(z, x) + -2*Div(z, y)
│ │ │  
│ │ │  o4 : WeilDivisor on R
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i5 : D == trim(D)
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,20 +19,20 @@
│ │ │ │  removed and where the ideals displayed to the user are trimmed.
│ │ │ │  i1 : R = QQ[x,y,z]/ideal(x*y-z^2);
│ │ │ │  i2 : D = divisor({1,0,-2}, {ideal(x, z), ideal(x-z,y-z), ideal(y+z, z)});
│ │ │ │  
│ │ │ │  o2 : WeilDivisor on R
│ │ │ │  i3 : cleanSupport(D)
│ │ │ │  
│ │ │ │ -o3 = -2*Div(y+z, z) + Div(x, z)
│ │ │ │ +o3 = Div(x, z) + -2*Div(y+z, z)
│ │ │ │  
│ │ │ │  o3 : WeilDivisor on R
│ │ │ │  i4 : trim(D)
│ │ │ │  
│ │ │ │ -o4 = -2*Div(z, y) + Div(z, x)
│ │ │ │ +o4 = Div(z, x) + -2*Div(z, y)
│ │ │ │  
│ │ │ │  o4 : WeilDivisor on R
│ │ │ │  i5 : D == trim(D)
│ │ │ │  
│ │ │ │  o5 = true
│ │ │ │  ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: **********
│ │ │ │      * _t_r_i_m_(_B_a_s_i_c_D_i_v_i_s_o_r_) -- trims the ideals displayed to the user and removes
│ │ ├── ./usr/share/doc/Macaulay2/WeylAlgebras/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=18
│ │ │  ZXh0cmFjdFZhcnNBbGdlYnJh
│ │ │  #:len=1106
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAidW5kZXJseWluZyBwb2x5bm9taWFsIHJp
│ │ │  bmcgaW4gdGhlIG9yZGluYXJ5IHZhcmlhYmxlcyBvZiBhIFdleWwgYWxnZWJyYSIsICJsaW5lbnVt
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/dump/rawdocumentation.dump
│ │ │ @@ -1,11 +1,11 @@
│ │ │  # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026
│ │ │  #:version=1.1
│ │ │  #:file=rawdocumentation-dcba-8.db
│ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644
│ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644
│ │ │  #:format=standard
│ │ │  # End of header
│ │ │  #:len=23
│ │ │  V2V5bEdyb3VwRWxlbWVudCAqIFJvb3Q=
│ │ │  #:len=994
│ │ │  bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiYXBwbHkgYW4gZWxlbWVudCBvZiBhIFdl
│ │ │  eWwgZ3JvdXAgdG8gYSByb290IiwgImxpbmVudW0iID0+IDI5NjMsIElucHV0cyA9PiB7U1BBTntU
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_above__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             |  2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            |  1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            |  3 |        | -1 |      
│ │ │ +     |                                            | -1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ -     |  2 |                                            | -2 |        | -1 |  
│ │ │ -     | -1 |                                            |  1 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ +     | -1 |                                            | -2 |        |  1 |  
│ │ │ +     |  0 |                                            |  3 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ -     |       |  1 |                                            |  3 |       
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ +     |       |  2 |                                            | -2 |       
│ │ │ +     |       | -1 |                                            |  1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_eval_lp__Root__System_cm__Weight_cm__Root_rp.out
│ │ │ @@ -4,33 +4,33 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : v=weight(R,{1,2})
│ │ │  
│ │ │  o3 = | 1 |
│ │ │       | 2 |
│ │ │  
│ │ │         2
│ │ │  o3 : ZZ
│ │ │  
│ │ │  i4 : eval(R,v,L#0)
│ │ │  
│ │ │ -o4 = 2
│ │ │ +o4 = 3
│ │ │  
│ │ │  i5 : eval(R,v,L#1)
│ │ │  
│ │ │ -o5 = 3
│ │ │ +o5 = 1
│ │ │  
│ │ │  i6 : eval(R,v,L#2)
│ │ │  
│ │ │ -o6 = 1
│ │ │ +o6 = 2
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_eval_lp__Root__System_cm__Z__Z_cm__Root_rp.out
│ │ │ @@ -4,25 +4,25 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
│ │ │  
│ │ │  i3 : eval(R,1,L#0)
│ │ │  
│ │ │ -o3 = 0
│ │ │ +o3 = 1
│ │ │  
│ │ │  i4 : eval(R,1,L#1)
│ │ │  
│ │ │  o4 = 1
│ │ │  
│ │ │  i5 : eval(R,1,L#2)
│ │ │  
│ │ │ -o5 = 1
│ │ │ +o5 = 0
│ │ │  
│ │ │  i6 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_rp.out
│ │ │ @@ -20,26 +20,26 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o3 : WeylGroupElement
│ │ │  
│ │ │  i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  1 |       |  2 |       |  1 |       | 0 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        |  1 |       | -1 |       | -1 |       | 1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
│ │ │  
│ │ │  i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, {, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
│ │ │  
│ │ │  i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, 1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, 0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        |  1 |       | 0 |                                              | -1 |        |  2 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  1 |       | 1 |                                              |  3 |        | -1 |       |  0 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ -                                             |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2
│ │ │ +                                             | -1 |        |  2 |       | -1
│ │ │ +                                             |  3 |        | -1 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2,
│ │ │ +     |                                            |  3 |        |  1 |      
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ -     | -1 |
│ │ │ -     |  0 |
│ │ │ +     | -1 |}}}}
│ │ │ +     |  1 |
│ │ │ +     |  1 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.out
│ │ │ @@ -26,30 +26,30 @@
│ │ │                                             | -2 |
│ │ │                                             |  1 |
│ │ │  
│ │ │  o4 : WeylGroupElement
│ │ │  
│ │ │  i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            |  1 |        | 0 |       | -1 |                                              | -2 |        | -1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            |  1 |        | 1 |       |  2 |                                              |  3 |        |  0 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
│ │ │  
│ │ │  i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │                                               | -3 |        |  1 |       |  2
│ │ │                                               |  1 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1,
│ │ │       |                                            | -1 |        | -1 |      
│ │ │ -     |                                            |  2 |        |  2 |      
│ │ │ +     |                                            |  2 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, |  2 |}}}}
│ │ │ -     |  1 |       | -1 |
│ │ │ -     |  1 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o6 : List
│ │ │  
│ │ │  i7 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_list__Weyl__Group__Elements_lp__Root__System_cm__Z__Z_rp.out
│ │ │ @@ -1,13 +1,13 @@
│ │ │  -- -*- M2-comint -*- hash: 1330776363660
│ │ │  
│ │ │  i1 : listWeylGroupElements(rootSystemG2,4)
│ │ │  
│ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, |  4 |},
│ │ │ -                                            | -3 |  
│ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |},
│ │ │ +                                            |  2 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     WeylGroupElement{RootSystem{...8...}, | -5 |}}
│ │ │ -                                           |  2 |
│ │ │ +     WeylGroupElement{RootSystem{...8...}, |  4 |}}
│ │ │ +                                           | -3 |
│ │ │  
│ │ │  o1 : List
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_poincare__Series_lp__Hasse__Diagram_cm__Ring__Element_rp.out
│ │ │ @@ -4,16 +4,16 @@
│ │ │  
│ │ │  o1 = RootSystem{...8...}
│ │ │  
│ │ │  o1 : RootSystem
│ │ │  
│ │ │  i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │  
│ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ -                                                          | -1 |        | -1 |       |  2 |                                              |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       | 1 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ +                                                          | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |       | 1 |                                            |  1 |        | 1 |       |  2 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │  
│ │ │  o2 : HasseDiagram
│ │ │  
│ │ │  i3 : ZZ[x]
│ │ │  
│ │ │  o3 = ZZ[x]
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_cm__Parabolic_rp.out
│ │ │ @@ -10,15 +10,15 @@
│ │ │  
│ │ │  o2 = set {1, 2}
│ │ │  
│ │ │  o2 : Parabolic
│ │ │  
│ │ │  i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ -          |  1 |  | -1 |  |  2 |
│ │ │ -          | -1 |  |  0 |  | -1 |
│ │ │ +o3 = set {| -1 |, |  1 |, |  2 |}
│ │ │ +          |  2 |  |  1 |  | -1 |
│ │ │ +          | -1 |  | -1 |  |  0 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
│ │ │  
│ │ │  i4 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_positive__Roots_lp__Root__System_rp.out
│ │ │ @@ -1,11 +1,11 @@
│ │ │  -- -*- M2-comint -*- hash: 1330744940387
│ │ │  
│ │ │  i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {| 1 |, |  1 |, |  2 |, |  0 |, | -1 |, | -1 |}
│ │ │ +          | 0 |  |  1 |  | -1 |  | -1 |  |  1 |  |  2 |
│ │ │ +          | 1 |  | -1 |  |  0 |  |  2 |  |  1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
│ │ │  
│ │ │  i2 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/example-output/_under__Bruhat_lp__Basic__List_rp.out
│ │ │ @@ -36,34 +36,34 @@
│ │ │                                             | -2 |
│ │ │                                             | -1 |
│ │ │  
│ │ │  o3 : List
│ │ │  
│ │ │  i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ +                                             | -3 |        | -1 |       | -1
│ │ │ +                                             |  1 |        |  2 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ +     |                                            | -1 |        |  1 |      
│ │ │ +     |                                            | -2 |        |  1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ -     |  1 |                                            |  2 |        | -1 |  
│ │ │ -     | -1 |                                            | -3 |        |  2 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ +     |  2 |                                            |  2 |        |  1 |  
│ │ │ +     | -1 |                                            | -1 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │ +         |  0 |                                            |  2 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ +     |       |  1 |                                            |  2 |       
│ │ │ +     |       | -1 |                                            | -3 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ -     |  1 |       | -1 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
│ │ │  
│ │ │  i5 :
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_above__Bruhat_lp__Basic__List_rp.html
│ │ │ @@ -121,37 +121,37 @@
│ │ │  o3 : List
    
│ │ │              
│ │ │            
│ │ │            
│ │ │              
│ │ │                
    i4 : aboveBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             | -2 |        |  1 |       | -1
│ │ │ -                                             |  3 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, |  1
│ │ │ +                                             |  1 |        |  2 |       |  1
│ │ │ +                                             |  2 |        | -1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ -     |                                            |  1 |        |  1 |      
│ │ │ -     |                                            | -2 |        |  1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {2,
│ │ │ +     |                                            |  3 |        | -1 |      
│ │ │ +     |                                            | -1 |        |  2 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |},
│ │ │ -     |  2 |                                            | -2 |        | -1 |  
│ │ │ -     | -1 |                                            |  1 |        |  2 |  
│ │ │ +     |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  1 |},
│ │ │ +     | -1 |                                            | -2 |        |  1 |  
│ │ │ +     |  0 |                                            |  3 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ -         |  1 |                                            |  2 |        | -1
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1
│ │ │ +         | -1 |                                            |  1 |        |  1
│ │ │ +         |  0 |                                            | -2 |        |  1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0,
│ │ │ -     |       |  1 |                                            |  3 |       
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ +     |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1,
│ │ │ +     |       |  2 |                                            | -2 |       
│ │ │ +     |       | -1 |                                            |  1 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  0 |}, {2, |  2 |}}}}
│ │ │ -     | -1 |       | -1 |
│ │ │ -     |  2 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
    
│ │ │              
│ │ │            
│ │ │          
│ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,37 +48,37 @@ │ │ │ │ WeylGroupElement{RootSystem{...8...}, | 1 |}} │ │ │ │ | 2 | │ │ │ │ | -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : aboveBruhat(L1) │ │ │ │ │ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 |}, {1, | 2 │ │ │ │ - | -2 | | 1 | | -1 │ │ │ │ - | 3 | | -1 | | 0 │ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, | 1 │ │ │ │ + | 1 | | 2 | | 1 │ │ │ │ + | 2 | | -1 | | -1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, {2, │ │ │ │ - | | 1 | | 1 | │ │ │ │ - | | -2 | | 1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 0 |}, {2, │ │ │ │ + | | 3 | | -1 | │ │ │ │ + | | -1 | | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, | 0 |}, │ │ │ │ - | 2 | | -2 | | -1 | │ │ │ │ - | -1 | | 1 | | 2 | │ │ │ │ + | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 |}, │ │ │ │ + | -1 | | -2 | | 1 | │ │ │ │ + | 0 | | 3 | | -1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 │ │ │ │ - | 1 | | 1 | | 2 │ │ │ │ - | 1 | | 2 | | -1 │ │ │ │ + {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 │ │ │ │ + | -1 | | 1 | | 1 │ │ │ │ + | 0 | | -2 | | 1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, │ │ │ │ - | | 1 | | 3 | │ │ │ │ - | | -1 | | -1 | │ │ │ │ + |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, │ │ │ │ + | | 2 | | -2 | │ │ │ │ + | | -1 | | 1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 0 |}, {2, | 2 |}}}} │ │ │ │ - | -1 | | -1 | │ │ │ │ - | 2 | | 0 | │ │ │ │ + | 0 |}, {2, | -1 |}}}} │ │ │ │ + | -1 | | 1 | │ │ │ │ + | 2 | | 1 | │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _a_b_o_v_e_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- The Weyl group elements just under the ones in │ │ │ │ the list for the Bruhat order │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_eval_lp__Root__System_cm__Weight_cm__Root_rp.html │ │ │ @@ -85,16 +85,16 @@ │ │ │ o1 : RootSystem │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : v=weight(R,{1,2})
│ │ │ @@ -106,29 +106,29 @@
│ │ │  o3 : ZZ
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : eval(R,v,L#0)
│ │ │  
│ │ │ -o4 = 2
    │ │ │ +o4 = 3 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : eval(R,v,L#1)
│ │ │  
│ │ │ -o5 = 3
    │ │ │ +o5 = 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i6 : eval(R,v,L#2)
│ │ │  
│ │ │ -o6 = 1
    │ │ │ +o6 = 2 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -18,32 +18,32 @@ │ │ │ │ i1 : R=rootSystemA(2) │ │ │ │ │ │ │ │ o1 = RootSystem{...8...} │ │ │ │ │ │ │ │ o1 : RootSystem │ │ │ │ i2 : L=toList(positiveRoots(R)) │ │ │ │ │ │ │ │ -o2 = {| -1 |, | 1 |, | 2 |} │ │ │ │ - | 2 | | 1 | | -1 | │ │ │ │ +o2 = {| 1 |, | 2 |, | -1 |} │ │ │ │ + | 1 | | -1 | | 2 | │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : v=weight(R,{1,2}) │ │ │ │ │ │ │ │ o3 = | 1 | │ │ │ │ | 2 | │ │ │ │ │ │ │ │ 2 │ │ │ │ o3 : ZZ │ │ │ │ i4 : eval(R,v,L#0) │ │ │ │ │ │ │ │ -o4 = 2 │ │ │ │ +o4 = 3 │ │ │ │ i5 : eval(R,v,L#1) │ │ │ │ │ │ │ │ -o5 = 3 │ │ │ │ +o5 = 1 │ │ │ │ i6 : eval(R,v,L#2) │ │ │ │ │ │ │ │ -o6 = 1 │ │ │ │ +o6 = 2 │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _e_v_a_l_(_R_o_o_t_S_y_s_t_e_m_,_W_e_i_g_h_t_,_R_o_o_t_) -- evaluate the dual of a root at a Weight │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/WeylGroups.m2:2488:0. │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_eval_lp__Root__System_cm__Z__Z_cm__Root_rp.html │ │ │ @@ -85,39 +85,39 @@ │ │ │ o1 : RootSystem │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : L=toList(positiveRoots(R))
│ │ │  
│ │ │ -o2 = {| -1 |, | 1 |, |  2 |}
│ │ │ -      |  2 |  | 1 |  | -1 |
│ │ │ +o2 = {| 1 |, |  2 |, | -1 |}
│ │ │ +      | 1 |  | -1 |  |  2 |
│ │ │  
│ │ │  o2 : List
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : eval(R,1,L#0)
│ │ │  
│ │ │ -o3 = 0
    │ │ │ +o3 = 1 │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : eval(R,1,L#1)
│ │ │  
│ │ │  o4 = 1
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : eval(R,1,L#2)
│ │ │  
│ │ │ -o5 = 1
    │ │ │ +o5 = 0 │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │
    │ │ │

    Ways to use this method:

    │ │ │ ├── html2text {} │ │ │ │ @@ -19,26 +19,26 @@ │ │ │ │ i1 : R=rootSystemA(2) │ │ │ │ │ │ │ │ o1 = RootSystem{...8...} │ │ │ │ │ │ │ │ o1 : RootSystem │ │ │ │ i2 : L=toList(positiveRoots(R)) │ │ │ │ │ │ │ │ -o2 = {| -1 |, | 1 |, | 2 |} │ │ │ │ - | 2 | | 1 | | -1 | │ │ │ │ +o2 = {| 1 |, | 2 |, | -1 |} │ │ │ │ + | 1 | | -1 | | 2 | │ │ │ │ │ │ │ │ o2 : List │ │ │ │ i3 : eval(R,1,L#0) │ │ │ │ │ │ │ │ -o3 = 0 │ │ │ │ +o3 = 1 │ │ │ │ i4 : eval(R,1,L#1) │ │ │ │ │ │ │ │ o4 = 1 │ │ │ │ i5 : eval(R,1,L#2) │ │ │ │ │ │ │ │ -o5 = 1 │ │ │ │ +o5 = 0 │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _e_v_a_l_(_R_o_o_t_S_y_s_t_e_m_,_Z_Z_,_R_o_o_t_) -- evaluate the dual of a root at a fundamental │ │ │ │ weight │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/WeylGroups.m2:2512:0. │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_hasse__Diagram__To__Graph_lp__Hasse__Diagram_rp.html │ │ │ @@ -112,40 +112,40 @@ │ │ │ o3 : WeylGroupElement │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : myInterval=intervalBruhat(w1,w2)
│ │ │  
│ │ │ -o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, |  1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, |  2 |}, {4, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  2 |       |  1 |       | 0 |       |  1 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        |  1 |       | -1 |       | -1 |                                              | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        | -1 |       | -1 |       | 1 |       |  1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  1 |       |  0 |       |  2 |                                              |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}, {2, |  1 |}, {4, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {3, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{1, |  0 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {3, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  1 |       |  2 |                                              | -3 |        |  1 |       |  2 |       |  1 |       | 0 |                                            |  2 |        | 0 |       |  2 |       | -1 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            | -1 |        | -1 |       |  2 |                                            |  1 |        | 0 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |       | -1 |                                              |  1 |        |  1 |       | -1 |       | -1 |       | 1 |                                            | -1 |        | 1 |       | -1 |       |  2 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            | -1 |        |  2 |       | -1 |                                            |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o4 : HasseDiagram
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : hasseDiagramToGraph(myInterval)
│ │ │  
│ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, {, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, {, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, {{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, {{, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}}
│ │ │  
│ │ │  o5 : HasseGraph
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    It is also possible to ask for reduced decompositions as labels by changing the option "labels" as below.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition")
│ │ │  
│ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, 1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, 0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, 1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, 0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}}
│ │ │  
│ │ │  o6 : HasseGraph
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -35,72 +35,73 @@ │ │ │ │ | 1 | │ │ │ │ │ │ │ │ o3 : WeylGroupElement │ │ │ │ i4 : myInterval=intervalBruhat(w1,w2) │ │ │ │ │ │ │ │ o4 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 0 │ │ │ │ |}, {1, | 1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 │ │ │ │ -|}, {{0, | -1 |}, {1, | 1 |}, {3, | 1 |}, {4, | -1 |}}}, {WeylGroupElement │ │ │ │ -{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, | -1 |}, {3, | 0 |}}}, │ │ │ │ -{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {2, | 2 |}, {4, │ │ │ │ -| 0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | 0 |}, {2, │ │ │ │ -| 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, | 0 |}, {2, | │ │ │ │ --1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | 0 │ │ │ │ +|}, {{0, | -1 |}, {1, | -1 |}, {2, | 1 |}, {4, | 1 |}}}, {WeylGroupElement │ │ │ │ +{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | -1 |}, {4, | 0 |}}}, │ │ │ │ +{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}, {1, | -1 |}, {3, │ │ │ │ +| 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, │ │ │ │ +| 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | 0 |}, {2, | │ │ │ │ +2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{1, | 0 |}, {2, | - │ │ │ │ +1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{2, | 1 |}, {3, | 0 │ │ │ │ |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 |}, {3, | - │ │ │ │ -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {3, | 2 │ │ │ │ -|}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 |}}}, │ │ │ │ +1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 |}}}, │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | -1 |}}}, │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{0, | 0 |}}}, │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}}, { │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 2 |}, {}}}} │ │ │ │ | -2 | | -1 | │ │ │ │ | 1 | | 2 | | -3 | │ │ │ │ -| 2 | | 1 | | 0 | | 1 | │ │ │ │ +| 1 | | 2 | | 1 | | 0 | │ │ │ │ | 2 | | 0 | | 2 | | -1 | │ │ │ │ -| -1 | | 1 | | -1 | | -1 | │ │ │ │ -| -3 | | -1 | | -1 | | │ │ │ │ --1 | | -1 | | 2 | | 1 │ │ │ │ +| -1 | | -1 | | 1 | | -1 | │ │ │ │ +| -1 | | 2 | | -1 | | │ │ │ │ +-3 | | -1 | | -1 | | - │ │ │ │ +1 | | -1 | | 2 | | 1 │ │ │ │ | | 0 | | -1 | | 3 | │ │ │ │ -| 1 | | 1 | | -1 | | │ │ │ │ -2 | | -1 | | -2 | | - │ │ │ │ -1 | | 1 | | 1 | │ │ │ │ +| 1 | | 1 | | -2 | │ │ │ │ +| -1 | | 1 | | 1 | │ │ │ │ | -2 | | -1 | | 1 | | │ │ │ │ 1 | | -1 | │ │ │ │ | 1 | | 2 | │ │ │ │ | -1 | | -1 | | 1 | │ │ │ │ -| -1 | | -1 | | 1 | | 1 | │ │ │ │ +| 1 | | -1 | | -1 | | 1 | │ │ │ │ | -1 | | 1 | | -1 | | 2 | │ │ │ │ -| 2 | | 1 | | 0 | | 2 | │ │ │ │ -| 2 | | 2 | | 0 | | │ │ │ │ --1 | | 2 | | -1 | | 1 │ │ │ │ +| 2 | | 2 | | 1 | | 0 | │ │ │ │ +| 3 | | -1 | | 0 | | │ │ │ │ +2 | | 2 | | 0 | | - │ │ │ │ +1 | | 2 | | -1 | | 1 │ │ │ │ | | 1 | | 2 | | -2 | │ │ │ │ -| -1 | | 1 | | 3 | | │ │ │ │ --1 | | 0 | | 3 | | │ │ │ │ -0 | | -2 | | 1 | │ │ │ │ +| -1 | | 1 | | 3 | │ │ │ │ +| 0 | | -2 | | 1 | │ │ │ │ | 1 | | 2 | | 2 | | │ │ │ │ -1 | | 2 | │ │ │ │ │ │ │ │ o4 : HasseDiagram │ │ │ │ i5 : hasseDiagramToGraph(myInterval) │ │ │ │ │ │ │ │ -o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 3}, {, 4}}}, │ │ │ │ -{, {{, 1}, {, 2}, {, 3}}}, {, {{, 0}, {, 2}, {, 4}}}}, {{, {{, 0}, {, 2}}}, {, │ │ │ │ -{{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}, {, {{, 0}, {, 3}}}}, { │ │ │ │ +o5 = HasseGraph{{{, {{, 0}, {, 1}, {, 2}}}}, {{, {{, 0}, {, 1}, {, 2}, {, 4}}}, │ │ │ │ +{, {{, 2}, {, 3}, {, 4}}}, {, {{, 0}, {, 1}, {, 3}}}}, {{, {{, 0}, {, 3}}}, {, │ │ │ │ +{{, 0}, {, 2}}}, {, {{, 1}, {, 2}}}, {, {{, 2}, {, 3}}}, {, {{, 1}, {, 3}}}}, { │ │ │ │ {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}, {, {{, 0}}}}, {{, {}}}} │ │ │ │ │ │ │ │ o5 : HasseGraph │ │ │ │ It is also possible to ask for reduced decompositions as labels by changing the │ │ │ │ option "labels" as below. │ │ │ │ i6 : hasseDiagramToGraph(myInterval,"labels"=>"reduced decomposition") │ │ │ │ │ │ │ │ -o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{2, 0}, {121, │ │ │ │ -1}, {12321, 3}, {232, 4}}}, {1232, {{12321, 1}, {2, 2}, {3, 3}}}, {1213, {{232, │ │ │ │ -0}, {1, 2}, {3, 4}}}}, {{213, {{3, 0}, {1, 2}}}, {232, {{3, 1}, {2, 2}}}, {123, │ │ │ │ -{{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, 3}}}, {121, {{2, 0}, {1, 3}}}}, { │ │ │ │ -{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, {{2, {}}}} │ │ │ │ +o6 = HasseGraph{{{12132, {{3, 0}, {121, 1}, {2, 2}}}}, {{2132, {{232, 0}, {2, │ │ │ │ +1}, {121, 2}, {12321, 4}}}, {1232, {{12321, 2}, {2, 3}, {3, 4}}}, {1213, {{3, │ │ │ │ +0}, {232, 1}, {1, 3}}}}, {{121, {{2, 0}, {1, 3}}}, {213, {{3, 0}, {1, 2}}}, │ │ │ │ +{232, {{3, 1}, {2, 2}}}, {123, {{12321, 2}, {3, 3}}}, {132, {{121, 1}, {232, │ │ │ │ +3}}}}, {{21, {{1, 0}}}, {32, {{232, 0}}}, {23, {{3, 0}}}, {12, {{121, 0}}}}, { │ │ │ │ +{2, {}}}} │ │ │ │ │ │ │ │ o6 : HasseGraph │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _h_a_s_s_e_D_i_a_g_r_a_m_T_o_G_r_a_p_h_(_H_a_s_s_e_D_i_a_g_r_a_m_) -- turning a hasse diagram into a graph │ │ │ │ (intended for graphic representation) │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Left__Coset_cm__Weyl__Group__Left__Coset_rp.html │ │ │ @@ -115,41 +115,41 @@ │ │ │ o4 : WeylGroupElement │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : myInterval=intervalBruhat(w1 % P,w2 % P)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -3 |        | 0 |       |  1 |                                              |  3 |        |  1 |       |  1 |                                            | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        | 1 |       |  1 |                                              | -2 |        | -1 |       |  1 |                                            |  3 |        | -1 |       |  0 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -3 |        |  1 |       | 0 |                                              | -1 |        |  2 |       | -1 |                                            |  3 |        |  1 |       |  1 |                                              | -2 |        | -1 |                                            |  1 |        |  1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  1 |       | 1 |                                              |  3 |        | -1 |       |  0 |                                            | -2 |        | -1 |       |  1 |                                              |  3 |        |  0 |                                            | -2 |        |  1 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2, | -1
│ │ │ -                                             |  3 |        |  1 |       |  1
│ │ │ -                                             | -2 |        | -1 |       |  1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2
│ │ │ +                                             | -1 |        |  2 |       | -1
│ │ │ +                                             |  3 |        | -1 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  3 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, |  1 |}, {2,
│ │ │ +     |                                            |  3 |        |  1 |      
│ │ │ +     |                                            | -2 |        | -1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  2 |}}}}
│ │ │ -     | -1 |
│ │ │ -     |  0 |
│ │ │ +     | -1 |}}}}
│ │ │ +     |  1 |
│ │ │ +     |  1 |
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -40,50 +40,50 @@ │ │ │ │ o4 = WeylGroupElement{RootSystem{...8...}, | -1 |} │ │ │ │ | -2 | │ │ │ │ | 1 | │ │ │ │ │ │ │ │ o4 : WeylGroupElement │ │ │ │ i5 : myInterval=intervalBruhat(w1 % P,w2 % P) │ │ │ │ │ │ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 1 |}, │ │ │ │ -{1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 |}, │ │ │ │ -{2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, │ │ │ │ -{2, | 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 │ │ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | - │ │ │ │ +1 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | - │ │ │ │ +1 |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 │ │ │ │ +|}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 │ │ │ │ |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{0, | -1 |}}}, │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}}, { │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 2 |}, {}}}} │ │ │ │ - | -3 | | 0 | │ │ │ │ -| 1 | | 3 | | 1 | │ │ │ │ -| 1 | | -1 | | 2 | | │ │ │ │ --1 | | -2 | | -1 | │ │ │ │ + | -3 | | 1 | │ │ │ │ +| 0 | | -1 | | 2 | | │ │ │ │ +-1 | | 3 | | 1 | | 1 │ │ │ │ +| | -2 | | -1 | │ │ │ │ | 1 | | 1 | | 1 | | │ │ │ │ 1 | | -1 | │ │ │ │ - | 1 | | 1 | │ │ │ │ -| 1 | | -2 | | -1 | │ │ │ │ -| 1 | | 3 | | -1 | | │ │ │ │ -0 | | 3 | | 0 | │ │ │ │ + | 1 | | 1 | │ │ │ │ +| 1 | | 3 | | -1 | | │ │ │ │ +0 | | -2 | | -1 | | 1 │ │ │ │ +| | 3 | | 0 | │ │ │ │ | -2 | | 1 | | 2 | | │ │ │ │ -1 | | 2 | │ │ │ │ │ │ │ │ o5 : HasseDiagram │ │ │ │ Each row of the Hasse diagram contains the elements of a certain length │ │ │ │ together with their links to the next row. │ │ │ │ i6 : myInterval#1 │ │ │ │ │ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 |}, {2, | -1 │ │ │ │ - | 3 | | 1 | | 1 │ │ │ │ - | -2 | | -1 | | 1 │ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, | 2 │ │ │ │ + | -1 | | 2 | | -1 │ │ │ │ + | 3 | | -1 | | 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, │ │ │ │ - | | -1 | | 2 | │ │ │ │ - | | 3 | | -1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{1, | 1 |}, {2, │ │ │ │ + | | 3 | | 1 | │ │ │ │ + | | -2 | | -1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 2 |}}}} │ │ │ │ - | -1 | │ │ │ │ - | 0 | │ │ │ │ + | -1 |}}}} │ │ │ │ + | 1 | │ │ │ │ + | 1 | │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_n_t_e_r_v_a_l_B_r_u_h_a_t_(_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_,_W_e_y_l_G_r_o_u_p_L_e_f_t_C_o_s_e_t_) -- elements between │ │ │ │ two given ones for the Bruhat order on a quotient of a Weyl group │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_interval__Bruhat_lp__Weyl__Group__Right__Coset_cm__Weyl__Group__Right__Coset_rp.html │ │ │ @@ -115,41 +115,41 @@ │ │ │ o4 : WeylGroupElement │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i5 : myInterval=intervalBruhat(P % w1,P % w2)
│ │ │  
│ │ │ -o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1, | -1 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, |  0 |}, {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{1, | 1 |}, {2, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ -                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       |  1 |       | -1 |                                              | -1 |        |  2 |       | -1 |                                            | -3 |        | -1 |       | -1 |                                            |  1 |        | 0 |       | -1 |                                              | -2 |        | -1 |                                            | -2 |        | -1 |                                            |  1 |        |  1 |                                              | -1 |
│ │ │ -                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  2 |       |  1 |       |  0 |                                              |  3 |        | -1 |       |  0 |                                            |  2 |        |  2 |       |  0 |                                            |  1 |        | 1 |       |  2 |                                              |  3 |        |  0 |                                            |  1 |        |  2 |                                            |  2 |        | -1 |                                              |  2 |
│ │ │ +o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  0 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1, |  0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, |  0 |}, {2, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  3 |}, {{0, |  0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {}}}}
│ │ │ +                                                          | -2 |        | -1 |       |  2 |                                              | -3 |        |  1 |       |  2 |                                            | -1 |        | -1 |       | -1 |       |  1 |                                              |  1 |        | -1 |       | 0 |                                            | -1 |        | -1 |       |  2 |                                            | -3 |        | -1 |       | -1 |                                              |  1 |        |  1 |                                            | -2 |        | -1 |                                            | -2 |        | -1 |                                              | -1 |
│ │ │ +                                                          |  1 |        |  2 |       | -1 |                                              |  1 |        |  1 |       | -1 |                                            |  2 |        |  0 |       |  2 |       |  1 |                                              |  1 |        |  2 |       | 1 |                                            |  3 |        |  0 |       | -1 |                                            |  2 |        |  2 |       |  0 |                                              |  2 |        | -1 |                                            |  3 |        |  0 |                                            |  1 |        |  2 |                                              |  2 |
│ │ │  
│ │ │  o5 : HasseDiagram
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │

    Each row of the Hasse diagram contains the elements of a certain length together with their links to the next row.

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i6 : myInterval#1
│ │ │  
│ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, | -1 |}, {1, | -1
│ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{1, | -1 |}, {2, | -1
│ │ │                                               | -3 |        |  1 |       |  2
│ │ │                                               |  1 |        |  1 |       | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  0 |}, {1,
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, |  2 |}, {1,
│ │ │       |                                            | -1 |        | -1 |      
│ │ │ -     |                                            |  2 |        |  2 |      
│ │ │ +     |                                            |  2 |        |  0 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, |  2 |}}}}
│ │ │ -     |  1 |       | -1 |
│ │ │ -     |  1 |       |  0 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o6 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -42,58 +42,58 @@ │ │ │ │ | -2 | │ │ │ │ | 1 | │ │ │ │ │ │ │ │ o4 : WeylGroupElement │ │ │ │ i5 : myInterval=intervalBruhat(P % w1,P % w2) │ │ │ │ │ │ │ │ o5 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | 0 │ │ │ │ -|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | - │ │ │ │ -1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 │ │ │ │ -|}, {1, | -1 |}, {2, | 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | - │ │ │ │ -1 |}, {{0, | -1 |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 │ │ │ │ -|}, {{0, | 0 |}, {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | - │ │ │ │ -3 |}, {{1, | 1 |}, {2, | 0 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 │ │ │ │ -|}, {{0, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{0, | 0 │ │ │ │ -|}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}}}}, { │ │ │ │ +|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{1, | - │ │ │ │ +1 |}, {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 2 │ │ │ │ +|}, {1, | 0 |}, {2, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | - │ │ │ │ +3 |}, {{0, | 0 |}, {2, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | - │ │ │ │ +1 |}, {{0, | 2 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 │ │ │ │ +|}, {{1, | 0 |}, {2, | 2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | - │ │ │ │ +2 |}, {{0, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 2 │ │ │ │ +|}}}, {WeylGroupElement{RootSystem{...8...}, | 3 |}, {{0, | 0 |}}}}, { │ │ │ │ {WeylGroupElement{RootSystem{...8...}, | 2 |}, {}}}} │ │ │ │ | -2 | | -1 | │ │ │ │ | 2 | | -3 | | 1 | │ │ │ │ | 2 | | -1 | | -1 | | │ │ │ │ -1 | | -1 | | -1 | | │ │ │ │ -2 | | -1 | | -3 | | - │ │ │ │ -1 | | -1 | | 1 | | 0 | │ │ │ │ -| -1 | | -2 | | -1 | │ │ │ │ -| -2 | | -1 | | 1 | | │ │ │ │ -1 | | -1 | │ │ │ │ +-1 | | 1 | | 1 | | │ │ │ │ +-1 | | 0 | | -1 | | - │ │ │ │ +1 | | 2 | | -3 | | - │ │ │ │ +1 | | -1 | | 1 | | │ │ │ │ +1 | | -2 | | -1 | │ │ │ │ +| -2 | | -1 | | -1 | │ │ │ │ | 1 | | 2 | │ │ │ │ | -1 | | 1 | | 1 | │ │ │ │ +| -1 | | 2 | | 0 | | │ │ │ │ +2 | | 1 | | 1 | | │ │ │ │ +2 | | 1 | | 3 | | 0 | │ │ │ │ | -1 | | 2 | | 2 | | │ │ │ │ -1 | | 0 | | 3 | | - │ │ │ │ -1 | | 0 | | 2 | | 2 │ │ │ │ -| | 0 | | 1 | | 1 | │ │ │ │ -| 2 | | 3 | | 0 | │ │ │ │ -| 1 | | 2 | | 2 | | │ │ │ │ --1 | | 2 | │ │ │ │ +0 | | 2 | | -1 | │ │ │ │ +| 3 | | 0 | | 1 | | │ │ │ │ +2 | | 2 | │ │ │ │ │ │ │ │ o5 : HasseDiagram │ │ │ │ Each row of the Hasse diagram contains the elements of a certain length │ │ │ │ together with their links to the next row. │ │ │ │ i6 : myInterval#1 │ │ │ │ │ │ │ │ -o6 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | -1 |}, {1, | -1 │ │ │ │ +o6 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{1, | -1 |}, {2, | -1 │ │ │ │ | -3 | | 1 | | 2 │ │ │ │ | 1 | | 1 | | -1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 0 |}, {1, │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 2 |}, {1, │ │ │ │ | | -1 | | -1 | │ │ │ │ - | | 2 | | 2 | │ │ │ │ + | | 2 | | 0 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | -1 |}, {2, | 2 |}}}} │ │ │ │ - | 1 | | -1 | │ │ │ │ - | 1 | | 0 | │ │ │ │ + | 0 |}, {2, | -1 |}}}} │ │ │ │ + | -1 | | 1 | │ │ │ │ + | 2 | | 1 | │ │ │ │ │ │ │ │ o6 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _i_n_t_e_r_v_a_l_B_r_u_h_a_t_(_W_e_y_l_G_r_o_u_p_R_i_g_h_t_C_o_s_e_t_,_W_e_y_l_G_r_o_u_p_R_i_g_h_t_C_o_s_e_t_) -- elements │ │ │ │ between two given ones for the Bruhat order on a quotient of a Weyl group │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_list__Weyl__Group__Elements_lp__Root__System_cm__Z__Z_rp.html │ │ │ @@ -75,19 +75,19 @@ │ │ │
    │ │ │

    Description

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : listWeylGroupElements(rootSystemG2,4)
│ │ │  
│ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, |  4 |},
│ │ │ -                                            | -3 |  
│ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |},
│ │ │ +                                            |  2 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     WeylGroupElement{RootSystem{...8...}, | -5 |}}
│ │ │ -                                           |  2 |
│ │ │ +     WeylGroupElement{RootSystem{...8...}, |  4 |}}
│ │ │ +                                           | -3 |
│ │ │  
│ │ │  o1 : List
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -12,19 +12,19 @@ │ │ │ │ o R, an instance of the type _R_o_o_t_S_y_s_t_e_m, │ │ │ │ o k, an _i_n_t_e_g_e_r, │ │ │ │ * Outputs: │ │ │ │ o a _l_i_s_t, of all elements of length k in the Weyl group of R │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : listWeylGroupElements(rootSystemG2,4) │ │ │ │ │ │ │ │ -o1 = {WeylGroupElement{RootSystem{...8...}, | 4 |}, │ │ │ │ - | -3 | │ │ │ │ +o1 = {WeylGroupElement{RootSystem{...8...}, | -5 |}, │ │ │ │ + | 2 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - WeylGroupElement{RootSystem{...8...}, | -5 |}} │ │ │ │ - | 2 | │ │ │ │ + WeylGroupElement{RootSystem{...8...}, | 4 |}} │ │ │ │ + | -3 | │ │ │ │ │ │ │ │ o1 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _l_i_s_t_W_e_y_l_G_r_o_u_p_E_l_e_m_e_n_t_s_(_R_o_o_t_S_y_s_t_e_m_,_Z_Z_) -- list all elements of a given │ │ │ │ length in a Weyl group │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_poincare__Series_lp__Hasse__Diagram_cm__Ring__Element_rp.html │ │ │ @@ -84,16 +84,16 @@ │ │ │ o1 : RootSystem │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │  
│ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ -                                                          | -1 |        | -1 |       |  2 |                                              |  1 |        | 1 |       |  2 |                                            | -2 |        | -1 |       | 1 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1 |}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ +                                                          | -1 |        |  2 |       | -1 |                                              | -2 |        | -1 |       | 1 |                                            |  1 |        | 1 |       |  2 |                                              | -1 |        |  2 |                                            |  2 |        | -1 |                                              | 1 |
│ │ │  
│ │ │  o2 : HasseDiagram
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : ZZ[x]
│ │ │ ├── html2text {}
│ │ │ │ @@ -19,25 +19,25 @@
│ │ │ │  i1 : R=rootSystemA(2)
│ │ │ │  
│ │ │ │  o1 = RootSystem{...8...}
│ │ │ │  
│ │ │ │  o1 : RootSystem
│ │ │ │  i2 : H=intervalBruhat(neutralWeylGroupElement R, longestWeylGroupElement R)
│ │ │ │  
│ │ │ │ -o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2
│ │ │ │ -|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1
│ │ │ │ -|}, {1, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  2
│ │ │ │ -|}, {1, | 1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -
│ │ │ │ +o2 = HasseDiagram{{{WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, | -
│ │ │ │ +1 |}, {1, |  2 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |
│ │ │ │ +2 |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | 1
│ │ │ │ +|}, {1, | -1 |}}}}, {{WeylGroupElement{RootSystem{...8...}, |  2 |}, {{0, | -
│ │ │ │  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{0, |  2 |}}}}, {
│ │ │ │  {WeylGroupElement{RootSystem{...8...}, | 1 |}, {}}}}
│ │ │ │ -                                                          | -1 |        | -1 |
│ │ │ │ -|  2 |                                              |  1 |        | 1 |       |
│ │ │ │ -2 |                                            | -2 |        | -1 |       | 1 |
│ │ │ │ -| -1 |        |  2 |                                            |  2 |        |
│ │ │ │ --1 |                                              | 1 |
│ │ │ │ +                                                          | -1 |        |  2 |
│ │ │ │ +| -1 |                                              | -2 |        | -1 |
│ │ │ │ +| 1 |                                            |  1 |        | 1 |       |  2
│ │ │ │ +|                                              | -1 |        |  2 |
│ │ │ │ +|  2 |        | -1 |                                              | 1 |
│ │ │ │  
│ │ │ │  o2 : HasseDiagram
│ │ │ │  i3 : ZZ[x]
│ │ │ │  
│ │ │ │  o3 = ZZ[x]
│ │ │ │  
│ │ │ │  o3 : PolynomialRing
│ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_positive__Roots_lp__Root__System_cm__Parabolic_rp.html
│ │ │ @@ -93,17 +93,17 @@
│ │ │  o2 : Parabolic
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i3 : positiveRoots(R,P)
│ │ │  
│ │ │ -o3 = set {|  1 |, |  2 |, | -1 |}
│ │ │ -          |  1 |  | -1 |  |  2 |
│ │ │ -          | -1 |  |  0 |  | -1 |
│ │ │ +o3 = set {| -1 |, |  1 |, |  2 |}
│ │ │ +          |  2 |  |  1 |  | -1 |
│ │ │ +          | -1 |  | -1 |  |  0 |
│ │ │            |  0 |  |  0 |  |  0 |
│ │ │  
│ │ │  o3 : Set
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -23,17 +23,17 @@ │ │ │ │ i2 : P=parabolic(R,set{1,2}) │ │ │ │ │ │ │ │ o2 = set {1, 2} │ │ │ │ │ │ │ │ o2 : Parabolic │ │ │ │ i3 : positiveRoots(R,P) │ │ │ │ │ │ │ │ -o3 = set {| 1 |, | 2 |, | -1 |} │ │ │ │ - | 1 | | -1 | | 2 | │ │ │ │ - | -1 | | 0 | | -1 | │ │ │ │ +o3 = set {| -1 |, | 1 |, | 2 |} │ │ │ │ + | 2 | | 1 | | -1 | │ │ │ │ + | -1 | | -1 | | 0 | │ │ │ │ | 0 | | 0 | | 0 | │ │ │ │ │ │ │ │ o3 : Set │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_,_P_a_r_a_b_o_l_i_c_) -- the set of all positive roots in a │ │ │ │ parabolic sugroups │ │ │ │ =============================================================================== │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_positive__Roots_lp__Root__System_rp.html │ │ │ @@ -74,17 +74,17 @@ │ │ │
    │ │ │

    Description

    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i1 : positiveRoots(rootSystemA(3))
│ │ │  
│ │ │ -o1 = set {|  0 |, | -1 |, | -1 |, | 1 |, |  1 |, |  2 |}
│ │ │ -          | -1 |  |  1 |  |  2 |  | 0 |  |  1 |  | -1 |
│ │ │ -          |  2 |  |  1 |  | -1 |  | 1 |  | -1 |  |  0 |
│ │ │ +o1 = set {| 1 |, |  1 |, |  2 |, |  0 |, | -1 |, | -1 |}
│ │ │ +          | 0 |  |  1 |  | -1 |  | -1 |  |  1 |  |  2 |
│ │ │ +          | 1 |  | -1 |  |  0 |  |  2 |  |  1 |  | -1 |
│ │ │  
│ │ │  o1 : Set
    │ │ │
    │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -10,17 +10,17 @@ │ │ │ │ * Inputs: │ │ │ │ o R, an instance of the type _R_o_o_t_S_y_s_t_e_m, │ │ │ │ * Outputs: │ │ │ │ o a _s_e_t, of all positive roots of R │ │ │ │ ********** DDeessccrriippttiioonn ********** │ │ │ │ i1 : positiveRoots(rootSystemA(3)) │ │ │ │ │ │ │ │ -o1 = set {| 0 |, | -1 |, | -1 |, | 1 |, | 1 |, | 2 |} │ │ │ │ - | -1 | | 1 | | 2 | | 0 | | 1 | | -1 | │ │ │ │ - | 2 | | 1 | | -1 | | 1 | | -1 | | 0 | │ │ │ │ +o1 = set {| 1 |, | 1 |, | 2 |, | 0 |, | -1 |, | -1 |} │ │ │ │ + | 0 | | 1 | | -1 | | -1 | | 1 | | 2 | │ │ │ │ + | 1 | | -1 | | 0 | | 2 | | 1 | | -1 | │ │ │ │ │ │ │ │ o1 : Set │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _p_o_s_i_t_i_v_e_R_o_o_t_s_(_R_o_o_t_S_y_s_t_e_m_) -- the set of all positive roots │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/WeylGroups.m2:2604:0. │ │ ├── ./usr/share/doc/Macaulay2/WeylGroups/html/_under__Bruhat_lp__Basic__List_rp.html │ │ │ @@ -121,37 +121,37 @@ │ │ │ o3 : List │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ 
    i4 : underBruhat(L1)
│ │ │  
│ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |}, {1, |  2
│ │ │ -                                             |  2 |        |  1 |       | -1
│ │ │ -                                             | -1 |        | -1 |       |  0
│ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0 |}, {2, |  2
│ │ │ +                                             | -3 |        | -1 |       | -1
│ │ │ +                                             |  1 |        |  2 |       |  0
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1,
│ │ │ -     |                                            | -1 |        |  2 |      
│ │ │ -     |                                            |  2 |        | -1 |      
│ │ │ +     |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1, | -1 |}, {2,
│ │ │ +     |                                            | -1 |        |  1 |      
│ │ │ +     |                                            | -2 |        |  1 |      
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, |  0 |},
│ │ │ -     |  1 |                                            |  2 |        | -1 |  
│ │ │ -     | -1 |                                            | -3 |        |  2 |  
│ │ │ +     | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, |  1 |},
│ │ │ +     |  2 |                                            |  2 |        |  1 |  
│ │ │ +     | -1 |                                            | -1 |        | -1 |  
│ │ │       ------------------------------------------------------------------------
│ │ │ -     {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, |  1 |}, {{0, |  0
│ │ │ -         |  1 |                                            | -3 |        | -1
│ │ │ -         |  1 |                                            |  1 |        |  2
│ │ │ +     {1, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1
│ │ │ +         | -1 |                                            | -1 |        |  2
│ │ │ +         |  0 |                                            |  2 |        | -1
│ │ │       ------------------------------------------------------------------------
│ │ │ -     |}, {2, |  2 |}}}, {WeylGroupElement{RootSystem{...8...}, |  2 |}, {{1,
│ │ │ -     |       | -1 |                                            | -1 |       
│ │ │ -     |       |  0 |                                            | -2 |       
│ │ │ +     |}, {1, |  1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1,
│ │ │ +     |       |  1 |                                            |  2 |       
│ │ │ +     |       | -1 |                                            | -3 |       
│ │ │       ------------------------------------------------------------------------
│ │ │ -     | -1 |}, {2, | -1 |}}}}
│ │ │ -     |  1 |       |  2 |
│ │ │ -     |  1 |       | -1 |
│ │ │ +     |  0 |}, {2, | -1 |}}}}
│ │ │ +     | -1 |       |  1 |
│ │ │ +     |  2 |       |  1 |
│ │ │  
│ │ │  o4 : List
    │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │
    │ │ │ ├── html2text {} │ │ │ │ @@ -48,37 +48,37 @@ │ │ │ │ WeylGroupElement{RootSystem{...8...}, | 1 |}} │ │ │ │ | -2 | │ │ │ │ | -1 | │ │ │ │ │ │ │ │ o3 : List │ │ │ │ i4 : underBruhat(L1) │ │ │ │ │ │ │ │ -o4 = {{WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, {1, | 2 │ │ │ │ - | 2 | | 1 | | -1 │ │ │ │ - | -1 | | -1 | | 0 │ │ │ │ +o4 = {{WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 0 |}, {2, | 2 │ │ │ │ + | -3 | | -1 | | -1 │ │ │ │ + | 1 | | 2 | | 0 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 |}, {1, │ │ │ │ - | | -1 | | 2 | │ │ │ │ - | | 2 | | -1 | │ │ │ │ + |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, | -1 |}, {2, │ │ │ │ + | | -1 | | 1 | │ │ │ │ + | | -2 | | 1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, | 0 |}, │ │ │ │ - | 1 | | 2 | | -1 | │ │ │ │ - | -1 | | -3 | | 2 | │ │ │ │ + | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -3 |}, {{0, | 1 |}, │ │ │ │ + | 2 | | 2 | | 1 | │ │ │ │ + | -1 | | -1 | | -1 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - {2, | -1 |}}}, {WeylGroupElement{RootSystem{...8...}, | 1 |}, {{0, | 0 │ │ │ │ - | 1 | | -3 | | -1 │ │ │ │ - | 1 | | 1 | | 2 │ │ │ │ + {1, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | -2 |}, {{0, | -1 │ │ │ │ + | -1 | | -1 | | 2 │ │ │ │ + | 0 | | 2 | | -1 │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - |}, {2, | 2 |}}}, {WeylGroupElement{RootSystem{...8...}, | 2 |}, {{1, │ │ │ │ - | | -1 | | -1 | │ │ │ │ - | | 0 | | -2 | │ │ │ │ + |}, {1, | 1 |}}}, {WeylGroupElement{RootSystem{...8...}, | -1 |}, {{1, │ │ │ │ + | | 1 | | 2 | │ │ │ │ + | | -1 | | -3 | │ │ │ │ ------------------------------------------------------------------------ │ │ │ │ - | -1 |}, {2, | -1 |}}}} │ │ │ │ - | 1 | | 2 | │ │ │ │ - | 1 | | -1 | │ │ │ │ + | 0 |}, {2, | -1 |}}}} │ │ │ │ + | -1 | | 1 | │ │ │ │ + | 2 | | 1 | │ │ │ │ │ │ │ │ o4 : List │ │ │ │ ********** WWaayyss ttoo uussee tthhiiss mmeetthhoodd:: ********** │ │ │ │ * _u_n_d_e_r_B_r_u_h_a_t_(_B_a_s_i_c_L_i_s_t_) -- Weyl group elements just under the ones in the │ │ │ │ list for the Bruhat order │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ -# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:32 2026 │ │ │ +# GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=8 │ │ │ Y29ub3JtYWw= │ │ │ #:len=1038 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiQ29tcHV0ZXMgdGhlIGNvbm9ybWFsIHZh │ │ │ cmlldHkiLCAibGluZW51bSIgPT4gMTY2NCwgSW5wdXRzID0+IHtTUEFOe1RUeyJJIn0sIiwgIixT │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/example-output/_map__Stratify.out │ │ │ @@ -122,15 +122,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc) │ │ │ + -- used 1.87135s (cpu); 1.05633s (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 4.97937s (cpu); 2.77946s (thread); 0s (gc) │ │ │ + -- used 6.1229s (cpu); 2.59684s (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 5.88833s (cpu); 3.18288s (thread); 0s (gc) │ │ │ + -- used 6.84432s (cpu); 2.93899s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ i28 : peek last ms │ │ ├── ./usr/share/doc/Macaulay2/WhitneyStratifications/html/_map__Stratify.html │ │ │ @@ -297,15 +297,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc) │ │ │ + -- used 1.87135s (cpu); 1.05633s (thread); 0s (gc) │ │ │ │ │ │ o23 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o23 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -323,15 +323,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 4.97937s (cpu); 2.77946s (thread); 0s (gc) │ │ │ + -- used 6.1229s (cpu); 2.59684s (thread); 0s (gc) │ │ │ │ │ │ o25 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o25 : List │ │ │ │ │ │ │ │ │ │ │ │ @@ -349,15 +349,15 @@ │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ for j2.... │ │ │ loop over components of JY=ideal 1 │ │ │ loop over components of JY=ideal 1 │ │ │ - -- used 5.88833s (cpu); 3.18288s (thread); 0s (gc) │ │ │ + -- used 6.84432s (cpu); 2.93899s (thread); 0s (gc) │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ o27 : List │ │ │ │ │ │ │ │ │ │ │ │ ├── html2text {} │ │ │ │ @@ -184,15 +184,15 @@ │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ for j2.... │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ loop over components of JY=ideal 1 │ │ │ │ - -- used 2.05196s (cpu); 1.19594s (thread); 0s (gc) │ │ │ │ + -- used 1.87135s (cpu); 1.05633s (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 4.97937s (cpu); 2.77946s (thread); 0s (gc) │ │ │ │ + -- used 6.1229s (cpu); 2.59684s (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 5.88833s (cpu); 3.18288s (thread); 0s (gc) │ │ │ │ + -- used 6.84432s (cpu); 2.93899s (thread); 0s (gc) │ │ │ │ │ │ │ │ o27 = {MutableHashTable{...5...}, MutableHashTable{...3...}} │ │ │ │ │ │ │ │ o27 : List │ │ │ │ i28 : peek last ms │ │ │ │ │ │ │ │ o28 = MutableHashTable{0 => {ideal (P, M1)} } │ │ ├── ./usr/share/doc/Macaulay2/WittVectors/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=50 │ │ │ ZmluZEZyb2Jlbml1c0xpZnRDb25zdHJhaW50cyguLi4sSG9tb2dlbmVvdXM9Pi4uLik= │ │ │ #:len=354 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMTEyOCwgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHtbZmluZEZyb2Jlbml1c0xpZnRDb25zdHJhaW50cyxI │ │ ├── ./usr/share/doc/Macaulay2/XML/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=25 │ │ │ Z2V0QXR0cmlidXRlcyhMaWJ4bWxOb2RlKQ== │ │ │ #:len=465 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHtIZWFkbGluZSA9PiAiZ2V0IHRoZSBsaXN0IG9mIGF0dHJpYnV0 │ │ │ ZXMgb2YgYW4gWE1MIG5vZGUiLCBEZXNjcmlwdGlvbiA9PiAxOihUQUJMRXsiY2xhc3MiID0+ICJl │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/dump/rawdocumentation.dump │ │ │ @@ -1,11 +1,11 @@ │ │ │ # GDBM dump file created by GDBM version 1.26. 30/07/2025 on Sat Oct 24 01:47:31 2026 │ │ │ #:version=1.1 │ │ │ #:file=rawdocumentation-dcba-8.db │ │ │ -#:uid=999,user=sbuild,gid=999,group=sbuild,mode=644 │ │ │ +#:uid=995,user=sbuild,gid=995,group=sbuild,mode=644 │ │ │ #:format=standard │ │ │ # End of header │ │ │ #:len=33 │ │ │ Z2ZhblBvbHlub21pYWxTZXRVbmlvbihMaXN0LExpc3Qp │ │ │ #:len=309 │ │ │ bmV3IEhhc2hUYWJsZSBmcm9tIHsibGluZW51bSIgPT4gMzY0Mywgc3ltYm9sIERvY3VtZW50VGFn │ │ │ ID0+IG5ldyBEb2N1bWVudFRhZyBmcm9tIHsoZ2ZhblBvbHlub21pYWxTZXRVbmlvbixMaXN0LExp │ │ ├── ./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/M2-16342-0/172 │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20526-0/172 │ │ │ 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/M2-16342-0/172 │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174 │ │ │ +using temporary file /tmp/M2-20526-0/172 │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20526-0/174 │ │ │ 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/M2-16342-0/174 │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176 │ │ │ +using temporary file /tmp/M2-20526-0/174 │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20526-0/176 │ │ │ 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/M2-16342-0/176 │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178 │ │ │ +using temporary file /tmp/M2-20526-0/176 │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20526-0/178 │ │ │ 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/M2-16342-0/178 │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180 │ │ │ +using temporary file /tmp/M2-20526-0/178 │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20526-0/180 │ │ │ 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/M2-16342-0/180 │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182 │ │ │ +using temporary file /tmp/M2-20526-0/180 │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20526-0/182 │ │ │ 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/M2-16342-0/182 │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184 │ │ │ +using temporary file /tmp/M2-20526-0/182 │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20526-0/184 │ │ │ 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/M2-16342-0/184 │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186 │ │ │ +using temporary file /tmp/M2-20526-0/184 │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20526-0/186 │ │ │ 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/M2-16342-0/186 │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188 │ │ │ +using temporary file /tmp/M2-20526-0/186 │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20526-0/188 │ │ │ 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/M2-16342-0/188 │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190 │ │ │ +using temporary file /tmp/M2-20526-0/188 │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20526-0/190 │ │ │ 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/M2-16342-0/190 │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192 │ │ │ +using temporary file /tmp/M2-20526-0/190 │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20526-0/192 │ │ │ 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/M2-16342-0/192 │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194 │ │ │ +using temporary file /tmp/M2-20526-0/192 │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20526-0/194 │ │ │ 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/M2-16342-0/194 │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196 │ │ │ +using temporary file /tmp/M2-20526-0/194 │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20526-0/196 │ │ │ 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/M2-16342-0/196 │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198 │ │ │ +using temporary file /tmp/M2-20526-0/196 │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20526-0/198 │ │ │ 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/M2-16342-0/198 │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200 │ │ │ +using temporary file /tmp/M2-20526-0/198 │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20526-0/200 │ │ │ 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/M2-16342-0/200 │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202 │ │ │ +using temporary file /tmp/M2-20526-0/200 │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20526-0/202 │ │ │ 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/M2-16342-0/202 │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204 │ │ │ +using temporary file /tmp/M2-20526-0/202 │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20526-0/204 │ │ │ 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/M2-16342-0/204 │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206 │ │ │ +using temporary file /tmp/M2-20526-0/204 │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20526-0/206 │ │ │ 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/M2-16342-0/206 │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208 │ │ │ +using temporary file /tmp/M2-20526-0/206 │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20526-0/208 │ │ │ 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: │ │ │ @@ -251,47 +251,47 @@ │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ --eco value: │ │ │ Use Eco-n example instead of reading input. │ │ │ -j value: │ │ │ Number of threads │ │ │ --resume: │ │ │ Continue enumeration from saved state... │ │ │ -using temporary file /tmp/M2-16342-0/208 │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210 │ │ │ +using temporary file /tmp/M2-20526-0/208 │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20526-0/210 │ │ │ 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. │ │ │ │ │ │ --twosetswithrings: │ │ │ Switch to mode where input is: ring polynomialset (same)ring polynomialset. │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/210 │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212 │ │ │ +using temporary file /tmp/M2-20526-0/210 │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20526-0/212 │ │ │ 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/M2-16342-0/212 │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214 │ │ │ +using temporary file /tmp/M2-20526-0/212 │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20526-0/214 │ │ │ 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/M2-16342-0/214 │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216 │ │ │ +using temporary file /tmp/M2-20526-0/214 │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20526-0/216 │ │ │ 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. │ │ │ @@ -304,25 +304,25 @@ │ │ │ │ │ │ --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/M2-16342-0/216 │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218 │ │ │ +using temporary file /tmp/M2-20526-0/216 │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20526-0/218 │ │ │ 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/M2-16342-0/218 │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220 │ │ │ +using temporary file /tmp/M2-20526-0/218 │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20526-0/220 │ │ │ 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: │ │ │ @@ -331,72 +331,72 @@ │ │ │ --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/M2-16342-0/220 │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222 │ │ │ +using temporary file /tmp/M2-20526-0/220 │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20526-0/222 │ │ │ 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/M2-16342-0/222 │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224 │ │ │ +using temporary file /tmp/M2-20526-0/222 │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20526-0/224 │ │ │ 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/M2-16342-0/224 │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226 │ │ │ +using temporary file /tmp/M2-20526-0/224 │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20526-0/226 │ │ │ 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/M2-16342-0/226 │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228 │ │ │ +using temporary file /tmp/M2-20526-0/226 │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20526-0/228 │ │ │ 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. │ │ │ --generatedcones: │ │ │ Instead of a list of reduced Groebner bases, read in a list of closed generated cones given in modern format (like _cone). In this case --restrict is ignored. │ │ │ --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/M2-16342-0/228 │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230 │ │ │ +using temporary file /tmp/M2-20526-0/228 │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20526-0/230 │ │ │ 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/M2-16342-0/230 │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232 │ │ │ +using temporary file /tmp/M2-20526-0/230 │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20526-0/232 │ │ │ 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/M2-16342-0/232 │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234 │ │ │ +using temporary file /tmp/M2-20526-0/232 │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20526-0/234 │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/234 │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236 │ │ │ +using temporary file /tmp/M2-20526-0/234 │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20526-0/236 │ │ │ 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/M2-16342-0/236 │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238 │ │ │ +using temporary file /tmp/M2-20526-0/236 │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20526-0/238 │ │ │ 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/M2-16342-0/238 │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240 │ │ │ +using temporary file /tmp/M2-20526-0/238 │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20526-0/240 │ │ │ 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. │ │ │ @@ -408,16 +408,16 @@ │ │ │ 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/M2-16342-0/240 │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242 │ │ │ +using temporary file /tmp/M2-20526-0/240 │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20526-0/242 │ │ │ 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 │ │ │ │ │ │ @@ -442,48 +442,48 @@ │ │ │ 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/M2-16342-0/242 │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244 │ │ │ +using temporary file /tmp/M2-20526-0/242 │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20526-0/244 │ │ │ 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/M2-16342-0/244 │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246 │ │ │ +using temporary file /tmp/M2-20526-0/244 │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20526-0/246 │ │ │ This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal. │ │ │ Options: │ │ │ -using temporary file /tmp/M2-16342-0/246 │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248 │ │ │ +using temporary file /tmp/M2-20526-0/246 │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20526-0/248 │ │ │ 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/M2-16342-0/248 │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250 │ │ │ +using temporary file /tmp/M2-20526-0/248 │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20526-0/250 │ │ │ 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/M2-16342-0/250 │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252 │ │ │ +using temporary file /tmp/M2-20526-0/250 │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20526-0/252 │ │ │ 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: │ │ │ @@ -491,24 +491,24 @@ │ │ │ --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/M2-16342-0/252 │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254 │ │ │ +using temporary file /tmp/M2-20526-0/252 │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20526-0/254 │ │ │ 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/M2-16342-0/254 │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256 │ │ │ +using temporary file /tmp/M2-20526-0/254 │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20526-0/256 │ │ │ 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. │ │ │ @@ -528,21 +528,21 @@ │ │ │ --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/M2-16342-0/256 │ │ │ +using temporary file /tmp/M2-20526-0/256 │ │ │ │ │ │ i6 : QQ[x,y]; │ │ │ │ │ │ i7 : gfan {x,y}; │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258 │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20526-0/258 │ │ │ Q[x1,x2] │ │ │ {{ │ │ │ x2, │ │ │ x1} │ │ │ } │ │ │ -using temporary file /tmp/M2-16342-0/258 │ │ │ +using temporary file /tmp/M2-20526-0/258 │ │ │ │ │ │ i8 : │ │ ├── ./usr/share/doc/Macaulay2/gfanInterface/html/___Installation_spand_sp__Configuration_spof_spgfan__Interface.html │ │ │ @@ -114,15 +114,15 @@ │ │ │

    │ │ │
    │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
    │ │ │ 
    i5 : loadPackage("gfanInterface", Configuration => { "keepfiles" => true, "verbose" => true}, Reload => true);
│ │ │   -- warning: reloading gfanInterface; recreate instances of types from this package
│ │ │ - -- running: /usr/bin/gfan gfan --help < /tmp/M2-16342-0/172
│ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20526-0/172
│ │ │  This is a program for computing all reduced Groebner bases of a polynomial ideal. It takes the ring and a generating set for the ideal as input. By default the enumeration is done by an almost memoryless reverse search. If the ideal is symmetric the symmetry option is useful and enumeration will be done up to symmetry using a breadth first search. The program needs a starting Groebner basis to do its computations. If the -g option is not specified it will compute one using Buchberger's algorithm.
│ │ │  Options:
│ │ │  -g:
│ │ │   Tells the program that the input is already a Groebner basis (with the initial term of each polynomial being the first ones listed). Use this option if it takes too much time to compute the starting (standard degree lexicographic) Groebner basis and the input is already a Groebner basis.
│ │ │  
│ │ │  --symmetry:
│ │ │   Tells the program to read in generators for a group of symmetries (subgroup of $S_n$) after having read in the ideal. The program checks that the ideal stays fixed when permuting the variables with respect to elements in the group. The program uses breadth first search to compute the set of reduced Groebner bases up to symmetry with respect to the specified subgroup.
│ │ │ @@ -133,16 +133,16 @@
│ │ │  --disableSymmetryTest:
│ │ │   When using --symmetry this option will disable the check that the group read off from the input actually is a symmetry group with respect to the input ideal.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │  --interrupt value:
│ │ │   Interrupt the enumeration after a specified number of facets have been computed (works for usual symmetric traversals, but may not work in general for non-symmetric traversals or for traversals restricted to fans).
│ │ │ -using temporary file /tmp/M2-16342-0/172
│ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174
│ │ │ +using temporary file /tmp/M2-20526-0/172
│ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20526-0/174
│ │ │  This program computes a reduced lexicographic Groebner basis of the polynomial ideal given as input. The default behavior is to use Buchberger's algorithm. The ordering of the variables is $a>b>c...$ (assuming that the ring is Q[a,b,c,...]).
│ │ │  Options:
│ │ │  -w:
│ │ │   Compute a Groebner basis with respect to a degree lexicographic order with $a>b>c...$ instead. The degrees are given by a weight vector which is read from the input after the generating set has been read.
│ │ │  
│ │ │  -r:
│ │ │   Use the reverse lexicographic order (or the reverse lexicographic order as a tie breaker if -w is used). The input must be homogeneous if the pure reverse lexicographic order is chosen. Ignored if -W is used.
│ │ │ @@ -151,69 +151,69 @@
│ │ │   Do a Groebner walk. The input must be a minimal Groebner basis. If -W is used -w is ignored.
│ │ │  
│ │ │  -g:
│ │ │   Do a generic Groebner walk. The input must be homogeneous and must be a minimal Groebner basis with respect to the reverse lexicographic term order. The target term order is always lexicographic. The -W option must be used.
│ │ │  
│ │ │  --parameters value:
│ │ │   With this option you can specify how many variables to treat as parameters instead of variables. This makes it possible to do computations where the coefficient field is the field of rational functions in the parameters.
│ │ │ -using temporary file /tmp/M2-16342-0/174
│ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176
│ │ │ +using temporary file /tmp/M2-20526-0/174
│ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20526-0/176
│ │ │  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/M2-16342-0/176
│ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178
│ │ │ +using temporary file /tmp/M2-20526-0/176
│ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20526-0/178
│ │ │  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/M2-16342-0/178
│ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180
│ │ │ +using temporary file /tmp/M2-20526-0/178
│ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20526-0/180
│ │ │  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/M2-16342-0/180
│ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182
│ │ │ +using temporary file /tmp/M2-20526-0/180
│ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20526-0/182
│ │ │  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/M2-16342-0/182
│ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184
│ │ │ +using temporary file /tmp/M2-20526-0/182
│ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20526-0/184
│ │ │  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/M2-16342-0/184
│ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186
│ │ │ +using temporary file /tmp/M2-20526-0/184
│ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20526-0/186
│ │ │  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/M2-16342-0/186
│ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188
│ │ │ +using temporary file /tmp/M2-20526-0/186
│ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20526-0/188
│ │ │  This program homogenises a list of polynomials by introducing an extra variable. The name of the variable to be introduced is read from the input after the list of polynomials. Without the -w option the homogenisation is done with respect to total degree.
│ │ │  Example:
│ │ │  Input:
│ │ │  Q[x,y]{y-1}
│ │ │  z
│ │ │  Output:
│ │ │  Q[x,y,z]{y-z}
│ │ │ @@ -221,30 +221,30 @@
│ │ │  -i:
│ │ │   Treat input as an ideal. This will make the program compute the homogenisation of the input ideal. This is done by computing a degree Groebner basis and homogenising it.
│ │ │  -w:
│ │ │   Specify a homogenisation vector. The length of the vector must be the same as the number of variables in the ring. The vector is read from the input after the list of polynomials.
│ │ │  
│ │ │  -H:
│ │ │   Let the name of the new variable be H rather than reading in a name from the input.
│ │ │ -using temporary file /tmp/M2-16342-0/188
│ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190
│ │ │ +using temporary file /tmp/M2-20526-0/188
│ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20526-0/190
│ │ │  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/M2-16342-0/190
│ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192
│ │ │ +using temporary file /tmp/M2-20526-0/190
│ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20526-0/192
│ │ │  This is a program for doing interactive walks in the Groebner fan of an ideal. The input is a Groebner basis defining the starting Groebner cone of the walk. The program will list all flippable facets of the Groebner cone and ask the user to choose one. The user types in the index (number) of the facet in the list. The program will walk through the selected facet and display the new Groebner basis and a list of new facet normals for the user to choose from. Since the program reads the user's choices through the the standard input it is recommended not to redirect the standard input for this program.
│ │ │  Options:
│ │ │  -L:
│ │ │   Latex mode. The program will try to show the current Groebner basis in a readable form by invoking LaTeX and xdvi.
│ │ │  
│ │ │  -x:
│ │ │   Exit immediately.
│ │ │ @@ -259,57 +259,57 @@
│ │ │   Tell the program to list the defining set of inequalities of the non-restricted Groebner cone as a set of vectors after having listed the current Groebner basis.
│ │ │  
│ │ │  -W:
│ │ │   Print weight vector. This will make the program print an interior vector of the current Groebner cone and a relative interior point for each flippable facet of the current Groebner cone.
│ │ │  
│ │ │  --tropical:
│ │ │   Traverse a tropical variety interactively.
│ │ │ -using temporary file /tmp/M2-16342-0/192
│ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194
│ │ │ +using temporary file /tmp/M2-20526-0/192
│ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20526-0/194
│ │ │  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/M2-16342-0/194
│ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196
│ │ │ +using temporary file /tmp/M2-20526-0/194
│ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20526-0/196
│ │ │  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/M2-16342-0/196
│ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198
│ │ │ +using temporary file /tmp/M2-20526-0/196
│ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20526-0/198
│ │ │  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/M2-16342-0/198
│ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200
│ │ │ +using temporary file /tmp/M2-20526-0/198
│ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20526-0/200
│ │ │  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/M2-16342-0/200
│ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202
│ │ │ +using temporary file /tmp/M2-20526-0/200
│ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20526-0/202
│ │ │  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/M2-16342-0/202
│ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204
│ │ │ +using temporary file /tmp/M2-20526-0/202
│ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20526-0/204
│ │ │  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/M2-16342-0/204
│ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206
│ │ │ +using temporary file /tmp/M2-20526-0/204
│ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20526-0/206
│ │ │  This program will generate the r*r minors of a d*n matrix of indeterminates.
│ │ │  Options:
│ │ │  -r value:
│ │ │   Specify r.
│ │ │  -d value:
│ │ │   Specify d.
│ │ │  -n value:
│ │ │ @@ -324,16 +324,16 @@
│ │ │   Do nothing but produce symmetry generators for the Pluecker ideal.
│ │ │  --symmetry:
│ │ │   Produces a list of generators for the group of symmetries keeping the set of minors fixed. (Only without --names).
│ │ │  --parametrize:
│ │ │   Parametrize the set of d times n matrices of Barvinok rank less than or equal to r-1 by a list of tropical polynomials.
│ │ │  --ultrametric:
│ │ │   Produce tropical equations cutting out the ultrametrics.
│ │ │ -using temporary file /tmp/M2-16342-0/206
│ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208
│ │ │ +using temporary file /tmp/M2-20526-0/206
│ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20526-0/208
│ │ │  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:
│ │ │ @@ -346,47 +346,47 @@
│ │ │   Use Gaukwa-n example instead of reading input.
│ │ │  --eco value:
│ │ │   Use Eco-n example instead of reading input.
│ │ │  -j value:
│ │ │   Number of threads
│ │ │  --resume:
│ │ │   Continue enumeration from saved state...
│ │ │ -using temporary file /tmp/M2-16342-0/208
│ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210
│ │ │ +using temporary file /tmp/M2-20526-0/208
│ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20526-0/210
│ │ │  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.
│ │ │  
│ │ │  --twosetswithrings:
│ │ │   Switch to mode where input is: ring polynomialset (same)ring polynomialset.
│ │ │  
│ │ │ -using temporary file /tmp/M2-16342-0/210
│ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212
│ │ │ +using temporary file /tmp/M2-20526-0/210
│ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20526-0/212
│ │ │  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/M2-16342-0/212
│ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214
│ │ │ +using temporary file /tmp/M2-20526-0/212
│ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20526-0/214
│ │ │  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/M2-16342-0/214
│ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216
│ │ │ +using temporary file /tmp/M2-20526-0/214
│ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20526-0/216
│ │ │  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.
│ │ │ @@ -399,25 +399,25 @@
│ │ │  
│ │ │  --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/M2-16342-0/216
│ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218
│ │ │ +using temporary file /tmp/M2-20526-0/216
│ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20526-0/218
│ │ │  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/M2-16342-0/218
│ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220
│ │ │ +using temporary file /tmp/M2-20526-0/218
│ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20526-0/220
│ │ │  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:
│ │ │ @@ -426,72 +426,72 @@
│ │ │  --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/M2-16342-0/220
│ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222
│ │ │ +using temporary file /tmp/M2-20526-0/220
│ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20526-0/222
│ │ │  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/M2-16342-0/222
│ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224
│ │ │ +using temporary file /tmp/M2-20526-0/222
│ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20526-0/224
│ │ │  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/M2-16342-0/224
│ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226
│ │ │ +using temporary file /tmp/M2-20526-0/224
│ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20526-0/226
│ │ │  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/M2-16342-0/226
│ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228
│ │ │ +using temporary file /tmp/M2-20526-0/226
│ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20526-0/228
│ │ │  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.
│ │ │  --generatedcones:
│ │ │   Instead of a list of reduced Groebner bases, read in a list of closed generated cones given in modern format (like _cone). In this case --restrict is ignored.
│ │ │  --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/M2-16342-0/228
│ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230
│ │ │ +using temporary file /tmp/M2-20526-0/228
│ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20526-0/230
│ │ │  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/M2-16342-0/230
│ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232
│ │ │ +using temporary file /tmp/M2-20526-0/230
│ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20526-0/232
│ │ │  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/M2-16342-0/232
│ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234
│ │ │ +using temporary file /tmp/M2-20526-0/232
│ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20526-0/234
│ │ │  This program evaluates a tropical polynomial function in a given set of points.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/234
│ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236
│ │ │ +using temporary file /tmp/M2-20526-0/234
│ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20526-0/236
│ │ │  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/M2-16342-0/236
│ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238
│ │ │ +using temporary file /tmp/M2-20526-0/236
│ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20526-0/238
│ │ │  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/M2-16342-0/238
│ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240
│ │ │ +using temporary file /tmp/M2-20526-0/238
│ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20526-0/240
│ │ │  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.
│ │ │ @@ -503,16 +503,16 @@
│ │ │   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/M2-16342-0/240
│ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242
│ │ │ +using temporary file /tmp/M2-20526-0/240
│ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20526-0/242
│ │ │  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
│ │ │  
│ │ │ @@ -537,48 +537,48 @@
│ │ │  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/M2-16342-0/242
│ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244
│ │ │ +using temporary file /tmp/M2-20526-0/242
│ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20526-0/244
│ │ │  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/M2-16342-0/244
│ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246
│ │ │ +using temporary file /tmp/M2-20526-0/244
│ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20526-0/246
│ │ │  This program computes the multiplicity of a tropical cone given a marked reduced Groebner basis for its initial ideal.
│ │ │  Options:
│ │ │ -using temporary file /tmp/M2-16342-0/246
│ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248
│ │ │ +using temporary file /tmp/M2-20526-0/246
│ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20526-0/248
│ │ │  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/M2-16342-0/248
│ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250
│ │ │ +using temporary file /tmp/M2-20526-0/248
│ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20526-0/250
│ │ │  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/M2-16342-0/250
│ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252
│ │ │ +using temporary file /tmp/M2-20526-0/250
│ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20526-0/252
│ │ │  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:
│ │ │ @@ -586,24 +586,24 @@
│ │ │  --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/M2-16342-0/252
│ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254
│ │ │ +using temporary file /tmp/M2-20526-0/252
│ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20526-0/254
│ │ │  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/M2-16342-0/254
│ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256
│ │ │ +using temporary file /tmp/M2-20526-0/254
│ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20526-0/256
│ │ │  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.
│ │ │ @@ -623,32 +623,32 @@
│ │ │  --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/M2-16342-0/256
    │ │ │ +using temporary file /tmp/M2-20526-0/256 │ │ │
    │ │ │ 
    i6 : QQ[x,y];
    │ │ │
    │ │ │ 
    i7 : gfan {x,y};
│ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258
│ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20526-0/258
│ │ │  Q[x1,x2]
│ │ │  {{
│ │ │  x2,
│ │ │  x1}
│ │ │  }
│ │ │ -using temporary file /tmp/M2-16342-0/258
    │ │ │ +using temporary file /tmp/M2-20526-0/258 │ │ │
    │ │ │
    │ │ │

    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/M2-16342-0/172 │ │ │ │ + -- running: /usr/bin/gfan gfan --help < /tmp/M2-20526-0/172 │ │ │ │ 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/M2-16342-0/172 │ │ │ │ - -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-16342-0/174 │ │ │ │ +using temporary file /tmp/M2-20526-0/172 │ │ │ │ + -- running: /usr/bin/gfan _buchberger --help < /tmp/M2-20526-0/174 │ │ │ │ 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/M2-16342-0/174 │ │ │ │ - -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-16342-0/176 │ │ │ │ +using temporary file /tmp/M2-20526-0/174 │ │ │ │ + -- running: /usr/bin/gfan _doesidealcontain --help < /tmp/M2-20526-0/176 │ │ │ │ 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/M2-16342-0/176 │ │ │ │ - -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-16342-0/178 │ │ │ │ +using temporary file /tmp/M2-20526-0/176 │ │ │ │ + -- running: /usr/bin/gfan _fancommonrefinement --help < /tmp/M2-20526-0/178 │ │ │ │ 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/M2-16342-0/178 │ │ │ │ - -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-16342-0/180 │ │ │ │ +using temporary file /tmp/M2-20526-0/178 │ │ │ │ + -- running: /usr/bin/gfan _fanlink --help < /tmp/M2-20526-0/180 │ │ │ │ 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/M2-16342-0/180 │ │ │ │ - -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-16342-0/182 │ │ │ │ +using temporary file /tmp/M2-20526-0/180 │ │ │ │ + -- running: /usr/bin/gfan _fanproduct --help < /tmp/M2-20526-0/182 │ │ │ │ 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/M2-16342-0/182 │ │ │ │ - -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-16342-0/184 │ │ │ │ +using temporary file /tmp/M2-20526-0/182 │ │ │ │ + -- running: /usr/bin/gfan _groebnercone --help < /tmp/M2-20526-0/184 │ │ │ │ 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/M2-16342-0/184 │ │ │ │ - -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-16342-0/186 │ │ │ │ +using temporary file /tmp/M2-20526-0/184 │ │ │ │ + -- running: /usr/bin/gfan _homogeneityspace --help < /tmp/M2-20526-0/186 │ │ │ │ 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/M2-16342-0/186 │ │ │ │ - -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-16342-0/188 │ │ │ │ +using temporary file /tmp/M2-20526-0/186 │ │ │ │ + -- running: /usr/bin/gfan _homogenize --help < /tmp/M2-20526-0/188 │ │ │ │ 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/M2-16342-0/188 │ │ │ │ - -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-16342-0/190 │ │ │ │ +using temporary file /tmp/M2-20526-0/188 │ │ │ │ + -- running: /usr/bin/gfan _initialforms --help < /tmp/M2-20526-0/190 │ │ │ │ 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/M2-16342-0/190 │ │ │ │ - -- running: /usr/bin/gfan _interactive --help < /tmp/M2-16342-0/192 │ │ │ │ +using temporary file /tmp/M2-20526-0/190 │ │ │ │ + -- running: /usr/bin/gfan _interactive --help < /tmp/M2-20526-0/192 │ │ │ │ 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/M2-16342-0/192 │ │ │ │ - -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-16342-0/194 │ │ │ │ +using temporary file /tmp/M2-20526-0/192 │ │ │ │ + -- running: /usr/bin/gfan _ismarkedgroebnerbasis --help < /tmp/M2-20526-0/194 │ │ │ │ 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/M2-16342-0/194 │ │ │ │ - -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-16342-0/196 │ │ │ │ +using temporary file /tmp/M2-20526-0/194 │ │ │ │ + -- running: /usr/bin/gfan _krulldimension --help < /tmp/M2-20526-0/196 │ │ │ │ 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/M2-16342-0/196 │ │ │ │ - -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-16342-0/198 │ │ │ │ +using temporary file /tmp/M2-20526-0/196 │ │ │ │ + -- running: /usr/bin/gfan _latticeideal --help < /tmp/M2-20526-0/198 │ │ │ │ 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/M2-16342-0/198 │ │ │ │ - -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-16342-0/200 │ │ │ │ +using temporary file /tmp/M2-20526-0/198 │ │ │ │ + -- running: /usr/bin/gfan _leadingterms --help < /tmp/M2-20526-0/200 │ │ │ │ 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/M2-16342-0/200 │ │ │ │ - -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-16342-0/202 │ │ │ │ +using temporary file /tmp/M2-20526-0/200 │ │ │ │ + -- running: /usr/bin/gfan _markpolynomialset --help < /tmp/M2-20526-0/202 │ │ │ │ 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/M2-16342-0/202 │ │ │ │ - -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-16342-0/204 │ │ │ │ +using temporary file /tmp/M2-20526-0/202 │ │ │ │ + -- running: /usr/bin/gfan _minkowskisum --help < /tmp/M2-20526-0/204 │ │ │ │ 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/M2-16342-0/204 │ │ │ │ - -- running: /usr/bin/gfan _minors --help < /tmp/M2-16342-0/206 │ │ │ │ +using temporary file /tmp/M2-20526-0/204 │ │ │ │ + -- running: /usr/bin/gfan _minors --help < /tmp/M2-20526-0/206 │ │ │ │ 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/M2-16342-0/206 │ │ │ │ - -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-16342-0/208 │ │ │ │ +using temporary file /tmp/M2-20526-0/206 │ │ │ │ + -- running: /usr/bin/gfan _mixedvolume --help < /tmp/M2-20526-0/208 │ │ │ │ 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. │ │ │ │ @@ -390,28 +390,28 @@ │ │ │ │ Use Gaukwa-n example instead of reading input. │ │ │ │ --eco value: │ │ │ │ Use Eco-n example instead of reading input. │ │ │ │ -j value: │ │ │ │ Number of threads │ │ │ │ --resume: │ │ │ │ Continue enumeration from saved state... │ │ │ │ -using temporary file /tmp/M2-16342-0/208 │ │ │ │ - -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-16342-0/210 │ │ │ │ +using temporary file /tmp/M2-20526-0/208 │ │ │ │ + -- running: /usr/bin/gfan _polynomialsetunion --help < /tmp/M2-20526-0/210 │ │ │ │ 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. │ │ │ │ │ │ │ │ --twosetswithrings: │ │ │ │ Switch to mode where input is: ring polynomialset (same)ring polynomialset. │ │ │ │ │ │ │ │ -using temporary file /tmp/M2-16342-0/210 │ │ │ │ - -- running: /usr/bin/gfan _render --help < /tmp/M2-16342-0/212 │ │ │ │ +using temporary file /tmp/M2-20526-0/210 │ │ │ │ + -- running: /usr/bin/gfan _render --help < /tmp/M2-20526-0/212 │ │ │ │ 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. │ │ │ │ @@ -419,16 +419,16 @@ │ │ │ │ -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/M2-16342-0/212 │ │ │ │ - -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-16342-0/214 │ │ │ │ +using temporary file /tmp/M2-20526-0/212 │ │ │ │ + -- running: /usr/bin/gfan _renderstaircase --help < /tmp/M2-20526-0/214 │ │ │ │ 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: │ │ │ │ @@ -441,16 +441,16 @@ │ │ │ │ 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/M2-16342-0/214 │ │ │ │ - -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-16342-0/216 │ │ │ │ +using temporary file /tmp/M2-20526-0/214 │ │ │ │ + -- running: /usr/bin/gfan _resultantfan --help < /tmp/M2-20526-0/216 │ │ │ │ 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. │ │ │ │ @@ -478,28 +478,28 @@ │ │ │ │ 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/M2-16342-0/216 │ │ │ │ - -- running: /usr/bin/gfan _saturation --help < /tmp/M2-16342-0/218 │ │ │ │ +using temporary file /tmp/M2-20526-0/216 │ │ │ │ + -- running: /usr/bin/gfan _saturation --help < /tmp/M2-20526-0/218 │ │ │ │ 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/M2-16342-0/218 │ │ │ │ - -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-16342-0/220 │ │ │ │ +using temporary file /tmp/M2-20526-0/218 │ │ │ │ + -- running: /usr/bin/gfan _secondaryfan --help < /tmp/M2-20526-0/220 │ │ │ │ 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 │ │ │ │ @@ -528,49 +528,49 @@ │ │ │ │ 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/M2-16342-0/220 │ │ │ │ - -- running: /usr/bin/gfan _stats --help < /tmp/M2-16342-0/222 │ │ │ │ +using temporary file /tmp/M2-20526-0/220 │ │ │ │ + -- running: /usr/bin/gfan _stats --help < /tmp/M2-20526-0/222 │ │ │ │ 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/M2-16342-0/222 │ │ │ │ - -- running: /usr/bin/gfan _substitute --help < /tmp/M2-16342-0/224 │ │ │ │ +using temporary file /tmp/M2-20526-0/222 │ │ │ │ + -- running: /usr/bin/gfan _substitute --help < /tmp/M2-20526-0/224 │ │ │ │ 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/M2-16342-0/224 │ │ │ │ - -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-16342-0/226 │ │ │ │ +using temporary file /tmp/M2-20526-0/224 │ │ │ │ + -- running: /usr/bin/gfan _tolatex --help < /tmp/M2-20526-0/226 │ │ │ │ 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/M2-16342-0/226 │ │ │ │ - -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-16342-0/228 │ │ │ │ +using temporary file /tmp/M2-20526-0/226 │ │ │ │ + -- running: /usr/bin/gfan _topolyhedralfan --help < /tmp/M2-20526-0/228 │ │ │ │ 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: │ │ │ │ @@ -581,54 +581,54 @@ │ │ │ │ generated cones given in modern format (like _cone). In this case --restrict is │ │ │ │ ignored. │ │ │ │ --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/M2-16342-0/228 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-16342-0/230 │ │ │ │ +using temporary file /tmp/M2-20526-0/228 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbasis --help < /tmp/M2-20526-0/230 │ │ │ │ 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/M2-16342-0/230 │ │ │ │ - -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-16342-0/232 │ │ │ │ +using temporary file /tmp/M2-20526-0/230 │ │ │ │ + -- running: /usr/bin/gfan _tropicalbruteforce --help < /tmp/M2-20526-0/232 │ │ │ │ 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/M2-16342-0/232 │ │ │ │ - -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-16342-0/234 │ │ │ │ +using temporary file /tmp/M2-20526-0/232 │ │ │ │ + -- running: /usr/bin/gfan _tropicalevaluation --help < /tmp/M2-20526-0/234 │ │ │ │ This program evaluates a tropical polynomial function in a given set of points. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/234 │ │ │ │ - -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-16342-0/236 │ │ │ │ +using temporary file /tmp/M2-20526-0/234 │ │ │ │ + -- running: /usr/bin/gfan _tropicalfunction --help < /tmp/M2-20526-0/236 │ │ │ │ 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/M2-16342-0/236 │ │ │ │ - -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-16342-0/238 │ │ │ │ +using temporary file /tmp/M2-20526-0/236 │ │ │ │ + -- running: /usr/bin/gfan _tropicalhypersurface --help < /tmp/M2-20526-0/238 │ │ │ │ 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/M2-16342-0/238 │ │ │ │ - -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-16342-0/240 │ │ │ │ +using temporary file /tmp/M2-20526-0/238 │ │ │ │ + -- running: /usr/bin/gfan _tropicalintersection --help < /tmp/M2-20526-0/240 │ │ │ │ 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: │ │ │ │ @@ -665,16 +665,16 @@ │ │ │ │ --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/M2-16342-0/240 │ │ │ │ - -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-16342-0/242 │ │ │ │ +using temporary file /tmp/M2-20526-0/240 │ │ │ │ + -- running: /usr/bin/gfan _tropicallifting --help < /tmp/M2-20526-0/242 │ │ │ │ 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 │ │ │ │ │ │ │ │ @@ -702,54 +702,54 @@ │ │ │ │ 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/M2-16342-0/242 │ │ │ │ - -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-16342-0/244 │ │ │ │ +using temporary file /tmp/M2-20526-0/242 │ │ │ │ + -- running: /usr/bin/gfan _tropicallinearspace --help < /tmp/M2-20526-0/244 │ │ │ │ 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/M2-16342-0/244 │ │ │ │ - -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-16342-0/246 │ │ │ │ +using temporary file /tmp/M2-20526-0/244 │ │ │ │ + -- running: /usr/bin/gfan _tropicalmultiplicity --help < /tmp/M2-20526-0/246 │ │ │ │ This program computes the multiplicity of a tropical cone given a marked │ │ │ │ reduced Groebner basis for its initial ideal. │ │ │ │ Options: │ │ │ │ -using temporary file /tmp/M2-16342-0/246 │ │ │ │ - -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-16342-0/248 │ │ │ │ +using temporary file /tmp/M2-20526-0/246 │ │ │ │ + -- running: /usr/bin/gfan _tropicalrank --help < /tmp/M2-20526-0/248 │ │ │ │ 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/M2-16342-0/248 │ │ │ │ - -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-16342-0/250 │ │ │ │ +using temporary file /tmp/M2-20526-0/248 │ │ │ │ + -- running: /usr/bin/gfan _tropicalstartingcone --help < /tmp/M2-20526-0/250 │ │ │ │ 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/M2-16342-0/250 │ │ │ │ - -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-16342-0/252 │ │ │ │ +using temporary file /tmp/M2-20526-0/250 │ │ │ │ + -- running: /usr/bin/gfan _tropicaltraverse --help < /tmp/M2-20526-0/252 │ │ │ │ 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 │ │ │ │ @@ -779,27 +779,27 @@ │ │ │ │ --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/M2-16342-0/252 │ │ │ │ - -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-16342-0/254 │ │ │ │ +using temporary file /tmp/M2-20526-0/252 │ │ │ │ + -- running: /usr/bin/gfan _tropicalweildivisor --help < /tmp/M2-20526-0/254 │ │ │ │ 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/M2-16342-0/254 │ │ │ │ - -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-16342-0/256 │ │ │ │ +using temporary file /tmp/M2-20526-0/254 │ │ │ │ + -- running: /usr/bin/gfan _overintegers --help < /tmp/M2-20526-0/256 │ │ │ │ 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, │ │ │ │ @@ -834,23 +834,23 @@ │ │ │ │ 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/M2-16342-0/256 │ │ │ │ +using temporary file /tmp/M2-20526-0/256 │ │ │ │ i6 : QQ[x,y]; │ │ │ │ i7 : gfan {x,y}; │ │ │ │ - -- running: /usr/bin/gfan _bases < /tmp/M2-16342-0/258 │ │ │ │ + -- running: /usr/bin/gfan _bases < /tmp/M2-20526-0/258 │ │ │ │ Q[x1,x2] │ │ │ │ {{ │ │ │ │ x2, │ │ │ │ x1} │ │ │ │ } │ │ │ │ -using temporary file /tmp/M2-16342-0/258 │ │ │ │ +using temporary file /tmp/M2-20526-0/258 │ │ │ │ Finally, if you want to be able to render Groebner fans and monomial staircases │ │ │ │ to .png files, you should install fig2dev. If it is installed in a non-standard │ │ │ │ location, then you may specify its path using _p_r_o_g_r_a_m_P_a_t_h_s. │ │ │ │ =============================================================================== │ │ │ │ The source of this document is in /build/reproducible-path/macaulay2- │ │ │ │ 1.26.06+ds/M2/Macaulay2/packages/gfanInterface.m2:2631:0. │ │ ├── ./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: 2d2d 2d2d 2b0a 7c69 3320 3a20 656c 6170 ----+.|i3 : elap │ │ │ │ 00017f80: 7365 6454 696d 6520 6275 726b 6552 6573 sedTime burkeRes │ │ │ │ 00017f90: 6f6c 7574 696f 6e28 4d2c 2037 2c20 4368 olution(M, 7, Ch │ │ │ │ 00017fa0: 6563 6b20 3d3e 2066 616c 7365 2920 2020 eck => false) │ │ │ │ -00017fb0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2032 |.| -- 2 │ │ │ │ -00017fc0: 2e34 3430 3938 7320 656c 6170 7365 6420 .44098s elapsed │ │ │ │ +00017fb0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2031 |.| -- 1 │ │ │ │ +00017fc0: 2e33 3337 3336 7320 656c 6170 7365 6420 .33736s elapsed │ │ │ │ 00017fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ff0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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 2020 2020 │ │ │ │ @@ -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 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00018220: 3420 3a20 656c 6170 7365 6454 696d 6520 4 : elapsedTime │ │ │ │ 00018230: 6275 726b 6552 6573 6f6c 7574 696f 6e28 burkeResolution( │ │ │ │ 00018240: 4d2c 2037 2c20 4368 6563 6b20 3d3e 2074 M, 7, Check => t │ │ │ │ 00018250: 7275 6529 2020 2020 2020 2020 2020 2020 rue) │ │ │ │ -00018260: 7c0a 7c20 2d2d 2032 2e33 3931 3435 7320 |.| -- 2.39145s │ │ │ │ -00018270: 656c 6170 7365 6420 2020 2020 2020 2020 elapsed │ │ │ │ +00018260: 7c0a 7c20 2d2d 2031 2e36 3034 3573 2065 |.| -- 1.6045s e │ │ │ │ +00018270: 6c61 7073 6564 2020 2020 2020 2020 2020 lapsed │ │ │ │ 00018280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182a0: 2020 2020 7c0a 7c20 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 2020 2020 7c0a 7c20 2020 2020 |.| │ │ ├── ./usr/share/info/AdjunctionForSurfaces.info.gz │ │ │ ├── AdjunctionForSurfaces.info │ │ │ │ @@ -741,16 +741,16 @@ │ │ │ │ 00002e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00002e70: 7c69 3130 203a 2065 6c61 7073 6564 5469 |i10 : elapsedTi │ │ │ │ 00002e80: 6d65 2066 493d 7265 7320 4920 2020 2020 me fI=res I │ │ │ │ 00002e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3235 3738 |.| -- .02578 │ │ │ │ -00002ec0: 3338 7320 656c 6170 7365 6420 2020 2020 38s elapsed │ │ │ │ +00002eb0: 2020 207c 0a7c 202d 2d20 2e30 3330 3337 |.| -- .03037 │ │ │ │ +00002ec0: 3734 7320 656c 6170 7365 6420 2020 2020 74s 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 3039 3138 3573 2065 6c61 7073 6564 .609185s elapsed │ │ │ │ +00006420: 2e35 3332 3632 3273 2065 6c61 7073 6564 .532622s 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 │ │ │ │ @@ -1651,16 +1651,16 @@ │ │ │ │ 00006720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00006750: 2d2d 2d2b 0a7c 6931 3720 3a20 656c 6170 ---+.|i17 : elap │ │ │ │ 00006760: 7365 6454 696d 6520 6261 7365 5074 733d sedTime basePts= │ │ │ │ 00006770: 7072 696d 6172 7944 6563 6f6d 706f 7369 primaryDecomposi │ │ │ │ 00006780: 7469 6f6e 2069 6465 616c 2048 3b20 7c0a tion ideal H; |. │ │ │ │ -00006790: 7c20 2d2d 2036 2e39 3831 3434 7320 656c | -- 6.98144s el │ │ │ │ -000067a0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00006790: 7c20 2d2d 2035 2e37 3230 3373 2065 6c61 | -- 5.7203s ela │ │ │ │ +000067a0: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 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 │ │ │ │ 00006810: 6c79 2061 7070 6c79 2862 6173 6550 7473 ly apply(basePts │ │ │ │ @@ -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: 3230 3035 3236 7320 656c 6170 7365 6420 200526s elapsed │ │ │ │ +0000a360: 3131 3830 3139 7320 656c 6170 7365 6420 118019s 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 3635 3536 3673 |.| -- .0565566s │ │ │ │ +0000a5e0: 7c0a 7c20 2d2d 202e 3037 3337 3235 3573 |.| -- .0737255s │ │ │ │ 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 2e39 3630 3431 7320 656c 6170 7365 1.96041s elapse │ │ │ │ +0000a920: 2031 2e37 3235 3337 7320 656c 6170 7365 1.72537s 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 3435 3030 3137 7320 2863 7075 293b 0.450017s (cpu); │ │ │ │ -00010f90: 2030 2e33 3736 3836 3373 2028 7468 7265 0.376863s (thre │ │ │ │ +00010f80: 302e 3532 3330 3332 7320 2863 7075 293b 0.523032s (cpu); │ │ │ │ +00010f90: 2030 2e33 3739 3330 3673 2028 7468 7265 0.379306s (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 │ │ │ │ @@ -4398,954 +4398,952 @@ │ │ │ │ 000112d0: 206f 6620 7661 7269 6162 6c65 7320 696e of variables in │ │ │ │ 000112e0: 0a41 2074 6869 7320 7275 6e73 206d 7563 .A this runs muc │ │ │ │ 000112f0: 6820 6661 7374 6572 2074 6861 6e20 7461 h faster than ta │ │ │ │ 00011300: 6b69 6e67 2061 2072 616e 646f 6d20 6d61 king a random ma │ │ │ │ 00011310: 7472 6978 204d 2e0a 0a2b 2d2d 2d2d 2d2d trix M...+------ │ │ │ │ 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 3638 3335 3173 2028 6370 ed 0.468351s (cp │ │ │ │ -000113a0: 7529 3b20 302e 3430 3337 3332 7320 2874 u); 0.403732s (t │ │ │ │ -000113b0: 6872 6561 6429 3b20 3073 2028 6763 297c hread); 0s (gc)| │ │ │ │ -000113c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00011340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00011350: 7c69 3135 203a 2074 696d 6520 6265 7474 |i15 : time bett │ │ │ │ +00011360: 6920 2846 203d 2070 7572 6552 6573 6f6c i (F = pureResol │ │ │ │ +00011370: 7574 696f 6e28 3131 2c34 2c7b 302c 322c ution(11,4,{0,2, │ │ │ │ +00011380: 347d 2929 207c 0a7c 202d 2d20 7573 6564 4})) |.| -- used │ │ │ │ +00011390: 2030 2e34 3735 3534 3473 2028 6370 7529 0.475544s (cpu) │ │ │ │ +000113a0: 3b20 302e 3339 3837 3473 2028 7468 7265 ; 0.39874s (thre │ │ │ │ +000113b0: 6164 293b 2030 7320 2867 6329 7c0a 7c20 ad); 0s (gc)|.| │ │ │ │ +000113c0: 2020 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 │ │ │ │ +000113f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00011400: 2020 2030 2031 2032 2020 2020 2020 2020 0 1 2 │ │ │ │ 00011410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011430: 0a7c 6f31 3520 3d20 746f 7461 6c3a 2033 .|o15 = total: 3 │ │ │ │ -00011440: 2036 2033 2020 2020 2020 2020 2020 2020 6 3 │ │ │ │ +00011420: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ +00011430: 203d 2074 6f74 616c 3a20 3320 3620 3320 = total: 3 6 3 │ │ │ │ +00011440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011470: 2020 2020 303a 2033 202e 202e 2020 2020 0: 3 . . │ │ │ │ +00011460: 207c 0a7c 2020 2020 2020 2020 2020 303a |.| 0: │ │ │ │ +00011470: 2033 202e 202e 2020 2020 2020 2020 2020 3 . . │ │ │ │ 00011480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000114a0: 0a7c 2020 2020 2020 2020 2020 313a 202e .| 1: . │ │ │ │ -000114b0: 2036 202e 2020 2020 2020 2020 2020 2020 6 . │ │ │ │ -000114c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000114d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000114e0: 2020 2020 323a 202e 202e 2033 2020 2020 2: . . 3 │ │ │ │ +00011490: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000114a0: 2020 2020 2031 3a20 2e20 3620 2e20 2020 1: . 6 . │ │ │ │ +000114b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000114c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000114d0: 0a7c 2020 2020 2020 2020 2020 323a 202e .| 2: . │ │ │ │ +000114e0: 202e 2033 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ 000114f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011500: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011510: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00011500: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00011510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011540: 2020 2020 2020 207c 0a7c 6f31 3520 3a20 |.|o15 : │ │ │ │ -00011550: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +00011530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00011540: 6f31 3520 3a20 4265 7474 6954 616c 6c79 o15 : BettiTally │ │ │ │ +00011550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011570: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011580: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00011570: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00011580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000115b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ -000115c0: 7269 6e67 2046 2020 2020 2020 2020 2020 ring F │ │ │ │ +000115a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000115b0: 3620 3a20 7269 6e67 2046 2020 2020 2020 6 : ring F │ │ │ │ +000115c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000115d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000115e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000115f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000115e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000115f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011620: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00011630: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00011610: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00011620: 2020 5a5a 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00011630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011650: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011660: 0a7c 6f31 3620 3d20 2d2d 5b61 202e 2e61 .|o16 = --[a ..a │ │ │ │ -00011670: 2020 5d20 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00011680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011690: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000116a0: 3131 2020 3020 2020 3135 2020 2020 2020 11 0 15 │ │ │ │ -000116b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000116d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00011650: 7c0a 7c6f 3136 203d 202d 2d5b 6120 2e2e |.|o16 = --[a .. │ │ │ │ +00011660: 6120 205d 2020 2020 2020 2020 2020 2020 a ] │ │ │ │ +00011670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011680: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00011690: 3131 2020 3020 2020 3135 2020 2020 2020 11 0 15 │ │ │ │ +000116a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000116b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000116c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000116d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000116e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000116f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011700: 2020 2020 2020 207c 0a7c 6f31 3620 3a20 |.|o16 : │ │ │ │ -00011710: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ -00011720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00011730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00011740: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000116f0: 2020 2020 207c 0a7c 6f31 3620 3a20 506f |.|o16 : Po │ │ │ │ +00011700: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020 lynomialRing │ │ │ │ +00011710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00011720: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00011730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00011740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011770: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00011780: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00011790: 202a 6e6f 7465 2064 6972 6563 7449 6d61 *note directIma │ │ │ │ -000117a0: 6765 436f 6d70 6c65 783a 2064 6972 6563 geComplex: direc │ │ │ │ -000117b0: 7449 6d61 6765 436f 6d70 6c65 782c 202d tImageComplex, - │ │ │ │ -000117c0: 2d20 6469 7265 6374 2069 6d61 6765 2063 - direct image c │ │ │ │ -000117d0: 6f6d 706c 6578 0a20 202a 202a 6e6f 7465 omplex. * *note │ │ │ │ +00011760: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ +00011770: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00011780: 7465 2064 6972 6563 7449 6d61 6765 436f te directImageCo │ │ │ │ +00011790: 6d70 6c65 783a 2064 6972 6563 7449 6d61 mplex: directIma │ │ │ │ +000117a0: 6765 436f 6d70 6c65 782c 202d 2d20 6469 geComplex, -- di │ │ │ │ +000117b0: 7265 6374 2069 6d61 6765 2063 6f6d 706c rect image compl │ │ │ │ +000117c0: 6578 0a20 202a 202a 6e6f 7465 2075 6e69 ex. * *note uni │ │ │ │ +000117d0: 7665 7273 616c 4578 7465 6e73 696f 6e3a versalExtension: │ │ │ │ 000117e0: 2075 6e69 7665 7273 616c 4578 7465 6e73 universalExtens │ │ │ │ -000117f0: 696f 6e3a 2075 6e69 7665 7273 616c 4578 ion: universalEx │ │ │ │ -00011800: 7465 6e73 696f 6e2c 202d 2d20 556e 6976 tension, -- Univ │ │ │ │ -00011810: 6572 7361 6c20 6578 7465 6e73 696f 6e20 ersal extension │ │ │ │ -00011820: 6f66 0a20 2020 2076 6563 746f 7220 6275 of. vector bu │ │ │ │ -00011830: 6e64 6c65 7320 6f6e 2050 5e31 0a0a 5761 ndles on P^1..Wa │ │ │ │ -00011840: 7973 2074 6f20 7573 6520 7075 7265 5265 ys to use pureRe │ │ │ │ -00011850: 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d 3d3d solution:.====== │ │ │ │ -00011860: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011870: 3d3d 3d3d 3d0a 0a20 202a 2022 7075 7265 =====.. * "pure │ │ │ │ -00011880: 5265 736f 6c75 7469 6f6e 284d 6174 7269 Resolution(Matri │ │ │ │ -00011890: 782c 4c69 7374 2922 0a20 202a 2022 7075 x,List)". * "pu │ │ │ │ -000118a0: 7265 5265 736f 6c75 7469 6f6e 2852 696e reResolution(Rin │ │ │ │ -000118b0: 672c 4c69 7374 2922 0a20 202a 2022 7075 g,List)". * "pu │ │ │ │ -000118c0: 7265 5265 736f 6c75 7469 6f6e 285a 5a2c reResolution(ZZ, │ │ │ │ -000118d0: 4c69 7374 2922 0a20 202a 2022 7075 7265 List)". * "pure │ │ │ │ -000118e0: 5265 736f 6c75 7469 6f6e 285a 5a2c 5a5a Resolution(ZZ,ZZ │ │ │ │ -000118f0: 2c4c 6973 7429 220a 0a46 6f72 2074 6865 ,List)"..For the │ │ │ │ -00011900: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00011910: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00011920: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00011930: 2070 7572 6552 6573 6f6c 7574 696f 6e3a pureResolution: │ │ │ │ -00011940: 2070 7572 6552 6573 6f6c 7574 696f 6e2c pureResolution, │ │ │ │ -00011950: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ -00011960: 6f64 2066 756e 6374 696f 6e3a 0a28 4d61 od function:.(Ma │ │ │ │ -00011970: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00011980: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +000117f0: 696f 6e2c 202d 2d20 556e 6976 6572 7361 ion, -- Universa │ │ │ │ +00011800: 6c20 6578 7465 6e73 696f 6e20 6f66 0a20 l extension of. │ │ │ │ +00011810: 2020 2076 6563 746f 7220 6275 6e64 6c65 vector bundle │ │ │ │ +00011820: 7320 6f6e 2050 5e31 0a0a 5761 7973 2074 s on P^1..Ways t │ │ │ │ +00011830: 6f20 7573 6520 7075 7265 5265 736f 6c75 o use pureResolu │ │ │ │ +00011840: 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d tion:.========== │ │ │ │ +00011850: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00011860: 3d0a 0a20 202a 2022 7075 7265 5265 736f =.. * "pureReso │ │ │ │ +00011870: 6c75 7469 6f6e 284d 6174 7269 782c 4c69 lution(Matrix,Li │ │ │ │ +00011880: 7374 2922 0a20 202a 2022 7075 7265 5265 st)". * "pureRe │ │ │ │ +00011890: 736f 6c75 7469 6f6e 2852 696e 672c 4c69 solution(Ring,Li │ │ │ │ +000118a0: 7374 2922 0a20 202a 2022 7075 7265 5265 st)". * "pureRe │ │ │ │ +000118b0: 736f 6c75 7469 6f6e 285a 5a2c 4c69 7374 solution(ZZ,List │ │ │ │ +000118c0: 2922 0a20 202a 2022 7075 7265 5265 736f )". * "pureReso │ │ │ │ +000118d0: 6c75 7469 6f6e 285a 5a2c 5a5a 2c4c 6973 lution(ZZ,ZZ,Lis │ │ │ │ +000118e0: 7429 220a 0a46 6f72 2074 6865 2070 726f t)"..For the pro │ │ │ │ +000118f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00011900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00011910: 6f62 6a65 6374 202a 6e6f 7465 2070 7572 object *note pur │ │ │ │ +00011920: 6552 6573 6f6c 7574 696f 6e3a 2070 7572 eResolution: pur │ │ │ │ +00011930: 6552 6573 6f6c 7574 696f 6e2c 2069 7320 eResolution, is │ │ │ │ +00011940: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +00011950: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +00011960: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00011970: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00011980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000119b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000119d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -000119e0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -000119f0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00011a00: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00011a10: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00011a20: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00011a30: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00011a40: 6765 732f 4247 472e 6d32 3a31 3232 360a ges/BGG.m2:1226. │ │ │ │ -00011a50: 3a30 2e0a 1f0a 4669 6c65 3a20 4247 472e :0....File: BGG. │ │ │ │ -00011a60: 696e 666f 2c20 4e6f 6465 3a20 5265 6775 info, Node: Regu │ │ │ │ -00011a70: 6c61 7269 7479 2c20 4e65 7874 3a20 7379 larity, Next: sy │ │ │ │ -00011a80: 6d45 7874 2c20 5072 6576 3a20 7075 7265 mExt, Prev: pure │ │ │ │ -00011a90: 5265 736f 6c75 7469 6f6e 2c20 5570 3a20 Resolution, Up: │ │ │ │ -00011aa0: 546f 700a 0a52 6567 756c 6172 6974 7920 Top..Regularity │ │ │ │ -00011ab0: 2d2d 204f 7074 696f 6e20 666f 7220 6469 -- Option for di │ │ │ │ -00011ac0: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ -00011ad0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ -00011ae0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011af0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4361 ************..Ca │ │ │ │ -00011b00: 7665 6174 0a3d 3d3d 3d3d 3d0a 0a43 7572 veat.======..Cur │ │ │ │ -00011b10: 7265 6e74 6c79 206e 6f74 2073 7570 706f rently not suppo │ │ │ │ -00011b20: 7274 6564 0a0a 4675 6e63 7469 6f6e 7320 rted..Functions │ │ │ │ -00011b30: 7769 7468 206f 7074 696f 6e61 6c20 6172 with optional ar │ │ │ │ -00011b40: 6775 6d65 6e74 206e 616d 6564 2052 6567 gument named Reg │ │ │ │ -00011b50: 756c 6172 6974 793a 0a3d 3d3d 3d3d 3d3d ularity:.======= │ │ │ │ +000119c0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000119d0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000119e0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000119f0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00011a00: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00011a10: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00011a20: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00011a30: 4247 472e 6d32 3a31 3232 360a 3a30 2e0a BGG.m2:1226.:0.. │ │ │ │ +00011a40: 1f0a 4669 6c65 3a20 4247 472e 696e 666f ..File: BGG.info │ │ │ │ +00011a50: 2c20 4e6f 6465 3a20 5265 6775 6c61 7269 , Node: Regulari │ │ │ │ +00011a60: 7479 2c20 4e65 7874 3a20 7379 6d45 7874 ty, Next: symExt │ │ │ │ +00011a70: 2c20 5072 6576 3a20 7075 7265 5265 736f , Prev: pureReso │ │ │ │ +00011a80: 6c75 7469 6f6e 2c20 5570 3a20 546f 700a lution, Up: Top. │ │ │ │ +00011a90: 0a52 6567 756c 6172 6974 7920 2d2d 204f .Regularity -- O │ │ │ │ +00011aa0: 7074 696f 6e20 666f 7220 6469 7265 6374 ption for direct │ │ │ │ +00011ab0: 496d 6167 6543 6f6d 706c 6578 0a2a 2a2a ImageComplex.*** │ │ │ │ +00011ac0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011ad0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00011ae0: 2a2a 2a2a 2a2a 2a2a 0a0a 4361 7665 6174 ********..Caveat │ │ │ │ +00011af0: 0a3d 3d3d 3d3d 3d0a 0a43 7572 7265 6e74 .======..Current │ │ │ │ +00011b00: 6c79 206e 6f74 2073 7570 706f 7274 6564 ly not supported │ │ │ │ +00011b10: 0a0a 4675 6e63 7469 6f6e 7320 7769 7468 ..Functions with │ │ │ │ +00011b20: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ +00011b30: 6e74 206e 616d 6564 2052 6567 756c 6172 nt named Regular │ │ │ │ +00011b40: 6974 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ity:.=========== │ │ │ │ +00011b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00011b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011b80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00011b90: 2022 6469 7265 6374 496d 6167 6543 6f6d "directImageCom │ │ │ │ -00011ba0: 706c 6578 282e 2e2e 2c52 6567 756c 6172 plex(...,Regular │ │ │ │ -00011bb0: 6974 793d 3e2e 2e2e 2922 202d 2d20 7365 ity=>...)" -- se │ │ │ │ -00011bc0: 6520 2a6e 6f74 6520 6469 7265 6374 496d e *note directIm │ │ │ │ -00011bd0: 6167 6543 6f6d 706c 6578 3a0a 2020 2020 ageComplex:. │ │ │ │ -00011be0: 6469 7265 6374 496d 6167 6543 6f6d 706c directImageCompl │ │ │ │ -00011bf0: 6578 2c20 2d2d 2064 6972 6563 7420 696d ex, -- direct im │ │ │ │ -00011c00: 6167 6520 636f 6d70 6c65 780a 0a46 6f72 age complex..For │ │ │ │ -00011c10: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00011c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00011c30: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00011c40: 6e6f 7465 2052 6567 756c 6172 6974 793a note Regularity: │ │ │ │ -00011c50: 2028 5265 6775 6c61 7269 7479 2954 6f70 (Regularity)Top │ │ │ │ -00011c60: 2c20 6973 2061 202a 6e6f 7465 2073 796d , is a *note sym │ │ │ │ -00011c70: 626f 6c3a 0a28 4d61 6361 756c 6179 3244 bol:.(Macaulay2D │ │ │ │ -00011c80: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ +00011b70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6469 =======.. * "di │ │ │ │ +00011b80: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ +00011b90: 282e 2e2e 2c52 6567 756c 6172 6974 793d (...,Regularity= │ │ │ │ +00011ba0: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ +00011bb0: 6f74 6520 6469 7265 6374 496d 6167 6543 ote directImageC │ │ │ │ +00011bc0: 6f6d 706c 6578 3a0a 2020 2020 6469 7265 omplex:. dire │ │ │ │ +00011bd0: 6374 496d 6167 6543 6f6d 706c 6578 2c20 ctImageComplex, │ │ │ │ +00011be0: 2d2d 2064 6972 6563 7420 696d 6167 6520 -- direct image │ │ │ │ +00011bf0: 636f 6d70 6c65 780a 0a46 6f72 2074 6865 complex..For the │ │ │ │ +00011c00: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ +00011c10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00011c20: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ +00011c30: 2052 6567 756c 6172 6974 793a 2028 5265 Regularity: (Re │ │ │ │ +00011c40: 6775 6c61 7269 7479 2954 6f70 2c20 6973 gularity)Top, is │ │ │ │ +00011c50: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ +00011c60: 0a28 4d61 6361 756c 6179 3244 6f63 2953 .(Macaulay2Doc)S │ │ │ │ +00011c70: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ +00011c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00011cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00011cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00011ce0: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00011cf0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00011d00: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00011d10: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00011d20: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00011d30: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00011d40: 6765 732f 4247 472e 6d32 3a38 3831 3a0a ges/BGG.m2:881:. │ │ │ │ -00011d50: 302e 0a1f 0a46 696c 653a 2042 4747 2e69 0....File: BGG.i │ │ │ │ -00011d60: 6e66 6f2c 204e 6f64 653a 2073 796d 4578 nfo, Node: symEx │ │ │ │ -00011d70: 742c 204e 6578 743a 2074 6174 6552 6573 t, Next: tateRes │ │ │ │ -00011d80: 6f6c 7574 696f 6e2c 2050 7265 763a 2052 olution, Prev: R │ │ │ │ -00011d90: 6567 756c 6172 6974 792c 2055 703a 2054 egularity, Up: T │ │ │ │ -00011da0: 6f70 0a0a 7379 6d45 7874 202d 2d20 7468 op..symExt -- th │ │ │ │ -00011db0: 6520 6669 7273 7420 6469 6666 6572 656e e first differen │ │ │ │ -00011dc0: 7469 616c 206f 6620 7468 6520 636f 6d70 tial of the comp │ │ │ │ -00011dd0: 6c65 7820 5228 4d29 0a2a 2a2a 2a2a 2a2a lex R(M).******* │ │ │ │ +00011cc0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00011cd0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00011ce0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00011cf0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00011d00: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00011d10: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00011d20: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00011d30: 4247 472e 6d32 3a38 3831 3a0a 302e 0a1f BGG.m2:881:.0... │ │ │ │ +00011d40: 0a46 696c 653a 2042 4747 2e69 6e66 6f2c .File: BGG.info, │ │ │ │ +00011d50: 204e 6f64 653a 2073 796d 4578 742c 204e Node: symExt, N │ │ │ │ +00011d60: 6578 743a 2074 6174 6552 6573 6f6c 7574 ext: tateResolut │ │ │ │ +00011d70: 696f 6e2c 2050 7265 763a 2052 6567 756c ion, Prev: Regul │ │ │ │ +00011d80: 6172 6974 792c 2055 703a 2054 6f70 0a0a arity, Up: Top.. │ │ │ │ +00011d90: 7379 6d45 7874 202d 2d20 7468 6520 6669 symExt -- the fi │ │ │ │ +00011da0: 7273 7420 6469 6666 6572 656e 7469 616c rst differential │ │ │ │ +00011db0: 206f 6620 7468 6520 636f 6d70 6c65 7820 of the complex │ │ │ │ +00011dc0: 5228 4d29 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a R(M).*********** │ │ │ │ +00011dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00011de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00011e00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00011e10: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00011e20: 2020 2073 796d 4578 7428 6d2c 4529 0a20 symExt(m,E). │ │ │ │ -00011e30: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -00011e40: 202a 206d 2c20 6120 2a6e 6f74 6520 6d61 * m, a *note ma │ │ │ │ -00011e50: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ -00011e60: 446f 6329 4d61 7472 6978 2c2c 2061 2070 Doc)Matrix,, a p │ │ │ │ -00011e70: 7265 7365 6e74 6174 696f 6e20 6d61 7472 resentation matr │ │ │ │ -00011e80: 6978 2066 6f72 2061 0a20 2020 2020 2020 ix for a. │ │ │ │ -00011e90: 2070 6f73 6974 6976 656c 7920 6772 6164 positively grad │ │ │ │ -00011ea0: 6564 206d 6f64 756c 6520 4d20 6f76 6572 ed module M over │ │ │ │ -00011eb0: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269 a polynomial ri │ │ │ │ -00011ec0: 6e67 0a20 2020 2020 202a 2045 2c20 6120 ng. * E, a │ │ │ │ -00011ed0: 2a6e 6f74 6520 706f 6c79 6e6f 6d69 616c *note polynomial │ │ │ │ -00011ee0: 2072 696e 673a 2028 4d61 6361 756c 6179 ring: (Macaulay │ │ │ │ -00011ef0: 3244 6f63 2950 6f6c 796e 6f6d 6961 6c52 2Doc)PolynomialR │ │ │ │ -00011f00: 696e 672c 2c20 6578 7465 7269 6f72 0a20 ing,, exterior. │ │ │ │ -00011f10: 2020 2020 2020 2061 6c67 6562 7261 0a20 algebra. │ │ │ │ -00011f20: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -00011f30: 2020 2a20 6120 2a6e 6f74 6520 6d61 7472 * a *note matr │ │ │ │ -00011f40: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00011f50: 6329 4d61 7472 6978 2c2c 2061 206d 6174 c)Matrix,, a mat │ │ │ │ -00011f60: 7269 7820 7265 7072 6573 656e 7469 6e67 rix representing │ │ │ │ -00011f70: 2074 6865 206d 6170 0a20 2020 2020 2020 the map. │ │ │ │ -00011f80: 204d 5f31 202a 2a20 6f6d 6567 615f 4520 M_1 ** omega_E │ │ │ │ -00011f90: 3c2d 2d20 4d5f 3020 2a2a 206f 6d65 6761 <-- M_0 ** omega │ │ │ │ -00011fa0: 5f45 0a0a 4465 7363 7269 7074 696f 6e0a _E..Description. │ │ │ │ -00011fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ -00011fc0: 7320 6675 6e63 7469 6f6e 2074 616b 6573 s function takes │ │ │ │ -00011fd0: 2061 7320 696e 7075 7420 6120 6d61 7472 as input a matr │ │ │ │ -00011fe0: 6978 206d 2077 6974 6820 6c69 6e65 6172 ix m with linear │ │ │ │ -00011ff0: 2065 6e74 7269 6573 2c20 7768 6963 6820 entries, which │ │ │ │ -00012000: 7765 2074 6869 6e6b 206f 660a 6173 2061 we think of.as a │ │ │ │ -00012010: 2070 7265 7365 6e74 6174 696f 6e20 6d61 presentation ma │ │ │ │ -00012020: 7472 6978 2066 6f72 2061 2070 6f73 6974 trix for a posit │ │ │ │ -00012030: 6976 656c 7920 6772 6164 6564 2053 2d6d ively graded S-m │ │ │ │ -00012040: 6f64 756c 6520 4d20 6d61 7472 6978 2072 odule M matrix r │ │ │ │ -00012050: 6570 7265 7365 6e74 696e 670a 7468 6520 epresenting.the │ │ │ │ -00012060: 6d61 7020 4d5f 3120 2a2a 206f 6d65 6761 map M_1 ** omega │ │ │ │ -00012070: 5f45 203c 2d2d 204d 5f30 202a 2a20 6f6d _E <-- M_0 ** om │ │ │ │ -00012080: 6567 615f 4520 7768 6963 6820 6973 2074 ega_E which is t │ │ │ │ -00012090: 6865 2066 6972 7374 2064 6966 6665 7265 he first differe │ │ │ │ -000120a0: 6e74 6961 6c20 6f66 0a74 6865 2063 6f6d ntial of.the com │ │ │ │ -000120b0: 706c 6578 2052 284d 292e 0a2b 2d2d 2d2d plex R(M)..+---- │ │ │ │ +00011df0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00011e00: 7361 6765 3a20 0a20 2020 2020 2020 2073 sage: . s │ │ │ │ +00011e10: 796d 4578 7428 6d2c 4529 0a20 202a 2049 ymExt(m,E). * I │ │ │ │ +00011e20: 6e70 7574 733a 0a20 2020 2020 202a 206d nputs:. * m │ │ │ │ +00011e30: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +00011e40: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00011e50: 4d61 7472 6978 2c2c 2061 2070 7265 7365 Matrix,, a prese │ │ │ │ +00011e60: 6e74 6174 696f 6e20 6d61 7472 6978 2066 ntation matrix f │ │ │ │ +00011e70: 6f72 2061 0a20 2020 2020 2020 2070 6f73 or a. pos │ │ │ │ +00011e80: 6974 6976 656c 7920 6772 6164 6564 206d itively graded m │ │ │ │ +00011e90: 6f64 756c 6520 4d20 6f76 6572 2061 2070 odule M over a p │ │ │ │ +00011ea0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20 olynomial ring. │ │ │ │ +00011eb0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ +00011ec0: 6520 706f 6c79 6e6f 6d69 616c 2072 696e e polynomial rin │ │ │ │ +00011ed0: 673a 2028 4d61 6361 756c 6179 3244 6f63 g: (Macaulay2Doc │ │ │ │ +00011ee0: 2950 6f6c 796e 6f6d 6961 6c52 696e 672c )PolynomialRing, │ │ │ │ +00011ef0: 2c20 6578 7465 7269 6f72 0a20 2020 2020 , exterior. │ │ │ │ +00011f00: 2020 2061 6c67 6562 7261 0a20 202a 204f algebra. * O │ │ │ │ +00011f10: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00011f20: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +00011f30: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +00011f40: 7472 6978 2c2c 2061 206d 6174 7269 7820 trix,, a matrix │ │ │ │ +00011f50: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ +00011f60: 206d 6170 0a20 2020 2020 2020 204d 5f31 map. M_1 │ │ │ │ +00011f70: 202a 2a20 6f6d 6567 615f 4520 3c2d 2d20 ** omega_E <-- │ │ │ │ +00011f80: 4d5f 3020 2a2a 206f 6d65 6761 5f45 0a0a M_0 ** omega_E.. │ │ │ │ +00011f90: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00011fa0: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6675 =======..This fu │ │ │ │ +00011fb0: 6e63 7469 6f6e 2074 616b 6573 2061 7320 nction takes as │ │ │ │ +00011fc0: 696e 7075 7420 6120 6d61 7472 6978 206d input a matrix m │ │ │ │ +00011fd0: 2077 6974 6820 6c69 6e65 6172 2065 6e74 with linear ent │ │ │ │ +00011fe0: 7269 6573 2c20 7768 6963 6820 7765 2074 ries, which we t │ │ │ │ +00011ff0: 6869 6e6b 206f 660a 6173 2061 2070 7265 hink of.as a pre │ │ │ │ +00012000: 7365 6e74 6174 696f 6e20 6d61 7472 6978 sentation matrix │ │ │ │ +00012010: 2066 6f72 2061 2070 6f73 6974 6976 656c for a positivel │ │ │ │ +00012020: 7920 6772 6164 6564 2053 2d6d 6f64 756c y graded S-modul │ │ │ │ +00012030: 6520 4d20 6d61 7472 6978 2072 6570 7265 e M matrix repre │ │ │ │ +00012040: 7365 6e74 696e 670a 7468 6520 6d61 7020 senting.the map │ │ │ │ +00012050: 4d5f 3120 2a2a 206f 6d65 6761 5f45 203c M_1 ** omega_E < │ │ │ │ +00012060: 2d2d 204d 5f30 202a 2a20 6f6d 6567 615f -- M_0 ** omega_ │ │ │ │ +00012070: 4520 7768 6963 6820 6973 2074 6865 2066 E which is the f │ │ │ │ +00012080: 6972 7374 2064 6966 6665 7265 6e74 6961 irst differentia │ │ │ │ +00012090: 6c20 6f66 0a74 6865 2063 6f6d 706c 6578 l of.the complex │ │ │ │ +000120a0: 2052 284d 292e 0a2b 2d2d 2d2d 2d2d 2d2d R(M)..+-------- │ │ │ │ +000120b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000120c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000120d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000120e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000120f0: 0a7c 6931 203a 2053 203d 205a 5a2f 3332 .|i1 : S = ZZ/32 │ │ │ │ -00012100: 3030 335b 785f 302e 2e78 5f32 5d3b 2020 003[x_0..x_2]; │ │ │ │ -00012110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012120: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000120d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +000120e0: 203a 2053 203d 205a 5a2f 3332 3030 335b : S = ZZ/32003[ │ │ │ │ +000120f0: 785f 302e 2e78 5f32 5d3b 2020 2020 2020 x_0..x_2]; │ │ │ │ +00012100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012110: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00012120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00012160: 203a 2045 203d 205a 5a2f 3332 3030 335b : E = ZZ/32003[ │ │ │ │ -00012170: 655f 302e 2e65 5f32 2c20 536b 6577 436f e_0..e_2, SkewCo │ │ │ │ -00012180: 6d6d 7574 6174 6976 653d 3e74 7275 655d mmutative=>true] │ │ │ │ -00012190: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ +00012140: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2045 -------+.|i2 : E │ │ │ │ +00012150: 203d 205a 5a2f 3332 3030 335b 655f 302e = ZZ/32003[e_0. │ │ │ │ +00012160: 2e65 5f32 2c20 536b 6577 436f 6d6d 7574 .e_2, SkewCommut │ │ │ │ +00012170: 6174 6976 653d 3e74 7275 655d 3b7c 0a2b ative=>true];|.+ │ │ │ │ +00012180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00012190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000121a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000121c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d -------+.|i3 : M │ │ │ │ -000121d0: 203d 2063 6f6b 6572 206d 6174 7269 7820 = coker matrix │ │ │ │ -000121e0: 7b7b 785f 305e 322c 2078 5f31 5e32 7d7d {{x_0^2, x_1^2}} │ │ │ │ -000121f0: 3b20 2020 2020 2020 2020 2020 207c 0a2b ; |.+ │ │ │ │ +000121b0: 2d2d 2d2b 0a7c 6933 203a 204d 203d 2063 ---+.|i3 : M = c │ │ │ │ +000121c0: 6f6b 6572 206d 6174 7269 7820 7b7b 785f oker matrix {{x_ │ │ │ │ +000121d0: 305e 322c 2078 5f31 5e32 7d7d 3b20 2020 0^2, x_1^2}}; │ │ │ │ +000121e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000121f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012230: 2d2d 2d2b 0a7c 6934 203a 206d 203d 2070 ---+.|i4 : m = p │ │ │ │ -00012240: 7265 7365 6e74 6174 696f 6e20 7472 756e resentation trun │ │ │ │ -00012250: 6361 7465 2872 6567 756c 6172 6974 7920 cate(regularity │ │ │ │ -00012260: 4d2c 4d29 3b20 2020 207c 0a7c 2020 2020 M,M); |.| │ │ │ │ +00012210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00012220: 0a7c 6934 203a 206d 203d 2070 7265 7365 .|i4 : m = prese │ │ │ │ +00012230: 6e74 6174 696f 6e20 7472 756e 6361 7465 ntation truncate │ │ │ │ +00012240: 2872 6567 756c 6172 6974 7920 4d2c 4d29 (regularity M,M) │ │ │ │ +00012250: 3b20 2020 207c 0a7c 2020 2020 2020 2020 ; |.| │ │ │ │ +00012260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000122a0: 0a7c 2020 2020 2020 2020 2020 2020 2034 .| 4 │ │ │ │ -000122b0: 2020 2020 2020 3820 2020 2020 2020 2020 8 │ │ │ │ -000122c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000122d0: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ -000122e0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ -000122f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012300: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012280: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012290: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ +000122a0: 2020 3820 2020 2020 2020 2020 2020 2020 8 │ │ │ │ +000122b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000122c0: 207c 0a7c 6f34 203a 204d 6174 7269 7820 |.|o4 : Matrix │ │ │ │ +000122d0: 5320 203c 2d2d 2053 2020 2020 2020 2020 S <-- S │ │ │ │ +000122e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000122f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00012300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012340: 2d2b 0a7c 6935 203a 2073 796d 4578 7428 -+.|i5 : symExt( │ │ │ │ -00012350: 6d2c 4529 2020 2020 2020 2020 2020 2020 m,E) │ │ │ │ -00012360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012370: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00012320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00012330: 6935 203a 2073 796d 4578 7428 6d2c 4529 i5 : symExt(m,E) │ │ │ │ +00012340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012360: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000123b0: 6f35 203d 207b 2d31 7d20 7c20 655f 3220 o5 = {-1} | e_2 │ │ │ │ -000123c0: 3020 2020 3020 2020 3020 2020 7c20 2020 0 0 0 | │ │ │ │ -000123d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000123e0: 2020 207c 0a7c 2020 2020 207b 2d31 7d20 |.| {-1} │ │ │ │ -000123f0: 7c20 655f 3120 655f 3220 3020 2020 3020 | e_1 e_2 0 0 │ │ │ │ -00012400: 2020 7c20 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00012410: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00012420: 207b 2d31 7d20 7c20 655f 3020 3020 2020 {-1} | e_0 0 │ │ │ │ -00012430: 655f 3220 3020 2020 7c20 2020 2020 2020 e_2 0 | │ │ │ │ -00012440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012450: 0a7c 2020 2020 207b 2d31 7d20 7c20 3020 .| {-1} | 0 │ │ │ │ -00012460: 2020 655f 3020 655f 3120 655f 3220 7c20 e_0 e_1 e_2 | │ │ │ │ -00012470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012480: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00012390: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +000123a0: 207b 2d31 7d20 7c20 655f 3220 3020 2020 {-1} | e_2 0 │ │ │ │ +000123b0: 3020 2020 3020 2020 7c20 2020 2020 2020 0 0 | │ │ │ │ +000123c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000123d0: 0a7c 2020 2020 207b 2d31 7d20 7c20 655f .| {-1} | e_ │ │ │ │ +000123e0: 3120 655f 3220 3020 2020 3020 2020 7c20 1 e_2 0 0 | │ │ │ │ +000123f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012400: 2020 2020 207c 0a7c 2020 2020 207b 2d31 |.| {-1 │ │ │ │ +00012410: 7d20 7c20 655f 3020 3020 2020 655f 3220 } | e_0 0 e_2 │ │ │ │ +00012420: 3020 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ +00012430: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012440: 2020 207b 2d31 7d20 7c20 3020 2020 655f {-1} | 0 e_ │ │ │ │ +00012450: 3020 655f 3120 655f 3220 7c20 2020 2020 0 e_1 e_2 | │ │ │ │ +00012460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012470: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000124c0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ -000124d0: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000124e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000124f0: 207c 0a7c 6f35 203a 204d 6174 7269 7820 |.|o5 : Matrix │ │ │ │ -00012500: 4520 203c 2d2d 2045 2020 2020 2020 2020 E <-- E │ │ │ │ -00012510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012520: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000124a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000124b0: 2020 2020 2020 2034 2020 2020 2020 3420 4 4 │ │ │ │ +000124c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000124d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000124e0: 6f35 203a 204d 6174 7269 7820 4520 203c o5 : Matrix E < │ │ │ │ +000124f0: 2d2d 2045 2020 2020 2020 2020 2020 2020 -- E │ │ │ │ +00012500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012510: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00012520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00012560: 4361 7665 6174 0a3d 3d3d 3d3d 3d0a 0a54 Caveat.======..T │ │ │ │ -00012570: 6869 7320 6675 6e63 7469 6f6e 2069 7320 his function is │ │ │ │ -00012580: 6120 7175 6963 6b2d 616e 642d 6469 7274 a quick-and-dirt │ │ │ │ -00012590: 7920 746f 6f6c 2077 6869 6368 2072 6571 y tool which req │ │ │ │ -000125a0: 7569 7265 7320 6c69 7474 6c65 2063 6f6d uires little com │ │ │ │ -000125b0: 7075 7461 7469 6f6e 2e0a 486f 7765 7665 putation..Howeve │ │ │ │ -000125c0: 7220 6966 2069 7420 6973 2063 616c 6c65 r if it is calle │ │ │ │ -000125d0: 6420 6f6e 2074 776f 2073 7563 6365 7373 d on two success │ │ │ │ -000125e0: 6976 6520 7472 756e 6361 7469 6f6e 7320 ive truncations │ │ │ │ -000125f0: 6f66 2061 206d 6f64 756c 652c 2074 6865 of a module, the │ │ │ │ -00012600: 6e20 7468 650a 6d61 7073 2069 7420 7072 n the.maps it pr │ │ │ │ -00012610: 6f64 7563 6573 206d 6179 204e 4f54 2063 oduces may NOT c │ │ │ │ -00012620: 6f6d 706f 7365 2074 6f20 7a65 726f 2062 ompose to zero b │ │ │ │ -00012630: 6563 6175 7365 2074 6865 2063 686f 6963 ecause the choic │ │ │ │ -00012640: 6520 6f66 2062 6173 6573 2069 7320 6e6f e of bases is no │ │ │ │ -00012650: 740a 636f 6e73 6973 7465 6e74 2e0a 0a53 t.consistent...S │ │ │ │ -00012660: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -00012670: 0a0a 2020 2a20 2a6e 6f74 6520 6267 673a .. * *note bgg: │ │ │ │ -00012680: 2062 6767 2c20 2d2d 2074 6865 2069 7468 bgg, -- the ith │ │ │ │ -00012690: 2064 6966 6665 7265 6e74 6961 6c20 6f66 differential of │ │ │ │ -000126a0: 2074 6865 2063 6f6d 706c 6578 2052 284d the complex R(M │ │ │ │ -000126b0: 290a 0a57 6179 7320 746f 2075 7365 2073 )..Ways to use s │ │ │ │ -000126c0: 796d 4578 743a 0a3d 3d3d 3d3d 3d3d 3d3d ymExt:.========= │ │ │ │ -000126d0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -000126e0: 2273 796d 4578 7428 4d61 7472 6978 2c50 "symExt(Matrix,P │ │ │ │ -000126f0: 6f6c 796e 6f6d 6961 6c52 696e 6729 220a olynomialRing)". │ │ │ │ -00012700: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00012710: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00012720: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00012730: 6374 202a 6e6f 7465 2073 796d 4578 743a ct *note symExt: │ │ │ │ -00012740: 2073 796d 4578 742c 2069 7320 6120 2a6e symExt, is a *n │ │ │ │ -00012750: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00012760: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00012770: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00012780: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00012540: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 ---------+..Cave │ │ │ │ +00012550: 6174 0a3d 3d3d 3d3d 3d0a 0a54 6869 7320 at.======..This │ │ │ │ +00012560: 6675 6e63 7469 6f6e 2069 7320 6120 7175 function is a qu │ │ │ │ +00012570: 6963 6b2d 616e 642d 6469 7274 7920 746f ick-and-dirty to │ │ │ │ +00012580: 6f6c 2077 6869 6368 2072 6571 7569 7265 ol which require │ │ │ │ +00012590: 7320 6c69 7474 6c65 2063 6f6d 7075 7461 s little computa │ │ │ │ +000125a0: 7469 6f6e 2e0a 486f 7765 7665 7220 6966 tion..However if │ │ │ │ +000125b0: 2069 7420 6973 2063 616c 6c65 6420 6f6e it is called on │ │ │ │ +000125c0: 2074 776f 2073 7563 6365 7373 6976 6520 two successive │ │ │ │ +000125d0: 7472 756e 6361 7469 6f6e 7320 6f66 2061 truncations of a │ │ │ │ +000125e0: 206d 6f64 756c 652c 2074 6865 6e20 7468 module, then th │ │ │ │ +000125f0: 650a 6d61 7073 2069 7420 7072 6f64 7563 e.maps it produc │ │ │ │ +00012600: 6573 206d 6179 204e 4f54 2063 6f6d 706f es may NOT compo │ │ │ │ +00012610: 7365 2074 6f20 7a65 726f 2062 6563 6175 se to zero becau │ │ │ │ +00012620: 7365 2074 6865 2063 686f 6963 6520 6f66 se the choice of │ │ │ │ +00012630: 2062 6173 6573 2069 7320 6e6f 740a 636f bases is not.co │ │ │ │ +00012640: 6e73 6973 7465 6e74 2e0a 0a53 6565 2061 nsistent...See a │ │ │ │ +00012650: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00012660: 2a20 2a6e 6f74 6520 6267 673a 2062 6767 * *note bgg: bgg │ │ │ │ +00012670: 2c20 2d2d 2074 6865 2069 7468 2064 6966 , -- the ith dif │ │ │ │ +00012680: 6665 7265 6e74 6961 6c20 6f66 2074 6865 ferential of the │ │ │ │ +00012690: 2063 6f6d 706c 6578 2052 284d 290a 0a57 complex R(M)..W │ │ │ │ +000126a0: 6179 7320 746f 2075 7365 2073 796d 4578 ays to use symEx │ │ │ │ +000126b0: 743a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d t:.============= │ │ │ │ +000126c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2273 796d ======.. * "sym │ │ │ │ +000126d0: 4578 7428 4d61 7472 6978 2c50 6f6c 796e Ext(Matrix,Polyn │ │ │ │ +000126e0: 6f6d 6961 6c52 696e 6729 220a 0a46 6f72 omialRing)"..For │ │ │ │ +000126f0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00012700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00012710: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00012720: 6e6f 7465 2073 796d 4578 743a 2073 796d note symExt: sym │ │ │ │ +00012730: 4578 742c 2069 7320 6120 2a6e 6f74 6520 Ext, is a *note │ │ │ │ +00012740: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00012750: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00012760: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00012770: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00012780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000127a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000127b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000127d0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -000127e0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -000127f0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00012800: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00012810: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00012820: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00012830: 7932 2f70 6163 6b61 6765 732f 4247 472e y2/packages/BGG. │ │ │ │ -00012840: 6d32 3a36 3937 3a0a 302e 0a1f 0a46 696c m2:697:.0....Fil │ │ │ │ -00012850: 653a 2042 4747 2e69 6e66 6f2c 204e 6f64 e: BGG.info, Nod │ │ │ │ -00012860: 653a 2074 6174 6552 6573 6f6c 7574 696f e: tateResolutio │ │ │ │ -00012870: 6e2c 204e 6578 743a 2075 6e69 7665 7273 n, Next: univers │ │ │ │ -00012880: 616c 4578 7465 6e73 696f 6e2c 2050 7265 alExtension, Pre │ │ │ │ -00012890: 763a 2073 796d 4578 742c 2055 703a 2054 v: symExt, Up: T │ │ │ │ -000128a0: 6f70 0a0a 7461 7465 5265 736f 6c75 7469 op..tateResoluti │ │ │ │ -000128b0: 6f6e 202d 2d20 6669 6e69 7465 2070 6965 on -- finite pie │ │ │ │ -000128c0: 6365 206f 6620 7468 6520 5461 7465 2072 ce of the Tate r │ │ │ │ -000128d0: 6573 6f6c 7574 696f 6e0a 2a2a 2a2a 2a2a esolution.****** │ │ │ │ +000127c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000127d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000127e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000127f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00012800: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00012810: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00012820: 6163 6b61 6765 732f 4247 472e 6d32 3a36 ackages/BGG.m2:6 │ │ │ │ +00012830: 3937 3a0a 302e 0a1f 0a46 696c 653a 2042 97:.0....File: B │ │ │ │ +00012840: 4747 2e69 6e66 6f2c 204e 6f64 653a 2074 GG.info, Node: t │ │ │ │ +00012850: 6174 6552 6573 6f6c 7574 696f 6e2c 204e ateResolution, N │ │ │ │ +00012860: 6578 743a 2075 6e69 7665 7273 616c 4578 ext: universalEx │ │ │ │ +00012870: 7465 6e73 696f 6e2c 2050 7265 763a 2073 tension, Prev: s │ │ │ │ +00012880: 796d 4578 742c 2055 703a 2054 6f70 0a0a ymExt, Up: Top.. │ │ │ │ +00012890: 7461 7465 5265 736f 6c75 7469 6f6e 202d tateResolution - │ │ │ │ +000128a0: 2d20 6669 6e69 7465 2070 6965 6365 206f - finite piece o │ │ │ │ +000128b0: 6620 7468 6520 5461 7465 2072 6573 6f6c f the Tate resol │ │ │ │ +000128c0: 7574 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ution.********** │ │ │ │ +000128d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000128e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000128f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00012900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00012910: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ -00012920: 2020 2020 2074 6174 6552 6573 6f6c 7574 tateResolut │ │ │ │ -00012930: 696f 6e28 6d2c 452c 6c2c 6829 0a20 202a ion(m,E,l,h). * │ │ │ │ -00012940: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00012950: 206d 2c20 6120 2a6e 6f74 6520 6d61 7472 m, a *note matr │ │ │ │ -00012960: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00012970: 6329 4d61 7472 6978 2c2c 2061 2070 7265 c)Matrix,, a pre │ │ │ │ -00012980: 7365 6e74 6174 696f 6e20 6d61 7472 6978 sentation matrix │ │ │ │ -00012990: 2066 6f72 2061 0a20 2020 2020 2020 206d for a. m │ │ │ │ -000129a0: 6f64 756c 650a 2020 2020 2020 2a20 452c odule. * E, │ │ │ │ -000129b0: 2061 202a 6e6f 7465 2070 6f6c 796e 6f6d a *note polynom │ │ │ │ -000129c0: 6961 6c20 7269 6e67 3a20 284d 6163 6175 ial ring: (Macau │ │ │ │ -000129d0: 6c61 7932 446f 6329 506f 6c79 6e6f 6d69 lay2Doc)Polynomi │ │ │ │ -000129e0: 616c 5269 6e67 2c2c 2065 7874 6572 696f alRing,, exterio │ │ │ │ -000129f0: 720a 2020 2020 2020 2020 616c 6765 6272 r. algebr │ │ │ │ -00012a00: 610a 2020 2020 2020 2a20 6c2c 2061 6e20 a. * l, an │ │ │ │ -00012a10: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00012a20: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00012a30: 2c20 6c6f 7765 7220 636f 686f 6d6f 6c6f , lower cohomolo │ │ │ │ -00012a40: 6769 6361 6c20 6465 6772 6565 0a20 2020 gical degree. │ │ │ │ -00012a50: 2020 202a 2068 2c20 616e 202a 6e6f 7465 * h, an *note │ │ │ │ -00012a60: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -00012a70: 6c61 7932 446f 6329 5a5a 2c2c 2075 7070 lay2Doc)ZZ,, upp │ │ │ │ -00012a80: 6572 2062 6f75 6e64 206f 6e20 7468 650a er bound on the. │ │ │ │ -00012a90: 2020 2020 2020 2020 636f 686f 6d6f 6c6f cohomolo │ │ │ │ -00012aa0: 6769 6361 6c20 6465 6772 6565 0a20 202a gical degree. * │ │ │ │ -00012ab0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00012ac0: 2a20 6120 2a6e 6f74 6520 636f 6d70 6c65 * a *note comple │ │ │ │ -00012ad0: 783a 2028 436f 6d70 6c65 7865 7329 436f x: (Complexes)Co │ │ │ │ -00012ae0: 6d70 6c65 782c 2c20 6120 6669 6e69 7465 mplex,, a finite │ │ │ │ -00012af0: 2070 6965 6365 206f 6620 7468 6520 5461 piece of the Ta │ │ │ │ -00012b00: 7465 0a20 2020 2020 2020 2072 6573 6f6c te. resol │ │ │ │ -00012b10: 7574 696f 6e0a 0a44 6573 6372 6970 7469 ution..Descripti │ │ │ │ -00012b20: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00012b30: 5468 6973 2066 756e 6374 696f 6e20 7461 This function ta │ │ │ │ -00012b40: 6b65 7320 6173 2069 6e70 7574 2061 2070 kes as input a p │ │ │ │ -00012b50: 7265 7365 6e74 6174 696f 6e20 6d61 7472 resentation matr │ │ │ │ -00012b60: 6978 206d 206f 6620 6120 6669 6e69 7465 ix m of a finite │ │ │ │ -00012b70: 6c79 2067 656e 6572 6174 6564 0a67 7261 ly generated.gra │ │ │ │ -00012b80: 6465 6420 532d 6d6f 6475 6c65 204d 2061 ded S-module M a │ │ │ │ -00012b90: 6e20 6578 7465 7269 6f72 2061 6c67 6562 n exterior algeb │ │ │ │ -00012ba0: 7261 2045 2061 6e64 2074 776f 2069 6e74 ra E and two int │ │ │ │ -00012bb0: 6567 6572 7320 6c20 616e 6420 682e 2049 egers l and h. I │ │ │ │ -00012bc0: 6620 7220 6973 2074 6865 0a72 6567 756c f r is the.regul │ │ │ │ -00012bd0: 6172 6974 7920 6f66 204d 2c20 7468 656e arity of M, then │ │ │ │ -00012be0: 2074 6869 7320 6675 6e63 7469 6f6e 2063 this function c │ │ │ │ -00012bf0: 6f6d 7075 7465 7320 7468 6520 7069 6563 omputes the piec │ │ │ │ -00012c00: 6520 6f66 2074 6865 2054 6174 6520 7265 e of the Tate re │ │ │ │ -00012c10: 736f 6c75 7469 6f6e 0a66 726f 6d20 636f solution.from co │ │ │ │ -00012c20: 686f 6d6f 6c6f 6769 6361 6c20 6465 6772 homological degr │ │ │ │ -00012c30: 6565 206c 2074 6f20 636f 686f 6d6f 6c6f ee l to cohomolo │ │ │ │ -00012c40: 6769 6361 6c20 6465 6772 6565 206d 6178 gical degree max │ │ │ │ -00012c50: 2872 2b32 2c68 292e 2046 6f72 2069 6e73 (r+2,h). For ins │ │ │ │ -00012c60: 7461 6e63 652c 0a66 6f72 2074 6865 2068 tance,.for the h │ │ │ │ -00012c70: 6f6d 6f67 656e 656f 7573 2063 6f6f 7264 omogeneous coord │ │ │ │ -00012c80: 696e 6174 6520 7269 6e67 206f 6620 6120 inate ring of a │ │ │ │ -00012c90: 706f 696e 7420 696e 2074 6865 2070 726f point in the pro │ │ │ │ -00012ca0: 6a65 6374 6976 6520 706c 616e 653a 0a2b jective plane:.+ │ │ │ │ +000128f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00012900: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00012910: 2074 6174 6552 6573 6f6c 7574 696f 6e28 tateResolution( │ │ │ │ +00012920: 6d2c 452c 6c2c 6829 0a20 202a 2049 6e70 m,E,l,h). * Inp │ │ │ │ +00012930: 7574 733a 0a20 2020 2020 202a 206d 2c20 uts:. * m, │ │ │ │ +00012940: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +00012950: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +00012960: 7472 6978 2c2c 2061 2070 7265 7365 6e74 trix,, a present │ │ │ │ +00012970: 6174 696f 6e20 6d61 7472 6978 2066 6f72 ation matrix for │ │ │ │ +00012980: 2061 0a20 2020 2020 2020 206d 6f64 756c a. modul │ │ │ │ +00012990: 650a 2020 2020 2020 2a20 452c 2061 202a e. * E, a * │ │ │ │ +000129a0: 6e6f 7465 2070 6f6c 796e 6f6d 6961 6c20 note polynomial │ │ │ │ +000129b0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +000129c0: 446f 6329 506f 6c79 6e6f 6d69 616c 5269 Doc)PolynomialRi │ │ │ │ +000129d0: 6e67 2c2c 2065 7874 6572 696f 720a 2020 ng,, exterior. │ │ │ │ +000129e0: 2020 2020 2020 616c 6765 6272 610a 2020 algebra. │ │ │ │ +000129f0: 2020 2020 2a20 6c2c 2061 6e20 2a6e 6f74 * l, an *not │ │ │ │ +00012a00: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00012a10: 756c 6179 3244 6f63 295a 5a2c 2c20 6c6f ulay2Doc)ZZ,, lo │ │ │ │ +00012a20: 7765 7220 636f 686f 6d6f 6c6f 6769 6361 wer cohomologica │ │ │ │ +00012a30: 6c20 6465 6772 6565 0a20 2020 2020 202a l degree. * │ │ │ │ +00012a40: 2068 2c20 616e 202a 6e6f 7465 2069 6e74 h, an *note int │ │ │ │ +00012a50: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ +00012a60: 446f 6329 5a5a 2c2c 2075 7070 6572 2062 Doc)ZZ,, upper b │ │ │ │ +00012a70: 6f75 6e64 206f 6e20 7468 650a 2020 2020 ound on the. │ │ │ │ +00012a80: 2020 2020 636f 686f 6d6f 6c6f 6769 6361 cohomologica │ │ │ │ +00012a90: 6c20 6465 6772 6565 0a20 202a 204f 7574 l degree. * Out │ │ │ │ +00012aa0: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00012ab0: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028 *note complex: ( │ │ │ │ +00012ac0: 436f 6d70 6c65 7865 7329 436f 6d70 6c65 Complexes)Comple │ │ │ │ +00012ad0: 782c 2c20 6120 6669 6e69 7465 2070 6965 x,, a finite pie │ │ │ │ +00012ae0: 6365 206f 6620 7468 6520 5461 7465 0a20 ce of the Tate. │ │ │ │ +00012af0: 2020 2020 2020 2072 6573 6f6c 7574 696f resolutio │ │ │ │ +00012b00: 6e0a 0a44 6573 6372 6970 7469 6f6e 0a3d n..Description.= │ │ │ │ +00012b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 ==========..This │ │ │ │ +00012b20: 2066 756e 6374 696f 6e20 7461 6b65 7320 function takes │ │ │ │ +00012b30: 6173 2069 6e70 7574 2061 2070 7265 7365 as input a prese │ │ │ │ +00012b40: 6e74 6174 696f 6e20 6d61 7472 6978 206d ntation matrix m │ │ │ │ +00012b50: 206f 6620 6120 6669 6e69 7465 6c79 2067 of a finitely g │ │ │ │ +00012b60: 656e 6572 6174 6564 0a67 7261 6465 6420 enerated.graded │ │ │ │ +00012b70: 532d 6d6f 6475 6c65 204d 2061 6e20 6578 S-module M an ex │ │ │ │ +00012b80: 7465 7269 6f72 2061 6c67 6562 7261 2045 terior algebra E │ │ │ │ +00012b90: 2061 6e64 2074 776f 2069 6e74 6567 6572 and two integer │ │ │ │ +00012ba0: 7320 6c20 616e 6420 682e 2049 6620 7220 s l and h. If r │ │ │ │ +00012bb0: 6973 2074 6865 0a72 6567 756c 6172 6974 is the.regularit │ │ │ │ +00012bc0: 7920 6f66 204d 2c20 7468 656e 2074 6869 y of M, then thi │ │ │ │ +00012bd0: 7320 6675 6e63 7469 6f6e 2063 6f6d 7075 s function compu │ │ │ │ +00012be0: 7465 7320 7468 6520 7069 6563 6520 6f66 tes the piece of │ │ │ │ +00012bf0: 2074 6865 2054 6174 6520 7265 736f 6c75 the Tate resolu │ │ │ │ +00012c00: 7469 6f6e 0a66 726f 6d20 636f 686f 6d6f tion.from cohomo │ │ │ │ +00012c10: 6c6f 6769 6361 6c20 6465 6772 6565 206c logical degree l │ │ │ │ +00012c20: 2074 6f20 636f 686f 6d6f 6c6f 6769 6361 to cohomologica │ │ │ │ +00012c30: 6c20 6465 6772 6565 206d 6178 2872 2b32 l degree max(r+2 │ │ │ │ +00012c40: 2c68 292e 2046 6f72 2069 6e73 7461 6e63 ,h). For instanc │ │ │ │ +00012c50: 652c 0a66 6f72 2074 6865 2068 6f6d 6f67 e,.for the homog │ │ │ │ +00012c60: 656e 656f 7573 2063 6f6f 7264 696e 6174 eneous coordinat │ │ │ │ +00012c70: 6520 7269 6e67 206f 6620 6120 706f 696e e ring of a poin │ │ │ │ +00012c80: 7420 696e 2074 6865 2070 726f 6a65 6374 t in the project │ │ │ │ +00012c90: 6976 6520 706c 616e 653a 0a2b 2d2d 2d2d ive plane:.+---- │ │ │ │ +00012ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ce0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a ---+.|i1 : S = Z │ │ │ │ -00012cf0: 5a2f 3332 3030 335b 785f 302e 2e78 5f32 Z/32003[x_0..x_2 │ │ │ │ -00012d00: 5d3b 2020 2020 2020 2020 2020 2020 2020 ]; │ │ │ │ -00012d10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00012cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00012cd0: 0a7c 6931 203a 2053 203d 205a 5a2f 3332 .|i1 : S = ZZ/32 │ │ │ │ +00012ce0: 3030 335b 785f 302e 2e78 5f32 5d3b 2020 003[x_0..x_2]; │ │ │ │ +00012cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012d00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00012d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 2d2d 2d2b ---------------+ │ │ │ │ -00012d50: 0a7c 6932 203a 2045 203d 205a 5a2f 3332 .|i2 : E = ZZ/32 │ │ │ │ -00012d60: 3030 335b 655f 302e 2e65 5f32 2c20 536b 003[e_0..e_2, Sk │ │ │ │ -00012d70: 6577 436f 6d6d 7574 6174 6976 653d 3e74 ewCommutative=>t │ │ │ │ -00012d80: 7275 655d 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d rue];|.+-------- │ │ │ │ +00012d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +00012d40: 203a 2045 203d 205a 5a2f 3332 3030 335b : E = ZZ/32003[ │ │ │ │ +00012d50: 655f 302e 2e65 5f32 2c20 536b 6577 436f e_0..e_2, SkewCo │ │ │ │ +00012d60: 6d6d 7574 6174 6976 653d 3e74 7275 655d mmutative=>true] │ │ │ │ +00012d70: 3b7c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ;|.+------------ │ │ │ │ +00012d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -00012dc0: 203a 206d 203d 206d 6174 7269 787b 7b78 : m = matrix{{x │ │ │ │ -00012dd0: 5f30 2c78 5f31 7d7d 3b20 2020 2020 2020 _0,x_1}}; │ │ │ │ +00012da0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 206d -------+.|i3 : m │ │ │ │ +00012db0: 203d 206d 6174 7269 787b 7b78 5f30 2c78 = matrix{{x_0,x │ │ │ │ +00012dc0: 5f31 7d7d 3b20 2020 2020 2020 2020 2020 _1}}; │ │ │ │ +00012dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00012de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012df0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00012e30: 2020 2020 2020 2031 2020 2020 2020 3220 1 2 │ │ │ │ -00012e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00012e60: 6f33 203a 204d 6174 7269 7820 5320 203c o3 : Matrix S < │ │ │ │ -00012e70: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ -00012e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012e90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00012e10: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012e20: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +00012e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e40: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +00012e50: 204d 6174 7269 7820 5320 203c 2d2d 2053 Matrix S <-- S │ │ │ │ +00012e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012e70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00012e80: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00012e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012ec0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -00012ed0: 2072 6567 756c 6172 6974 7920 636f 6b65 regularity coke │ │ │ │ -00012ee0: 7220 6d20 2020 2020 2020 2020 2020 2020 r m │ │ │ │ -00012ef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00012f00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00012eb0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2072 6567 -----+.|i4 : reg │ │ │ │ +00012ec0: 756c 6172 6974 7920 636f 6b65 7220 6d20 ularity coker m │ │ │ │ +00012ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012ee0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00012ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00012f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f30: 2020 2020 207c 0a7c 6f34 203d 2030 2020 |.|o4 = 0 │ │ │ │ +00012f20: 207c 0a7c 6f34 203d 2030 2020 2020 2020 |.|o4 = 0 │ │ │ │ +00012f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012f60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00012f50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00012f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00012f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00012fa0: 2d2b 0a7c 6935 203a 2054 203d 2074 6174 -+.|i5 : T = tat │ │ │ │ -00012fb0: 6552 6573 6f6c 7574 696f 6e28 6d2c 452c eResolution(m,E, │ │ │ │ -00012fc0: 2d32 2c34 2920 2020 2020 2020 2020 2020 -2,4) │ │ │ │ -00012fd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00012f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00012f90: 6935 203a 2054 203d 2074 6174 6552 6573 i5 : T = tateRes │ │ │ │ +00012fa0: 6f6c 7574 696f 6e28 6d2c 452c 2d32 2c34 olution(m,E,-2,4 │ │ │ │ +00012fb0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00012fc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00012fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00012ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00013010: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ -00013020: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00013030: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -00013040: 3120 207c 0a7c 6f35 203d 2045 2020 3c2d 1 |.|o5 = E <- │ │ │ │ -00013050: 2d20 4520 203c 2d2d 2045 2020 3c2d 2d20 - E <-- E <-- │ │ │ │ -00013060: 4520 203c 2d2d 2045 2020 3c2d 2d20 4520 E <-- E <-- E │ │ │ │ -00013070: 203c 2d2d 2045 2020 207c 0a7c 2020 2020 <-- E |.| │ │ │ │ +00012ff0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00013000: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +00013010: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00013020: 2020 2020 2031 2020 2020 2020 3120 207c 1 1 | │ │ │ │ +00013030: 0a7c 6f35 203d 2045 2020 3c2d 2d20 4520 .|o5 = E <-- E │ │ │ │ +00013040: 203c 2d2d 2045 2020 3c2d 2d20 4520 203c <-- E <-- E < │ │ │ │ +00013050: 2d2d 2045 2020 3c2d 2d20 4520 203c 2d2d -- E <-- E <-- │ │ │ │ +00013060: 2045 2020 207c 0a7c 2020 2020 2020 2020 E |.| │ │ │ │ +00013070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000130a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000130b0: 0a7c 2020 2020 2030 2020 2020 2020 3120 .| 0 1 │ │ │ │ -000130c0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -000130d0: 2020 2034 2020 2020 2020 3520 2020 2020 4 5 │ │ │ │ -000130e0: 2036 2020 207c 0a7c 2020 2020 2020 2020 6 |.| │ │ │ │ +00013090: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000130a0: 2020 2030 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ +000130b0: 2032 2020 2020 2020 3320 2020 2020 2034 2 3 4 │ │ │ │ +000130c0: 2020 2020 2020 3520 2020 2020 2036 2020 5 6 │ │ │ │ +000130d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000130e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000130f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013110: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00013120: 203a 2043 6f6d 706c 6578 2020 2020 2020 : Complex │ │ │ │ -00013130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013150: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00013100: 2020 2020 2020 207c 0a7c 6f35 203a 2043 |.|o5 : C │ │ │ │ +00013110: 6f6d 706c 6578 2020 2020 2020 2020 2020 omplex │ │ │ │ +00013120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013130: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00013140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013180: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2062 -------+.|i6 : b │ │ │ │ -00013190: 6574 7469 2054 2020 2020 2020 2020 2020 etti T │ │ │ │ -000131a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013170: 2d2d 2d2b 0a7c 6936 203a 2062 6574 7469 ---+.|i6 : betti │ │ │ │ +00013180: 2054 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ +00013190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000131a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000131b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000131f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00013200: 2020 3020 3120 3220 3320 3420 3520 3620 0 1 2 3 4 5 6 │ │ │ │ -00013210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013220: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -00013230: 2074 6f74 616c 3a20 3120 3120 3120 3120 total: 1 1 1 1 │ │ │ │ -00013240: 3120 3120 3120 2020 2020 2020 2020 2020 1 1 1 │ │ │ │ -00013250: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00013260: 0a7c 2020 2020 2020 2020 2d34 3a20 3120 .| -4: 1 │ │ │ │ -00013270: 3120 3120 3120 3120 3120 3120 2020 2020 1 1 1 1 1 1 │ │ │ │ -00013280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013290: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000131d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000131e0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ +000131f0: 3120 3220 3320 3420 3520 3620 2020 2020 1 2 3 4 5 6 │ │ │ │ +00013200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013210: 2020 2020 207c 0a7c 6f36 203d 2074 6f74 |.|o6 = tot │ │ │ │ +00013220: 616c 3a20 3120 3120 3120 3120 3120 3120 al: 1 1 1 1 1 1 │ │ │ │ +00013230: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013240: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00013250: 2020 2020 2020 2d34 3a20 3120 3120 3120 -4: 1 1 1 │ │ │ │ +00013260: 3120 3120 3120 3120 2020 2020 2020 2020 1 1 1 1 │ │ │ │ +00013270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013280: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00013290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000132a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -000132d0: 203a 2042 6574 7469 5461 6c6c 7920 2020 : BettiTally │ │ │ │ -000132e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000132f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013300: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000132b0: 2020 2020 2020 207c 0a7c 6f36 203a 2042 |.|o6 : B │ │ │ │ +000132c0: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +000132d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000132e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000132f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013330: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2054 -------+.|i7 : T │ │ │ │ -00013340: 2e64 645f 3120 2020 2020 2020 2020 2020 .dd_1 │ │ │ │ -00013350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00013320: 2d2d 2d2b 0a7c 6937 203a 2054 2e64 645f ---+.|i7 : T.dd_ │ │ │ │ +00013330: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013350: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00013360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013380: 2020 2020 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2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ -00013430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013440: 2020 2020 207c 0a7c 6f37 203a 204d 6174 |.|o7 : Mat │ │ │ │ -00013450: 7269 7820 4520 203c 2d2d 2045 2020 2020 rix E <-- E │ │ │ │ -00013460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013470: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +000133f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00013400: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00013410: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013430: 207c 0a7c 6f37 203a 204d 6174 7269 7820 |.|o7 : Matrix │ │ │ │ +00013440: 4520 203c 2d2d 2045 2020 2020 2020 2020 E <-- E │ │ │ │ +00013450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013460: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00013470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000134a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000134b0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -000134c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -000134d0: 2073 796d 4578 743a 2073 796d 4578 742c symExt: symExt, │ │ │ │ -000134e0: 202d 2d20 7468 6520 6669 7273 7420 6469 -- the first di │ │ │ │ -000134f0: 6666 6572 656e 7469 616c 206f 6620 7468 fferential of th │ │ │ │ -00013500: 6520 636f 6d70 6c65 7820 5228 4d29 0a0a e complex R(M).. │ │ │ │ -00013510: 5761 7973 2074 6f20 7573 6520 7461 7465 Ways to use tate │ │ │ │ -00013520: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ -00013530: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00013540: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7461 =======.. * "ta │ │ │ │ -00013550: 7465 5265 736f 6c75 7469 6f6e 284d 6174 teResolution(Mat │ │ │ │ -00013560: 7269 782c 506f 6c79 6e6f 6d69 616c 5269 rix,PolynomialRi │ │ │ │ -00013570: 6e67 2c5a 5a2c 5a5a 2922 0a0a 466f 7220 ng,ZZ,ZZ)"..For │ │ │ │ -00013580: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00013590: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000135a0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -000135b0: 6f74 6520 7461 7465 5265 736f 6c75 7469 ote tateResoluti │ │ │ │ -000135c0: 6f6e 3a20 7461 7465 5265 736f 6c75 7469 on: tateResoluti │ │ │ │ -000135d0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d on, is a *note m │ │ │ │ -000135e0: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -000135f0: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00013600: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00013490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000134a0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +000134b0: 3d0a 0a20 202a 202a 6e6f 7465 2073 796d =.. * *note sym │ │ │ │ +000134c0: 4578 743a 2073 796d 4578 742c 202d 2d20 Ext: symExt, -- │ │ │ │ +000134d0: 7468 6520 6669 7273 7420 6469 6666 6572 the first differ │ │ │ │ +000134e0: 656e 7469 616c 206f 6620 7468 6520 636f ential of the co │ │ │ │ +000134f0: 6d70 6c65 7820 5228 4d29 0a0a 5761 7973 mplex R(M)..Ways │ │ │ │ +00013500: 2074 6f20 7573 6520 7461 7465 5265 736f to use tateReso │ │ │ │ +00013510: 6c75 7469 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d lution:.======== │ │ │ │ +00013520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00013530: 3d3d 3d0a 0a20 202a 2022 7461 7465 5265 ===.. * "tateRe │ │ │ │ +00013540: 736f 6c75 7469 6f6e 284d 6174 7269 782c solution(Matrix, │ │ │ │ +00013550: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c5a PolynomialRing,Z │ │ │ │ +00013560: 5a2c 5a5a 2922 0a0a 466f 7220 7468 6520 Z,ZZ)"..For the │ │ │ │ +00013570: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00013580: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00013590: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +000135a0: 7461 7465 5265 736f 6c75 7469 6f6e 3a20 tateResolution: │ │ │ │ +000135b0: 7461 7465 5265 736f 6c75 7469 6f6e 2c20 tateResolution, │ │ │ │ +000135c0: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +000135d0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ +000135e0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +000135f0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00013600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00013660: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00013670: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00013680: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00013690: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -000136a0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -000136b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -000136c0: 636b 6167 6573 2f42 4747 2e6d 323a 3735 ckages/BGG.m2:75 │ │ │ │ -000136d0: 333a 0a30 2e0a 1f0a 4669 6c65 3a20 4247 3:.0....File: BG │ │ │ │ -000136e0: 472e 696e 666f 2c20 4e6f 6465 3a20 756e G.info, Node: un │ │ │ │ -000136f0: 6976 6572 7361 6c45 7874 656e 7369 6f6e iversalExtension │ │ │ │ -00013700: 2c20 5072 6576 3a20 7461 7465 5265 736f , Prev: tateReso │ │ │ │ -00013710: 6c75 7469 6f6e 2c20 5570 3a20 546f 700a lution, Up: Top. │ │ │ │ -00013720: 0a75 6e69 7665 7273 616c 4578 7465 6e73 .universalExtens │ │ │ │ -00013730: 696f 6e20 2d2d 2055 6e69 7665 7273 616c ion -- Universal │ │ │ │ -00013740: 2065 7874 656e 7369 6f6e 206f 6620 7665 extension of ve │ │ │ │ -00013750: 6374 6f72 2062 756e 646c 6573 206f 6e20 ctor bundles on │ │ │ │ -00013760: 505e 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a P^1.************ │ │ │ │ +00013640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00013650: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00013660: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00013670: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00013680: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00013690: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +000136a0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +000136b0: 6573 2f42 4747 2e6d 323a 3735 333a 0a30 es/BGG.m2:753:.0 │ │ │ │ +000136c0: 2e0a 1f0a 4669 6c65 3a20 4247 472e 696e ....File: BGG.in │ │ │ │ +000136d0: 666f 2c20 4e6f 6465 3a20 756e 6976 6572 fo, Node: univer │ │ │ │ +000136e0: 7361 6c45 7874 656e 7369 6f6e 2c20 5072 salExtension, Pr │ │ │ │ +000136f0: 6576 3a20 7461 7465 5265 736f 6c75 7469 ev: tateResoluti │ │ │ │ +00013700: 6f6e 2c20 5570 3a20 546f 700a 0a75 6e69 on, Up: Top..uni │ │ │ │ +00013710: 7665 7273 616c 4578 7465 6e73 696f 6e20 versalExtension │ │ │ │ +00013720: 2d2d 2055 6e69 7665 7273 616c 2065 7874 -- Universal ext │ │ │ │ +00013730: 656e 7369 6f6e 206f 6620 7665 6374 6f72 ension of vector │ │ │ │ +00013740: 2062 756e 646c 6573 206f 6e20 505e 310a bundles on P^1. │ │ │ │ +00013750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00013760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013790: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000137a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -000137b0: 653a 200a 2020 2020 2020 2020 4520 3d20 e: . E = │ │ │ │ -000137c0: 756e 6976 6572 7361 6c45 7874 656e 7369 universalExtensi │ │ │ │ -000137d0: 6f6e 284c 612c 204c 6229 0a20 202a 2049 on(La, Lb). * I │ │ │ │ -000137e0: 6e70 7574 733a 0a20 2020 2020 202a 204c nputs:. * L │ │ │ │ -000137f0: 612c 2061 202a 6e6f 7465 206c 6973 743a a, a *note list: │ │ │ │ -00013800: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -00013810: 6973 742c 2c20 6f66 2069 6e74 6567 6572 ist,, of integer │ │ │ │ -00013820: 730a 2020 2020 2020 2a20 4c62 2c20 6120 s. * Lb, a │ │ │ │ -00013830: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ -00013840: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ -00013850: 206f 6620 696e 7465 6765 7273 0a20 202a of integers. * │ │ │ │ -00013860: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -00013870: 2a20 452c 2061 202a 6e6f 7465 206d 6f64 * E, a *note mod │ │ │ │ -00013880: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -00013890: 6f63 294d 6f64 756c 652c 2c20 7265 7072 oc)Module,, repr │ │ │ │ -000138a0: 6573 656e 7469 6e67 2074 6865 2065 7874 esenting the ext │ │ │ │ -000138b0: 656e 7369 6f6e 0a0a 4465 7363 7269 7074 ension..Descript │ │ │ │ -000138c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -000138d0: 0a45 7665 7279 2076 6563 746f 7220 6275 .Every vector bu │ │ │ │ -000138e0: 6e64 6c65 2045 206f 6e20 245c 5050 5e31 ndle E on $\PP^1 │ │ │ │ -000138f0: 2420 7370 6c69 7473 2061 7320 6120 7375 $ splits as a su │ │ │ │ -00013900: 6d20 6f66 206c 696e 6520 6275 6e64 6c65 m of line bundle │ │ │ │ -00013910: 7320 4f4f 2861 5f69 292e 2049 6620 4c61 s OO(a_i). If La │ │ │ │ -00013920: 0a69 7320 6120 6c69 7374 206f 6620 696e .is a list of in │ │ │ │ -00013930: 7465 6765 7273 2c20 7765 2077 7269 7465 tegers, we write │ │ │ │ -00013940: 2045 284c 6129 2066 6f72 2074 6865 2064 E(La) for the d │ │ │ │ -00013950: 6972 6563 7420 7375 6d20 6f66 2074 6865 irect sum of the │ │ │ │ -00013960: 206c 696e 6520 6275 6e64 6c65 0a4f 4f28 line bundle.OO( │ │ │ │ -00013970: 4c61 5f69 292e 2020 4769 7665 6e20 7477 La_i). Given tw │ │ │ │ -00013980: 6f20 7375 6368 2062 756e 646c 6573 2073 o such bundles s │ │ │ │ -00013990: 7065 6369 6669 6564 2062 7920 7468 6520 pecified by the │ │ │ │ -000139a0: 6c69 7374 7320 4c61 2061 6e64 204c 6220 lists La and Lb │ │ │ │ -000139b0: 7468 6973 2073 6372 6970 740a 636f 6e73 this script.cons │ │ │ │ -000139c0: 7472 7563 7473 2061 206d 6f64 756c 6520 tructs a module │ │ │ │ -000139d0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ -000139e0: 2075 6e69 7665 7273 616c 2065 7874 656e universal exten │ │ │ │ -000139f0: 7369 6f6e 206f 6620 4528 4c62 2920 6279 sion of E(Lb) by │ │ │ │ -00013a00: 2045 284c 6129 2e20 4974 0a69 7320 6465 E(La). It.is de │ │ │ │ -00013a10: 6669 6e65 6420 6f6e 2074 6865 2070 726f fined on the pro │ │ │ │ -00013a20: 6475 6374 2076 6172 6965 7479 2045 7874 duct variety Ext │ │ │ │ -00013a30: 5e31 2845 284c 6129 2c20 4528 4c62 2929 ^1(E(La), E(Lb)) │ │ │ │ -00013a40: 2078 2024 5c50 505e 3124 2c20 616e 640a x $\PP^1$, and. │ │ │ │ -00013a50: 7265 7072 6573 656e 7465 6420 6865 7265 represented here │ │ │ │ -00013a60: 2062 7920 6120 6772 6164 6564 206d 6f64 by a graded mod │ │ │ │ -00013a70: 756c 6520 6f76 6572 2074 6865 2063 6f6f ule over the coo │ │ │ │ -00013a80: 7264 696e 6174 6520 7269 6e67 2053 203d rdinate ring S = │ │ │ │ -00013a90: 2041 5b79 5f30 2c79 5f31 5d20 6f66 0a74 A[y_0,y_1] of.t │ │ │ │ -00013aa0: 6869 7320 7661 7269 6574 793b 2068 6572 his variety; her │ │ │ │ -00013ab0: 6520 4120 6973 2074 6865 2063 6f6f 7264 e A is the coord │ │ │ │ -00013ac0: 696e 6174 6520 7269 6e67 206f 6620 4578 inate ring of Ex │ │ │ │ -00013ad0: 745e 3128 4528 4c61 292c 2045 284c 6229 t^1(E(La), E(Lb) │ │ │ │ -00013ae0: 292c 2077 6869 6368 2069 7320 610a 706f ), which is a.po │ │ │ │ -00013af0: 6c79 6e6f 6d69 616c 2072 696e 672e 0a0a lynomial ring... │ │ │ │ -00013b00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013790: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +000137a0: 2020 2020 2020 2020 4520 3d20 756e 6976 E = univ │ │ │ │ +000137b0: 6572 7361 6c45 7874 656e 7369 6f6e 284c ersalExtension(L │ │ │ │ +000137c0: 612c 204c 6229 0a20 202a 2049 6e70 7574 a, Lb). * Input │ │ │ │ +000137d0: 733a 0a20 2020 2020 202a 204c 612c 2061 s:. * La, a │ │ │ │ +000137e0: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ +000137f0: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ +00013800: 2c20 6f66 2069 6e74 6567 6572 730a 2020 , of integers. │ │ │ │ +00013810: 2020 2020 2a20 4c62 2c20 6120 2a6e 6f74 * Lb, a *not │ │ │ │ +00013820: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +00013830: 7932 446f 6329 4c69 7374 2c2c 206f 6620 y2Doc)List,, of │ │ │ │ +00013840: 696e 7465 6765 7273 0a20 202a 204f 7574 integers. * Out │ │ │ │ +00013850: 7075 7473 3a0a 2020 2020 2020 2a20 452c puts:. * E, │ │ │ │ +00013860: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ +00013870: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00013880: 6f64 756c 652c 2c20 7265 7072 6573 656e odule,, represen │ │ │ │ +00013890: 7469 6e67 2074 6865 2065 7874 656e 7369 ting the extensi │ │ │ │ +000138a0: 6f6e 0a0a 4465 7363 7269 7074 696f 6e0a on..Description. │ │ │ │ +000138b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a45 7665 ===========..Eve │ │ │ │ +000138c0: 7279 2076 6563 746f 7220 6275 6e64 6c65 ry vector bundle │ │ │ │ +000138d0: 2045 206f 6e20 245c 5050 5e31 2420 7370 E on $\PP^1$ sp │ │ │ │ +000138e0: 6c69 7473 2061 7320 6120 7375 6d20 6f66 lits as a sum of │ │ │ │ +000138f0: 206c 696e 6520 6275 6e64 6c65 7320 4f4f line bundles OO │ │ │ │ +00013900: 2861 5f69 292e 2049 6620 4c61 0a69 7320 (a_i). If La.is │ │ │ │ +00013910: 6120 6c69 7374 206f 6620 696e 7465 6765 a list of intege │ │ │ │ +00013920: 7273 2c20 7765 2077 7269 7465 2045 284c rs, we write E(L │ │ │ │ +00013930: 6129 2066 6f72 2074 6865 2064 6972 6563 a) for the direc │ │ │ │ +00013940: 7420 7375 6d20 6f66 2074 6865 206c 696e t sum of the lin │ │ │ │ +00013950: 6520 6275 6e64 6c65 0a4f 4f28 4c61 5f69 e bundle.OO(La_i │ │ │ │ +00013960: 292e 2020 4769 7665 6e20 7477 6f20 7375 ). Given two su │ │ │ │ +00013970: 6368 2062 756e 646c 6573 2073 7065 6369 ch bundles speci │ │ │ │ +00013980: 6669 6564 2062 7920 7468 6520 6c69 7374 fied by the list │ │ │ │ +00013990: 7320 4c61 2061 6e64 204c 6220 7468 6973 s La and Lb this │ │ │ │ +000139a0: 2073 6372 6970 740a 636f 6e73 7472 7563 script.construc │ │ │ │ +000139b0: 7473 2061 206d 6f64 756c 6520 7265 7072 ts a module repr │ │ │ │ +000139c0: 6573 656e 7469 6e67 2074 6865 2075 6e69 esenting the uni │ │ │ │ +000139d0: 7665 7273 616c 2065 7874 656e 7369 6f6e versal extension │ │ │ │ +000139e0: 206f 6620 4528 4c62 2920 6279 2045 284c of E(Lb) by E(L │ │ │ │ +000139f0: 6129 2e20 4974 0a69 7320 6465 6669 6e65 a). It.is define │ │ │ │ +00013a00: 6420 6f6e 2074 6865 2070 726f 6475 6374 d on the product │ │ │ │ +00013a10: 2076 6172 6965 7479 2045 7874 5e31 2845 variety Ext^1(E │ │ │ │ +00013a20: 284c 6129 2c20 4528 4c62 2929 2078 2024 (La), E(Lb)) x $ │ │ │ │ +00013a30: 5c50 505e 3124 2c20 616e 640a 7265 7072 \PP^1$, and.repr │ │ │ │ +00013a40: 6573 656e 7465 6420 6865 7265 2062 7920 esented here by │ │ │ │ +00013a50: 6120 6772 6164 6564 206d 6f64 756c 6520 a graded module │ │ │ │ +00013a60: 6f76 6572 2074 6865 2063 6f6f 7264 696e over the coordin │ │ │ │ +00013a70: 6174 6520 7269 6e67 2053 203d 2041 5b79 ate ring S = A[y │ │ │ │ +00013a80: 5f30 2c79 5f31 5d20 6f66 0a74 6869 7320 _0,y_1] of.this │ │ │ │ +00013a90: 7661 7269 6574 793b 2068 6572 6520 4120 variety; here A │ │ │ │ +00013aa0: 6973 2074 6865 2063 6f6f 7264 696e 6174 is the coordinat │ │ │ │ +00013ab0: 6520 7269 6e67 206f 6620 4578 745e 3128 e ring of Ext^1( │ │ │ │ +00013ac0: 4528 4c61 292c 2045 284c 6229 292c 2077 E(La), E(Lb)), w │ │ │ │ +00013ad0: 6869 6368 2069 7320 610a 706f 6c79 6e6f hich is a.polyno │ │ │ │ +00013ae0: 6d69 616c 2072 696e 672e 0a0a 2b2d 2d2d mial ring...+--- │ │ │ │ +00013af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013b10: 2d2d 2d2d 2d2d 2d2d 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 2b0a --------------+. │ │ │ │ -00013b50: 7c69 3120 3a20 4d20 3d20 756e 6976 6572 |i1 : M = univer │ │ │ │ -00013b60: 7361 6c45 7874 656e 7369 6f6e 287b 2d32 salExtension({-2 │ │ │ │ -00013b70: 7d2c 207b 327d 2920 2020 2020 2020 2020 }, {2}) │ │ │ │ -00013b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013b90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013ba0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00013b40: 3a20 4d20 3d20 756e 6976 6572 7361 6c45 : M = universalE │ │ │ │ +00013b50: 7874 656e 7369 6f6e 287b 2d32 7d2c 207b xtension({-2}, { │ │ │ │ +00013b60: 327d 2920 2020 2020 2020 2020 2020 2020 2}) │ │ │ │ +00013b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013b80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013be0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013bf0: 7c6f 3120 3d20 636f 6b65 726e 656c 207b |o1 = cokernel { │ │ │ │ -00013c00: 322c 2030 7d20 7c20 785f 3020 785f 3120 2, 0} | x_0 x_1 │ │ │ │ -00013c10: 785f 3220 7c20 2020 2020 2020 2020 2020 x_2 | │ │ │ │ -00013c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013c30: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013c40: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013c50: 312c 2031 7d20 7c20 795f 3020 3020 2020 1, 1} | y_0 0 │ │ │ │ -00013c60: 3020 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -00013c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013c80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013c90: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013ca0: 312c 2031 7d20 7c20 795f 3120 795f 3020 1, 1} | y_1 y_0 │ │ │ │ -00013cb0: 3020 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -00013cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013cd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013ce0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013cf0: 312c 2031 7d20 7c20 3020 2020 795f 3120 1, 1} | 0 y_1 │ │ │ │ -00013d00: 795f 3020 7c20 2020 2020 2020 2020 2020 y_0 | │ │ │ │ -00013d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013d30: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00013d40: 312c 2031 7d20 7c20 3020 2020 3020 2020 1, 1} | 0 0 │ │ │ │ -00013d50: 795f 3120 7c20 2020 2020 2020 2020 2020 y_1 | │ │ │ │ -00013d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013d70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013d80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013bd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00013be0: 3d20 636f 6b65 726e 656c 207b 322c 2030 = cokernel {2, 0 │ │ │ │ +00013bf0: 7d20 7c20 785f 3020 785f 3120 785f 3220 } | x_0 x_1 x_2 │ │ │ │ +00013c00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013c20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013c30: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013c40: 7d20 7c20 795f 3020 3020 2020 3020 2020 } | y_0 0 0 │ │ │ │ +00013c50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013c70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013c80: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013c90: 7d20 7c20 795f 3120 795f 3020 3020 2020 } | y_1 y_0 0 │ │ │ │ +00013ca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013cc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013cd0: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013ce0: 7d20 7c20 3020 2020 795f 3120 795f 3020 } | 0 y_1 y_0 │ │ │ │ +00013cf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013d20: 2020 2020 2020 2020 2020 207b 312c 2031 {1, 1 │ │ │ │ +00013d30: 7d20 7c20 3020 2020 3020 2020 795f 3120 } | 0 0 y_1 │ │ │ │ +00013d40: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013dc0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013dd0: 7c20 2020 2020 205a 5a20 2020 2020 2020 | ZZ │ │ │ │ -00013de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013e00: 5a5a 2020 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ -00013e10: 2020 2035 2020 2020 2020 2020 2020 7c0a 5 |. │ │ │ │ -00013e20: 7c6f 3120 3a20 2d2d 2d5b 7820 2e2e 7820 |o1 : ---[x ..x │ │ │ │ -00013e30: 5d5b 7920 2e2e 7920 5d2d 6d6f 6475 6c65 ][y ..y ]-module │ │ │ │ -00013e40: 2c20 7175 6f74 6965 6e74 206f 6620 282d , quotient of (- │ │ │ │ -00013e50: 2d2d 5b78 202e 2e78 205d 5b79 202e 2e79 --[x ..x ][y ..y │ │ │ │ -00013e60: 205d 2920 2020 2020 2020 2020 2020 7c0a ]) |. │ │ │ │ -00013e70: 7c20 2020 2020 3130 3120 2030 2020 2032 | 101 0 2 │ │ │ │ -00013e80: 2020 2030 2020 2031 2020 2020 2020 2020 0 1 │ │ │ │ -00013e90: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ -00013ea0: 3031 2020 3020 2020 3220 2020 3020 2020 01 0 2 0 │ │ │ │ -00013eb0: 3120 2020 2020 2020 2020 2020 2020 7c0a 1 |. │ │ │ │ -00013ec0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00013db0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013dc0: 2020 205a 5a20 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00013dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013de0: 2020 2020 2020 2020 2020 2020 5a5a 2020 ZZ │ │ │ │ +00013df0: 2020 2020 2020 2020 2020 2020 2020 2035 5 │ │ │ │ +00013e00: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00013e10: 3a20 2d2d 2d5b 7820 2e2e 7820 5d5b 7920 : ---[x ..x ][y │ │ │ │ +00013e20: 2e2e 7920 5d2d 6d6f 6475 6c65 2c20 7175 ..y ]-module, qu │ │ │ │ +00013e30: 6f74 6965 6e74 206f 6620 282d 2d2d 5b78 otient of (---[x │ │ │ │ +00013e40: 202e 2e78 205d 5b79 202e 2e79 205d 2920 ..x ][y ..y ]) │ │ │ │ +00013e50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013e60: 2020 3130 3120 2030 2020 2032 2020 2030 101 0 2 0 │ │ │ │ +00013e70: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00013e80: 2020 2020 2020 2020 2020 2031 3031 2020 101 │ │ │ │ +00013e90: 3020 2020 3220 2020 3020 2020 3120 2020 0 2 0 1 │ │ │ │ +00013ea0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00013eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00013ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00013f10: 7c69 3220 3a20 4d20 3d20 756e 6976 6572 |i2 : M = univer │ │ │ │ -00013f20: 7361 6c45 7874 656e 7369 6f6e 287b 2d32 salExtension({-2 │ │ │ │ -00013f30: 2c2d 337d 2c20 7b32 2c33 7d29 2020 2020 ,-3}, {2,3}) │ │ │ │ -00013f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013f60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00013ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +00013f00: 3a20 4d20 3d20 756e 6976 6572 7361 6c45 : M = universalE │ │ │ │ +00013f10: 7874 656e 7369 6f6e 287b 2d32 2c2d 337d xtension({-2,-3} │ │ │ │ +00013f20: 2c20 7b32 2c33 7d29 2020 2020 2020 2020 , {2,3}) │ │ │ │ +00013f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013f40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00013f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00013f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013fa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00013fb0: 7c6f 3220 3d20 636f 6b65 726e 656c 207b |o2 = cokernel { │ │ │ │ -00013fc0: 322c 2030 7d20 7c20 785f 3079 5f31 2078 2, 0} | x_0y_1 x │ │ │ │ -00013fd0: 5f31 795f 3120 785f 3279 5f31 2078 5f33 _1y_1 x_2y_1 x_3 │ │ │ │ -00013fe0: 795f 3120 785f 3479 5f31 2078 5f35 795f y_1 x_4y_1 x_5y_ │ │ │ │ -00013ff0: 3120 785f 3679 5f31 2020 2020 2020 7c0a 1 x_6y_1 |. │ │ │ │ -00014000: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014010: 332c 2030 7d20 7c20 785f 3920 2020 2078 3, 0} | x_9 x │ │ │ │ -00014020: 5f31 3020 2020 785f 3131 2020 2078 5f31 _10 x_11 x_1 │ │ │ │ -00014030: 3220 2020 785f 3133 2020 2078 5f31 3420 2 x_13 x_14 │ │ │ │ -00014040: 2020 785f 3135 2020 2020 2020 2020 7c0a x_15 |. │ │ │ │ -00014050: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014060: 322c 2031 7d20 7c20 795f 3020 2020 2030 2, 1} | y_0 0 │ │ │ │ -00014070: 2020 2020 2020 3020 2020 2020 2030 2020 0 0 │ │ │ │ -00014080: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014090: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000140a0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000140b0: 322c 2031 7d20 7c20 795f 3120 2020 2079 2, 1} | y_1 y │ │ │ │ -000140c0: 5f30 2020 2020 3020 2020 2020 2030 2020 _0 0 0 │ │ │ │ -000140d0: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -000140e0: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000140f0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014100: 322c 2031 7d20 7c20 3020 2020 2020 2079 2, 1} | 0 y │ │ │ │ -00014110: 5f31 2020 2020 795f 3020 2020 2030 2020 _1 y_0 0 │ │ │ │ -00014120: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014130: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -00014140: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -00014150: 322c 2031 7d20 7c20 3020 2020 2020 2030 2, 1} | 0 0 │ │ │ │ -00014160: 2020 2020 2020 795f 3120 2020 2079 5f30 y_1 y_0 │ │ │ │ -00014170: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -00014180: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -00014190: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000141a0: 322c 2031 7d20 7c20 3020 2020 2020 2030 2, 1} | 0 0 │ │ │ │ -000141b0: 2020 2020 2020 3020 2020 2020 2079 5f31 0 y_1 │ │ │ │ -000141c0: 2020 2020 3020 2020 2020 2030 2020 2020 0 0 │ │ │ │ -000141d0: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -000141e0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ -000141f0: 322c 2031 7d20 7c20 3020 2020 2020 2030 2, 1} | 0 0 │ │ │ │ -00014200: 2020 2020 2020 3020 2020 2020 2030 2020 0 0 │ │ │ │ -00014210: 2020 2020 795f 3020 2020 2030 2020 2020 y_0 0 │ │ │ │ -00014220: 2020 3020 2020 2020 2020 2020 2020 7c0a 0 |. │ │ │ │ -00014230: 7c20 2020 2020 2020 2020 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See the docu │ │ │ │ -00014a20: 6d65 6e74 6174 696f 6e20 666f 7220 6469 mentation for di │ │ │ │ -00014a30: 7265 6374 496d 6167 6543 6f6d 706c 6578 rectImageComplex │ │ │ │ -00014a40: 2066 6f72 2061 0a63 6f6e 6a65 6374 7572 for a.conjectur │ │ │ │ -00014a50: 6520 6162 6f75 7420 7468 6520 6571 7561 e about the equa │ │ │ │ -00014a60: 7469 6f6e 7320 6f66 2074 6865 7365 2076 tions of these v │ │ │ │ -00014a70: 6172 6965 7469 6573 2e0a 0a53 6565 2061 arieties...See a │ │ │ │ -00014a80: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00014a90: 2a20 2a6e 6f74 6520 6469 7265 6374 496d * *note directIm │ │ │ │ -00014aa0: 6167 6543 6f6d 706c 6578 3a20 6469 7265 ageComplex: dire │ │ │ │ -00014ab0: 6374 496d 6167 6543 6f6d 706c 6578 2c20 ctImageComplex, │ │ │ │ -00014ac0: 2d2d 2064 6972 6563 7420 696d 6167 6520 -- direct image │ │ │ │ -00014ad0: 636f 6d70 6c65 780a 0a57 6179 7320 746f complex..Ways to │ │ │ │ -00014ae0: 2075 7365 2075 6e69 7665 7273 616c 4578 use universalEx │ │ │ │ -00014af0: 7465 6e73 696f 6e3a 0a3d 3d3d 3d3d 3d3d tension:.======= │ │ │ │ -00014b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00014b10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2275 ========.. * "u │ │ │ │ -00014b20: 6e69 7665 7273 616c 4578 7465 6e73 696f niversalExtensio │ │ │ │ -00014b30: 6e28 4c69 7374 2c4c 6973 7429 220a 0a46 n(List,List)"..F │ │ │ │ -00014b40: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00014b50: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00014b60: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00014b70: 202a 6e6f 7465 2075 6e69 7665 7273 616c *note universal │ │ │ │ -00014b80: 4578 7465 6e73 696f 6e3a 2075 6e69 7665 Extension: unive │ │ │ │ -00014b90: 7273 616c 4578 7465 6e73 696f 6e2c 2069 rsalExtension, i │ │ │ │ -00014ba0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ -00014bb0: 0a66 756e 6374 696f 6e3a 2028 4d61 6361 .function: (Maca │ │ │ │ -00014bc0: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00014bd0: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +00014990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 7420 ----------+..It │ │ │ │ +000149a0: 6973 2069 6e74 6572 6573 7469 6e67 2074 is interesting t │ │ │ │ +000149b0: 6f20 636f 6e73 6964 6572 2074 6865 206c o consider the l │ │ │ │ +000149c0: 6f63 6920 696e 2045 7874 2077 6865 7265 oci in Ext where │ │ │ │ +000149d0: 2074 6865 2065 7874 656e 7369 6f6e 2068 the extension h │ │ │ │ +000149e0: 6173 2061 0a70 6172 7469 6375 6c61 7220 as a.particular │ │ │ │ +000149f0: 7370 6c69 7474 696e 6720 7479 7065 2e20 splitting type. │ │ │ │ +00014a00: 5365 6520 7468 6520 646f 6375 6d65 6e74 See the document │ │ │ │ +00014a10: 6174 696f 6e20 666f 7220 6469 7265 6374 ation for direct │ │ │ │ +00014a20: 496d 6167 6543 6f6d 706c 6578 2066 6f72 ImageComplex for │ │ │ │ +00014a30: 2061 0a63 6f6e 6a65 6374 7572 6520 6162 a.conjecture ab │ │ │ │ +00014a40: 6f75 7420 7468 6520 6571 7561 7469 6f6e out the equation │ │ │ │ +00014a50: 7320 6f66 2074 6865 7365 2076 6172 6965 s of these varie │ │ │ │ +00014a60: 7469 6573 2e0a 0a53 6565 2061 6c73 6f0a ties...See also. │ │ │ │ +00014a70: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00014a80: 6f74 6520 6469 7265 6374 496d 6167 6543 ote directImageC │ │ │ │ +00014a90: 6f6d 706c 6578 3a20 6469 7265 6374 496d omplex: directIm │ │ │ │ +00014aa0: 6167 6543 6f6d 706c 6578 2c20 2d2d 2064 ageComplex, -- d │ │ │ │ +00014ab0: 6972 6563 7420 696d 6167 6520 636f 6d70 irect image comp │ │ │ │ +00014ac0: 6c65 780a 0a57 6179 7320 746f 2075 7365 lex..Ways to use │ │ │ │ +00014ad0: 2075 6e69 7665 7273 616c 4578 7465 6e73 universalExtens │ │ │ │ +00014ae0: 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ion:.=========== │ │ │ │ +00014af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00014b00: 3d3d 3d3d 0a0a 2020 2a20 2275 6e69 7665 ====.. * "unive │ │ │ │ +00014b10: 7273 616c 4578 7465 6e73 696f 6e28 4c69 rsalExtension(Li │ │ │ │ +00014b20: 7374 2c4c 6973 7429 220a 0a46 6f72 2074 st,List)"..For t │ │ │ │ +00014b30: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +00014b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00014b50: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +00014b60: 7465 2075 6e69 7665 7273 616c 4578 7465 te universalExte │ │ │ │ +00014b70: 6e73 696f 6e3a 2075 6e69 7665 7273 616c nsion: universal │ │ │ │ +00014b80: 4578 7465 6e73 696f 6e2c 2069 7320 6120 Extension, is a │ │ │ │ +00014b90: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ +00014ba0: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ +00014bb0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +00014bc0: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +00014bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014c20: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00014c30: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00014c40: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00014c50: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00014c60: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00014c70: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -00014c80: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00014c90: 732f 4247 472e 6d32 3a31 3131 330a 3a30 s/BGG.m2:1113.:0 │ │ │ │ -00014ca0: 2e0a 1f0a 5461 6720 5461 626c 653a 0a4e ....Tag Table:.N │ │ │ │ -00014cb0: 6f64 653a 2054 6f70 7f32 3339 0a4e 6f64 ode: 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│ │ │ │ -00014d90: 6f64 756c 655f 7270 7f34 3433 3539 0a4e odule_rp.44359.N │ │ │ │ -00014da0: 6f64 653a 2045 7874 6572 696f 727f 3537 ode: Exterior.57 │ │ │ │ -00014db0: 3537 350a 4e6f 6465 3a20 7072 6f6a 6563 575.Node: projec │ │ │ │ -00014dc0: 7469 7665 5072 6f64 7563 747f 3538 3231 tiveProduct.5821 │ │ │ │ -00014dd0: 310a 4e6f 6465 3a20 7075 7265 5265 736f 1.Node: pureReso │ │ │ │ -00014de0: 6c75 7469 6f6e 7f35 3937 3435 0a4e 6f64 lution.59745.Nod │ │ │ │ -00014df0: 653a 2052 6567 756c 6172 6974 797f 3732 e: Regularity.72 │ │ │ │ -00014e00: 3237 360a 4e6f 6465 3a20 7379 6d45 7874 276.Node: symExt │ │ │ │ -00014e10: 7f37 3330 3433 0a4e 6f64 653a 2074 6174 .73043.Node: tat │ │ │ │ -00014e20: 6552 6573 6f6c 7574 696f 6e7f 3735 3835 eResolution.7585 │ │ │ │ -00014e30: 310a 4e6f 6465 3a20 756e 6976 6572 7361 1.Node: universa │ │ │ │ -00014e40: 6c45 7874 656e 7369 6f6e 7f37 3935 3734 lExtension.79574 │ │ │ │ -00014e50: 0a1f 0a45 6e64 2054 6167 2054 6162 6c65 ...End Tag Table │ │ │ │ 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cohomologyTa │ │ │ │ +00014ce0: 626c 657f 3230 3736 380a 4e6f 6465 3a20 ble.20768.Node: │ │ │ │ +00014cf0: 6469 7265 6374 496d 6167 6543 6f6d 706c directImageCompl │ │ │ │ +00014d00: 6578 7f33 3030 3037 0a4e 6f64 653a 2064 ex.30007.Node: d │ │ │ │ +00014d10: 6972 6563 7449 6d61 6765 436f 6d70 6c65 irectImageComple │ │ │ │ +00014d20: 785f 6c70 436f 6d70 6c65 785f 7270 7f33 x_lpComplex_rp.3 │ │ │ │ +00014d30: 3134 3033 0a4e 6f64 653a 2064 6972 6563 1403.Node: direc │ │ │ │ +00014d40: 7449 6d61 6765 436f 6d70 6c65 785f 6c70 tImageComplex_lp │ │ │ │ +00014d50: 4d61 7472 6978 5f72 707f 3337 3839 310a Matrix_rp.37891. │ │ │ │ +00014d60: 4e6f 6465 3a20 6469 7265 6374 496d 6167 Node: directImag │ │ │ │ +00014d70: 6543 6f6d 706c 6578 5f6c 704d 6f64 756c eComplex_lpModul │ │ │ │ +00014d80: 655f 7270 7f34 3433 3539 0a4e 6f64 653a e_rp.44359.Node: │ │ │ │ +00014d90: 2045 7874 6572 696f 727f 3537 3537 350a Exterior.57575. │ │ │ │ +00014da0: 4e6f 6465 3a20 7072 6f6a 6563 7469 7665 Node: projective │ │ │ │ +00014db0: 5072 6f64 7563 747f 3538 3231 310a 4e6f Product.58211.No │ │ │ │ +00014dc0: 6465 3a20 7075 7265 5265 736f 6c75 7469 de: pureResoluti │ │ │ │ +00014dd0: 6f6e 7f35 3937 3435 0a4e 6f64 653a 2052 on.59745.Node: R │ │ │ │ +00014de0: 6567 756c 6172 6974 797f 3732 3235 360a egularity.72256. │ │ │ │ +00014df0: 4e6f 6465 3a20 7379 6d45 7874 7f37 3330 Node: symExt.730 │ │ │ │ +00014e00: 3233 0a4e 6f64 653a 2074 6174 6552 6573 23.Node: tateRes │ │ │ │ +00014e10: 6f6c 7574 696f 6e7f 3735 3833 310a 4e6f olution.75831.No │ │ │ │ +00014e20: 6465 3a20 756e 6976 6572 7361 6c45 7874 de: universalExt │ │ │ │ +00014e30: 656e 7369 6f6e 7f37 3935 3534 0a1f 0a45 ension.79554...E │ │ │ │ +00014e40: 6e64 2054 6167 2054 6162 6c65 0a nd Tag Table. │ │ ├── ./usr/share/info/Benchmark.info.gz │ │ │ ├── Benchmark.info │ │ │ │ @@ -201,45 +201,45 @@ │ │ │ │ 00000c80: 3120 3a20 7275 6e42 656e 6368 6d61 726b 1 : runBenchmark │ │ │ │ 00000c90: 7320 2272 6573 3339 2220 2020 2020 2020 s "res39" │ │ │ │ 00000ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000cc0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000cd0: 2d20 6265 6769 6e6e 696e 6720 636f 6d70 - beginning comp │ │ │ │ 00000ce0: 7574 6174 696f 6e20 5361 7420 4f63 7420 utation Sat Oct │ │ │ │ -00000cf0: 3234 2030 333a 3230 3a30 3620 5554 4320 24 03:20:06 UTC │ │ │ │ +00000cf0: 3234 2030 393a 3336 3a34 3620 5554 4320 24 09:36:46 UTC │ │ │ │ 00000d00: 3230 3236 2020 2020 2020 2020 2020 2020 2026 │ │ │ │ 00000d10: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000d20: 2d20 4c69 6e75 7820 7362 7569 6c64 2036 - Linux sbuild 6 │ │ │ │ -00000d30: 2e31 322e 3131 332b 6465 6231 332d 616d .12.113+deb13-am │ │ │ │ -00000d40: 6436 3420 2331 2053 4d50 2050 5245 454d d64 #1 SMP PREEM │ │ │ │ -00000d50: 5054 5f44 594e 414d 4943 2044 6562 6961 PT_DYNAMIC Debia │ │ │ │ -00000d60: 6e20 362e 3132 2e31 3133 2d31 7c0a 7c2d n 6.12.113-1|.|- │ │ │ │ -00000d70: 2d20 414d 4420 4550 5943 2037 3730 3250 - AMD EPYC 7702P │ │ │ │ -00000d80: 2036 342d 436f 7265 2050 726f 6365 7373 64-Core Process │ │ │ │ -00000d90: 6f72 2020 4175 7468 656e 7469 6341 4d44 or AuthenticAMD │ │ │ │ -00000da0: 2020 6370 7520 4d48 7a20 3139 3936 2e32 cpu MHz 1996.2 │ │ │ │ -00000db0: 3530 2020 2020 2020 2020 2020 7c0a 7c2d 50 |.|- │ │ │ │ +00000d30: 2e31 322e 3130 302b 6465 6231 332d 636c .12.100+deb13-cl │ │ │ │ +00000d40: 6f75 642d 616d 6436 3420 2331 2053 4d50 oud-amd64 #1 SMP │ │ │ │ +00000d50: 2050 5245 454d 5054 5f44 594e 414d 4943 PREEMPT_DYNAMIC │ │ │ │ +00000d60: 2044 6562 6961 6e20 2020 2020 7c0a 7c2d Debian |.|- │ │ │ │ +00000d70: 2d20 496e 7465 6c20 5865 6f6e 2050 726f - Intel Xeon Pro │ │ │ │ +00000d80: 6365 7373 6f72 2028 536b 796c 616b 652c cessor (Skylake, │ │ │ │ +00000d90: 2049 4252 5329 2020 4765 6e75 696e 6549 IBRS) GenuineI │ │ │ │ +00000da0: 6e74 656c 2020 6370 7520 4d48 7a20 3231 ntel cpu MHz 21 │ │ │ │ +00000db0: 3030 2e30 3238 2020 2020 2020 7c0a 7c2d 00.028 |.|- │ │ │ │ 00000dc0: 2d20 4d61 6361 756c 6179 3220 312e 3236 - Macaulay2 1.26 │ │ │ │ 00000dd0: 2e30 362c 2063 6f6d 7069 6c65 6420 7769 .06, compiled wi │ │ │ │ 00000de0: 7468 2067 6363 2031 362e 322e 3020 2020 th gcc 16.2.0 │ │ │ │ 00000df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000e00: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 00000e10: 2d20 7265 7333 393a 2072 6573 206f 6620 - res39: res of │ │ │ │ 00000e20: 6120 6765 6e65 7269 6320 3320 6279 2039 a generic 3 by 9 │ │ │ │ 00000e30: 206d 6174 7269 7820 6f76 6572 205a 5a2f matrix over ZZ/ │ │ │ │ -00000e40: 3130 313a 202e 3330 3331 3520 7365 636f 101: .30315 seco │ │ │ │ -00000e50: 6e64 7320 2020 2020 2020 2020 7c0a 7c2d nds |.|- │ │ │ │ +00000e40: 3130 313a 202e 3135 3035 3034 2073 6563 101: .150504 sec │ │ │ │ +00000e50: 6f6e 6473 2020 2020 2020 2020 7c0a 7c2d onds |.|- │ │ │ │ 00000e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c28 ------------|.|( │ │ │ │ -00000eb0: 3230 3236 2d31 302d 3135 2920 7838 365f 2026-10-15) x86_ │ │ │ │ -00000ec0: 3634 2047 4e55 2f4c 696e 7578 2020 2020 64 GNU/Linux │ │ │ │ -00000ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00000ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c36 ------------|.|6 │ │ │ │ +00000eb0: 2e31 322e 3130 302d 3120 2832 3032 362d .12.100-1 (2026- │ │ │ │ +00000ec0: 3037 2d33 3029 2078 3836 5f36 3420 474e 07-30) x86_64 GN │ │ │ │ +00000ed0: 552f 4c69 6e75 7820 2020 2020 2020 2020 U/Linux │ │ │ │ 00000ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00000ef0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 00000f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00000f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 ------------+..F │ │ ├── ./usr/share/info/Bertini.info.gz │ │ │ ├── Bertini.info │ │ │ │ @@ -2253,15 +2253,15 @@ │ │ │ │ 00008cc0: 616c 206e 756d 6265 720a 2020 2020 2020 al number. │ │ │ │ 00008cd0: 2020 6f72 2072 616e 646f 6d20 636f 6d70 or random comp │ │ │ │ 00008ce0: 6c65 7820 6e75 6d62 6572 0a20 2020 2020 lex number. │ │ │ │ 00008cf0: 202a 202a 6e6f 7465 2054 6f70 4469 7265 * *note TopDire │ │ │ │ 00008d00: 6374 6f72 793a 2054 6f70 4469 7265 6374 ctory: TopDirect │ │ │ │ 00008d10: 6f72 792c 203d 3e20 2e2e 2e2c 2064 6566 ory, => ..., def │ │ │ │ 00008d20: 6175 6c74 2076 616c 7565 0a20 2020 2020 ault value. │ │ │ │ -00008d30: 2020 2022 2f74 6d70 2f4d 322d 3233 3636 "/tmp/M2-2366 │ │ │ │ +00008d30: 2020 2022 2f74 6d70 2f4d 322d 3330 3130 "/tmp/M2-3010 │ │ │ │ 00008d40: 392d 302f 3022 2c20 4f70 7469 6f6e 2074 9-0/0", Option t │ │ │ │ 00008d50: 6f20 6368 616e 6765 2064 6972 6563 746f o change directo │ │ │ │ 00008d60: 7279 2066 6f72 2066 696c 6520 7374 6f72 ry for file stor │ │ │ │ 00008d70: 6167 652e 0a20 2020 2020 202a 202a 6e6f age.. * *no │ │ │ │ 00008d80: 7465 2056 6572 626f 7365 3a20 6265 7274 te Verbose: bert │ │ │ │ 00008d90: 696e 6954 7261 636b 486f 6d6f 746f 7079 iniTrackHomotopy │ │ │ │ 00008da0: 5f6c 705f 7064 5f70 645f 7064 5f63 6d56 _lp_pd_pd_pd_cmV │ │ │ │ @@ -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 3336 3639 2d30 2f30 222c p/M2-23669-0/0", │ │ │ │ +00013710: 702f 4d32 2d33 3031 3039 2d30 2f30 222c p/M2-30109-0/0", │ │ │ │ 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 3336 "/tmp/M2-236 │ │ │ │ -00015670: 3639 2d30 2f30 222c 204f 7074 696f 6e20 69-0/0", Option │ │ │ │ +00015660: 2020 2020 222f 746d 702f 4d32 2d33 3031 "/tmp/M2-301 │ │ │ │ +00015670: 3039 2d30 2f30 222c 204f 7074 696f 6e20 09-0/0", Option │ │ │ │ 00015680: 746f 2063 6861 6e67 6520 6469 7265 6374 to change direct │ │ │ │ 00015690: 6f72 7920 666f 7220 6669 6c65 2073 746f ory for file sto │ │ │ │ 000156a0: 7261 6765 2e0a 2020 2020 2020 2a20 5573 rage.. * Us │ │ │ │ 000156b0: 6552 6567 656e 6572 6174 696f 6e20 286d eRegeneration (m │ │ │ │ 000156c0: 6973 7369 6e67 2064 6f63 756d 656e 7461 issing documenta │ │ │ │ 000156d0: 7469 6f6e 2920 3d3e 202e 2e2e 2c20 6465 tion) => ..., de │ │ │ │ 000156e0: 6661 756c 7420 7661 6c75 6520 2d31 2c20 fault value -1, │ │ ├── ./usr/share/info/BettiCharacters.info.gz │ │ │ ├── BettiCharacters.info │ │ │ │ @@ -16843,15 +16843,15 @@ │ │ │ │ 00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 00041cc0: 3920 3a20 656c 6170 7365 6454 696d 6520 9 : elapsedTime │ │ │ │ 00041cd0: 6320 3d20 6368 6172 6163 7465 7220 4120 c = character A │ │ │ │ 00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00041d10: 2d2d 202e 3434 3931 3832 7320 656c 6170 -- .449182s elap │ │ │ │ +00041d10: 2d2d 202e 3234 3337 3532 7320 656c 6170 -- .243752s elap │ │ │ │ 00041d20: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 00041d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 00041d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -18058,16 +18058,16 @@ │ │ │ │ 00046890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000468a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000468b0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 656c ------+.|i7 : el │ │ │ │ 000468c0: 6170 7365 6454 696d 6520 633d 6368 6172 apsedTime c=char │ │ │ │ 000468d0: 6163 7465 7220 4120 2020 2020 2020 2020 acter A │ │ │ │ 000468e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000468f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046900: 2020 2020 2020 7c0a 7c20 2d2d 2031 2e30 |.| -- 1.0 │ │ │ │ -00046910: 3632 3635 7320 656c 6170 7365 6420 2020 6265s elapsed │ │ │ │ +00046900: 2020 2020 2020 7c0a 7c20 2d2d 202e 3338 |.| -- .38 │ │ │ │ +00046910: 3935 3333 7320 656c 6170 7365 6420 2020 9533s elapsed │ │ │ │ 00046920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046950: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00046960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19501,15 +19501,15 @@ │ │ │ │ 0004c2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c2f0: 2b0a 7c69 3230 203a 2065 6c61 7073 6564 +.|i20 : elapsed │ │ │ │ 0004c300: 5469 6d65 2061 3120 3d20 6368 6172 6163 Time a1 = charac │ │ │ │ 0004c310: 7465 7220 4131 2020 2020 2020 2020 2020 ter A1 │ │ │ │ 0004c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004c330: 2020 7c0a 7c20 2d2d 2031 2e30 3637 3132 |.| -- 1.06712 │ │ │ │ +0004c330: 2020 7c0a 7c20 2d2d 202e 3536 3339 3334 |.| -- .563934 │ │ │ │ 0004c340: 7320 656c 6170 7365 6420 2020 2020 2020 s elapsed │ │ │ │ 0004c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c370: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 0004c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -19555,15 +19555,15 @@ │ │ │ │ 0004c620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004c640: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 ----------+.|i21 │ │ │ │ 0004c650: 203a 2065 6c61 7073 6564 5469 6d65 2061 : elapsedTime a │ │ │ │ 0004c660: 3220 3d20 6368 6172 6163 7465 7220 4132 2 = character A2 │ │ │ │ 0004c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0004c690: 2d2d 2033 362e 3335 3339 7320 656c 6170 -- 36.3539s elap │ │ │ │ +0004c690: 2d2d 2032 312e 3333 3737 7320 656c 6170 -- 21.3377s elap │ │ │ │ 0004c6a0: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0004c6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 0004c6d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0004c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -20066,15 +20066,15 @@ │ │ │ │ 0004e610: 4f6e 4772 6164 6564 4d6f 6475 6c65 2020 OnGradedModule │ │ │ │ 0004e620: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ 0004e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004e640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004e650: 2d2b 0a7c 6933 3220 3a20 656c 6170 7365 -+.|i32 : elapse │ │ │ │ 0004e660: 6454 696d 6520 6220 3d20 6368 6172 6163 dTime b = charac │ │ │ │ 0004e670: 7465 7228 422c 3231 297c 0a7c 202d 2d20 ter(B,21)|.| -- │ │ │ │ -0004e680: 3136 2e34 3237 3773 2065 6c61 7073 6564 16.4277s elapsed │ │ │ │ +0004e680: 392e 3637 3732 7320 656c 6170 7365 6420 9.6772s elapsed │ │ │ │ 0004e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004e6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0004e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004e6c0: 2020 2020 2020 2020 207c 0a7c 6f33 3220 |.|o32 │ │ │ │ 0004e6d0: 3d20 4368 6172 6163 7465 7220 6f76 6572 = Character over │ │ │ │ 0004e6e0: 206b 6b20 2020 2020 2020 2020 2020 2020 kk │ │ │ │ 0004e6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ ├── ./usr/share/info/Bruns.info.gz │ │ │ ├── Bruns.info │ │ │ │ @@ -1095,17 +1095,17 @@ │ │ │ │ 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 2e34 3032 3236 -- used 0.40226 │ │ │ │ -000044e0: 3873 2028 6370 7529 3b20 302e 3234 3632 8s (cpu); 0.2462 │ │ │ │ -000044f0: 3439 7320 2874 6872 6561 6429 3b20 3073 49s (thread); 0s │ │ │ │ +000044d0: 202d 2d20 7573 6564 2030 2e33 3438 3435 -- used 0.34845 │ │ │ │ +000044e0: 3373 2028 6370 7529 3b20 302e 3232 3038 3s (cpu); 0.2208 │ │ │ │ +000044f0: 3831 7320 2874 6872 6561 6429 3b20 3073 81s (thread); 0s │ │ │ │ 00004500: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ 00004510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00004520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004560: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ ├── ./usr/share/info/ChainComplexExtras.info.gz │ │ │ ├── ChainComplexExtras.info │ │ │ │ @@ -6049,17 +6049,17 @@ │ │ │ │ 00017a00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a30: 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a 2074 ------+.|i13 : t │ │ │ │ 00017a40: 696d 6520 6d20 3d20 6d69 6e69 6d69 7a65 ime m = minimize │ │ │ │ 00017a50: 2028 455b 315d 293b 2020 2020 2020 2020 (E[1]); │ │ │ │ 00017a60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00017a70: 7c20 2d2d 2075 7365 6420 302e 3239 3635 | -- used 0.2965 │ │ │ │ -00017a80: 3733 7320 2863 7075 293b 2030 2e32 3137 73s (cpu); 0.217 │ │ │ │ -00017a90: 3036 3473 2028 7468 7265 6164 293b 2030 064s (thread); 0 │ │ │ │ +00017a70: 7c20 2d2d 2075 7365 6420 302e 3239 3735 | -- used 0.2975 │ │ │ │ +00017a80: 3934 7320 2863 7075 293b 2030 2e32 3232 94s (cpu); 0.222 │ │ │ │ +00017a90: 3238 3973 2028 7468 7265 6164 293b 2030 289s (thread); 0 │ │ │ │ 00017aa0: 7320 2867 6329 7c0a 2b2d 2d2d 2d2d 2d2d s (gc)|.+------- │ │ │ │ 00017ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00017ae0: 7c69 3134 203a 2069 7351 7561 7369 4973 |i14 : isQuasiIs │ │ │ │ 00017af0: 6f6d 6f72 7068 6973 6d20 6d20 2020 2020 omorphism m │ │ │ │ 00017b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -7810,32 +7810,32 @@ │ │ │ │ 0001e810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e820: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 -------+.|i8 : t │ │ │ │ 0001e830: 696d 6520 6d20 3d20 7265 736f 6c75 7469 ime m = resoluti │ │ │ │ 0001e840: 6f6e 4f66 4368 6169 6e43 6f6d 706c 6578 onOfChainComplex │ │ │ │ 0001e850: 2043 3b20 2020 2020 2020 2020 2020 2020 C; │ │ │ │ 0001e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e870: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0001e880: 6564 2030 2e30 3939 3530 3833 7320 2863 ed 0.0995083s (c │ │ │ │ -0001e890: 7075 293b 2030 2e30 3938 3736 3537 7320 pu); 0.0987657s │ │ │ │ -0001e8a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0001e8b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001e880: 6564 2030 2e30 3932 3139 7320 2863 7075 ed 0.09219s (cpu │ │ │ │ +0001e890: 293b 2030 2e30 3931 3637 3833 7320 2874 ); 0.0916783s (t │ │ │ │ +0001e8a0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8c0: 2020 2020 2020 207c 0a2b 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 2d2d 2d2d ---------------- │ │ │ │ 0001e900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e910: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2074 -------+.|i9 : t │ │ │ │ 0001e920: 696d 6520 6e20 3d20 6361 7274 616e 4569 ime n = cartanEi │ │ │ │ 0001e930: 6c65 6e62 6572 6752 6573 6f6c 7574 696f lenbergResolutio │ │ │ │ 0001e940: 6e20 433b 2020 2020 2020 2020 2020 2020 n C; │ │ │ │ 0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e960: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -0001e970: 6564 2030 2e31 3133 3236 3173 2028 6370 ed 0.113261s (cp │ │ │ │ -0001e980: 7529 3b20 302e 3131 3439 7320 2874 6872 u); 0.1149s (thr │ │ │ │ -0001e990: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +0001e970: 6564 2030 2e31 3132 3835 3873 2028 6370 ed 0.112858s (cp │ │ │ │ +0001e980: 7529 3b20 302e 3131 3631 3834 7320 2874 u); 0.116184s (t │ │ │ │ +0001e990: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0001e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ 0001e9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ ├── ./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 3732 3834 3173 2028 used 0.272841s ( │ │ │ │ -00004c60: 6370 7529 3b20 302e 3137 3635 3436 7320 cpu); 0.176546s │ │ │ │ +00004c50: 7573 6564 2030 2e32 3438 3135 3873 2028 used 0.248158s ( │ │ │ │ +00004c60: 6370 7529 3b20 302e 3136 3738 3937 7320 cpu); 0.167897s │ │ │ │ 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 2e34 3134 3636 3273 2028 used 0.414662s ( │ │ │ │ -000051b0: 6370 7529 3b20 302e 3332 3231 3836 7320 cpu); 0.322186s │ │ │ │ +000051a0: 7573 6564 2030 2e33 3536 3731 3173 2028 used 0.356711s ( │ │ │ │ +000051b0: 6370 7529 3b20 302e 3238 3138 3338 7320 cpu); 0.281838s │ │ │ │ 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 3535 3336 3233 -- used 0.553623 │ │ │ │ -00010fc0: 7320 2863 7075 293b 2030 2e33 3833 3036 s (cpu); 0.38306 │ │ │ │ -00010fd0: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ -00010fe0: 2867 6329 2020 2020 2020 2020 2020 207c (gc) | │ │ │ │ +00010fb0: 2d2d 2075 7365 6420 302e 3537 3839 3773 -- used 0.57897s │ │ │ │ +00010fc0: 2028 6370 7529 3b20 302e 3332 3130 3973 (cpu); 0.32109s │ │ │ │ +00010fd0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ +00010fe0: 6329 2020 2020 2020 2020 2020 2020 207c c) | │ │ │ │ 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 3331 3331 3373 2028 6370 7529 d 2.31313s (cpu) │ │ │ │ -00011370: 3b20 312e 3835 3435 3173 2028 7468 7265 ; 1.85451s (thre │ │ │ │ +00011360: 6420 322e 3031 3236 3973 2028 6370 7529 d 2.01269s (cpu) │ │ │ │ +00011370: 3b20 312e 3737 3439 3173 2028 7468 7265 ; 1.77491s (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 2e33 3032 3131 3273 2028 6370 7529 0.302112s (cpu) │ │ │ │ -000118f0: 3b20 302e 3230 3531 3735 7320 2874 6872 ; 0.205175s (thr │ │ │ │ +000118e0: 2030 2e32 3630 3333 3473 2028 6370 7529 0.260334s (cpu) │ │ │ │ +000118f0: 3b20 302e 3139 3036 3739 7320 2874 6872 ; 0.190679s (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 3837 .| -- used 0.087 │ │ │ │ -00011c90: 3934 3638 7320 2863 7075 293b 2030 2e30 9468s (cpu); 0.0 │ │ │ │ -00011ca0: 3837 3935 3573 2028 7468 7265 6164 293b 87955s (thread); │ │ │ │ -00011cb0: 2030 7320 2867 6329 2020 2020 2020 2020 0s (gc) │ │ │ │ +00011c80: 0a7c 202d 2d20 7573 6564 2030 2e30 3932 .| -- used 0.092 │ │ │ │ +00011c90: 3336 3131 7320 2863 7075 293b 2030 2e30 3611s (cpu); 0.0 │ │ │ │ +00011ca0: 3932 3336 3736 7320 2874 6872 6561 6429 923676s (thread) │ │ │ │ +00011cb0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00011cc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00011cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00011d00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00011d10: 3320 2020 2020 3220 2020 2020 2020 2020 3 2 │ │ │ │ 00011d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5545,16 +5545,16 @@ │ │ │ │ 00015a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015ab0: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520 --+.|i15 : time │ │ │ │ 00015ac0: 6373 6d4b 3d43 534d 2841 2c4b 2920 2020 csmK=CSM(A,K) │ │ │ │ 00015ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015ae0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -00015af0: 2075 7365 6420 302e 3339 3939 3339 7320 used 0.399939s │ │ │ │ -00015b00: 2863 7075 293b 2030 2e33 3130 3437 3373 (cpu); 0.310473s │ │ │ │ +00015af0: 2075 7365 6420 302e 3832 3537 3931 7320 used 0.825791s │ │ │ │ +00015b00: 2863 7075 293b 2030 2e33 3136 3235 3873 (cpu); 0.316258s │ │ │ │ 00015b10: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ 00015b20: 6329 7c0a 7c20 2020 2020 2020 2020 2020 c)|.| │ │ │ │ 00015b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015b50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 00015b60: 2020 2020 2032 2032 2020 2020 2032 2020 2 2 2 │ │ │ │ 00015b70: 2020 2020 2020 2032 2020 2020 3220 2020 2 2 │ │ │ │ @@ -5724,17 +5724,17 @@ │ │ │ │ 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 3832 3638 3673 2028 6370 ed 0.182686s (cp │ │ │ │ -00016630: 7529 3b20 302e 3130 3736 3831 7320 2874 u); 0.107681s (t │ │ │ │ -00016640: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00016620: 6564 2030 2e32 3035 3431 3573 2028 6370 ed 0.205415s (cp │ │ │ │ +00016630: 7529 3b20 302e 3039 3137 3236 3173 2028 u); 0.0917261s ( │ │ │ │ +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 │ │ │ │ @@ -6694,16 +6694,16 @@ │ │ │ │ 0001a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a260: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 696d -----+.|i4 : tim │ │ │ │ 0001a270: 6520 4575 6c65 7228 492c 496e 7075 7449 e Euler(I,InputI │ │ │ │ 0001a280: 7353 6d6f 6f74 683d 3e74 7275 6529 2020 sSmooth=>true) │ │ │ │ 0001a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a2b0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a2c0: 2030 2e30 3337 3639 3636 7320 2863 7075 0.0376966s (cpu │ │ │ │ -0001a2d0: 293b 2030 2e30 3336 3939 3531 7320 2874 ); 0.0369951s (t │ │ │ │ +0001a2c0: 2030 2e30 3638 3032 3133 7320 2863 7075 0.0680213s (cpu │ │ │ │ +0001a2d0: 293b 2030 2e30 3432 3631 3236 7320 2874 ); 0.0426126s (t │ │ │ │ 0001a2e0: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 0001a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a300: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001a310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6719,16 +6719,16 @@ │ │ │ │ 0001a3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a3f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 696d -----+.|i5 : tim │ │ │ │ 0001a400: 6520 4575 6c65 7220 4920 2020 2020 2020 e Euler I │ │ │ │ 0001a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a440: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -0001a450: 2030 2e32 3537 3335 3473 2028 6370 7529 0.257354s (cpu) │ │ │ │ -0001a460: 3b20 302e 3136 3434 3435 7320 2874 6872 ; 0.164445s (thr │ │ │ │ +0001a450: 2030 2e32 3531 3339 3873 2028 6370 7529 0.251398s (cpu) │ │ │ │ +0001a460: 3b20 302e 3134 3530 3832 7320 2874 6872 ; 0.145082s (thr │ │ │ │ 0001a470: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 0001a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a490: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001a4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -6922,18 +6922,18 @@ │ │ │ │ 0001b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b0b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ 0001b0c0: 3a20 7469 6d65 2045 756c 6572 284a 2c4d : time Euler(J,M │ │ │ │ 0001b0d0: 6574 686f 643d 3e44 6972 6563 7443 6f6d ethod=>DirectCom │ │ │ │ 0001b0e0: 706c 6574 6549 6e74 2920 2020 2020 2020 pleteInt) │ │ │ │ 0001b0f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001b100: 202d 2d20 7573 6564 2030 2e30 3739 3233 -- used 0.07923 │ │ │ │ -0001b110: 3233 7320 2863 7075 293b 2030 2e30 3735 23s (cpu); 0.075 │ │ │ │ -0001b120: 3436 3438 7320 2874 6872 6561 6429 3b20 4648s (thread); │ │ │ │ -0001b130: 3073 2028 6763 2920 2020 2020 2020 2020 0s (gc) │ │ │ │ +0001b100: 202d 2d20 7573 6564 2030 2e31 3030 3732 -- used 0.10072 │ │ │ │ +0001b110: 3773 2028 6370 7529 3b20 302e 3037 3433 7s (cpu); 0.0743 │ │ │ │ +0001b120: 3939 3173 2028 7468 7265 6164 293b 2030 991s (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 │ │ │ │ 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 │ │ │ │ @@ -6944,17 +6944,17 @@ │ │ │ │ 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: 3935 3536 3273 2028 6370 7529 3b20 302e 95562s (cpu); 0. │ │ │ │ -0001b270: 3130 3337 3035 7320 2874 6872 6561 6429 103705s (thread) │ │ │ │ -0001b280: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0001b260: 3836 3838 3873 2028 6370 7529 3b20 302e 86888s (cpu); 0. │ │ │ │ +0001b270: 3038 3538 3231 3673 2028 7468 7265 6164 0858216s (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 │ │ │ │ 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 │ │ │ │ @@ -7515,4721 +7515,4724 @@ │ │ │ │ 0001d5a0: 6573 2074 6865 2045 756c 6572 2063 6861 es the Euler cha │ │ │ │ 0001d5b0: 7261 6374 6572 6973 7469 6320 6f66 2061 racteristic of a │ │ │ │ 0001d5c0: 2063 6f6d 706c 6578 2061 6666 696e 6520 complex affine │ │ │ │ 0001d5d0: 7661 7269 6574 792e 0a0a 2b2d 2d2d 2d2d variety...+----- │ │ │ │ 0001d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d610: 2d2b 0a7c 6931 203a 206b 6b3d 5a5a 2f33 -+.|i1 : kk=ZZ/3 │ │ │ │ -0001d620: 3237 3439 3b20 2020 2020 2020 2020 2020 2749; │ │ │ │ +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 7c0a 2b2d 2d2d |.+--- │ │ │ │ +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 2d2b 0a7c 6932 203a 2052 3d6b 6b5b ---+.|i2 : R=kk[ │ │ │ │ -0001d690: 785f 312e 2e78 5f33 5d20 2020 2020 2020 x_1..x_3] │ │ │ │ +0001d680: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 523d ------+.|i2 : R= │ │ │ │ +0001d690: 6b6b 5b78 5f31 2e2e 785f 335d 2020 2020 kk[x_1..x_3] │ │ │ │ 0001d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001d6c0: 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 |.| │ │ │ │ 0001d6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d6f0: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ -0001d700: 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 │ │ │ │ 0001d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d720: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001d730: 7c20 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 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -0001d770: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ -0001d780: 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: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +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 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0001d7e0: 2049 3d69 6465 616c 2878 5f31 5e32 2b78 I=ideal(x_1^2+x │ │ │ │ -0001d7f0: 5f32 5e32 2b78 5f33 5e32 2d31 2920 2020 _2^2+x_3^2-1) │ │ │ │ -0001d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d810: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 │ │ │ │ -0001d840: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001d850: 2020 2020 2020 2020 2020 3220 2020 2032 2 2 │ │ │ │ -0001d860: 2020 2020 3220 2020 2020 2020 2020 2020 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 │ │ │ │ -0001d880: 2020 2020 7c0a 7c6f 3320 3d20 6964 6561 |.|o3 = idea │ │ │ │ -0001d890: 6c28 7820 202b 2078 2020 2b20 7820 202d l(x + x + x - │ │ │ │ -0001d8a0: 2031 2920 2020 2020 2020 2020 2020 2020 1) │ │ │ │ -0001d8b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001d8c0: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -0001d8d0: 2032 2020 2020 3320 2020 2020 2020 2020 2 3 │ │ │ │ -0001d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d8f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ +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 │ │ │ │ -0001d920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001d930: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ -0001d940: 5220 2020 2020 2020 2020 2020 2020 2020 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 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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: 2d2b 0a7c 6934 203a 2074 696d 6520 4575 -+.|i4 : time Eu │ │ │ │ -0001d9b0: 6c65 7241 6666 696e 6520 4920 2020 2020 lerAffine I │ │ │ │ -0001d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001d9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0001d9e0: 2075 7365 6420 302e 3034 3731 3038 7320 used 0.047108s │ │ │ │ -0001d9f0: 2863 7075 293b 2030 2e30 3436 3733 3439 (cpu); 0.0467349 │ │ │ │ -0001da00: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0001da10: 6763 297c 0a7c 2020 2020 2020 2020 2020 gc)|.| │ │ │ │ -0001da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 3036 3135 3731 -- used 0.061571 │ │ │ │ +0001da00: 3473 2028 6370 7529 3b20 302e 3035 3430 4s (cpu); 0.0540 │ │ │ │ +0001da10: 3635 3773 2028 7468 7265 6164 293b 2030 657s (thread); 0 │ │ │ │ +0001da20: 7320 2867 6329 7c0a 7c20 2020 2020 2020 s (gc)|.| │ │ │ │ 0001da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001da50: 3420 3d20 3220 2020 2020 2020 2020 2020 4 = 2 │ │ │ │ -0001da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001da60: 7c0a 7c6f 3420 3d20 3220 2020 2020 2020 |.|o4 = 2 │ │ │ │ 0001da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001da80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0001da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 2b0a --------------+. │ │ │ │ -0001dac0: 0a4f 6273 6572 7665 2074 6861 7420 7468 .Observe that th │ │ │ │ -0001dad0: 6520 616c 676f 7269 7468 6d20 6973 2061 e algorithm is a │ │ │ │ -0001dae0: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ -0001daf0: 6c67 6f72 6974 686d 2061 6e64 206d 6179 lgorithm and may │ │ │ │ -0001db00: 2067 6976 6520 6120 7772 6f6e 670a 616e give a wrong.an │ │ │ │ -0001db10: 7377 6572 2077 6974 6820 6120 736d 616c swer with a smal │ │ │ │ -0001db20: 6c20 6275 7420 6e6f 6e7a 6572 6f20 7072 l but nonzero pr │ │ │ │ -0001db30: 6f62 6162 696c 6974 792e 2052 6561 6420 obability. Read │ │ │ │ -0001db40: 6d6f 7265 2075 6e64 6572 202a 6e6f 7465 more under *note │ │ │ │ -0001db50: 0a70 726f 6261 6269 6c69 7374 6963 2061 .probabilistic a │ │ │ │ -0001db60: 6c67 6f72 6974 686d 3a20 7072 6f62 6162 lgorithm: probab │ │ │ │ -0001db70: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ -0001db80: 6d2c 2e0a 0a57 6179 7320 746f 2075 7365 m,...Ways to use │ │ │ │ -0001db90: 2045 756c 6572 4166 6669 6e65 3a0a 3d3d EulerAffine:.== │ │ │ │ -0001dba0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001dbb0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 756c ======.. * "Eul │ │ │ │ -0001dbc0: 6572 4166 6669 6e65 2849 6465 616c 2922 erAffine(Ideal)" │ │ │ │ -0001dbd0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0001dbe0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0001dbf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0001dc00: 6563 7420 2a6e 6f74 6520 4575 6c65 7241 ect *note EulerA │ │ │ │ -0001dc10: 6666 696e 653a 2045 756c 6572 4166 6669 ffine: EulerAffi │ │ │ │ -0001dc20: 6e65 2c20 6973 2061 202a 6e6f 7465 206d ne, is a *note m │ │ │ │ -0001dc30: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -0001dc40: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0001dc50: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ -0001dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dc70: 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,...---------- │ │ │ │ 0001dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0001dcb0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0001dcc0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0001dcd0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0001dce0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0001dcf0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0001dd00: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0001dd10: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ -0001dd20: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ -0001dd30: 3a32 3534 313a 302e 0a1f 0a46 696c 653a :2541:0....File: │ │ │ │ -0001dd40: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ -0001dd50: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ -0001dd60: 653a 2049 6e64 734f 6653 6d6f 6f74 682c e: IndsOfSmooth, │ │ │ │ -0001dd70: 204e 6578 743a 2049 6e70 7574 4973 536d Next: InputIsSm │ │ │ │ -0001dd80: 6f6f 7468 2c20 5072 6576 3a20 4575 6c65 ooth, Prev: Eule │ │ │ │ -0001dd90: 7241 6666 696e 652c 2055 703a 2054 6f70 rAffine, Up: Top │ │ │ │ -0001dda0: 0a0a 496e 6473 4f66 536d 6f6f 7468 0a2a ..IndsOfSmooth.* │ │ │ │ -0001ddb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 ***********..Des │ │ │ │ -0001ddc0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0001ddd0: 3d3d 3d3d 0a0a 5468 6520 6f70 7469 6f6e ====..The option │ │ │ │ -0001dde0: 2049 6e64 734f 6653 6d6f 6f74 6820 6973 IndsOfSmooth is │ │ │ │ -0001ddf0: 206f 6e6c 7920 7573 6564 2062 7920 7468 only used by th │ │ │ │ -0001de00: 6520 636f 6d6d 616e 6473 202a 6e6f 7465 e commands *note │ │ │ │ -0001de10: 2043 534d 3a20 4353 4d2c 2c20 616e 6420 CSM: CSM,, and │ │ │ │ -0001de20: 2a6e 6f74 650a 4575 6c65 723a 2045 756c *note.Euler: Eul │ │ │ │ -0001de30: 6572 2c20 696e 2063 6f6d 6269 6e61 7469 er, in combinati │ │ │ │ -0001de40: 6f6e 2077 6974 6820 7468 6520 6f70 7469 on with the opti │ │ │ │ -0001de50: 6f6e 204d 6574 686f 643d 3e44 6972 6563 on Method=>Direc │ │ │ │ -0001de60: 7443 6f6d 706c 6574 496e 742e 2057 6865 tCompletInt. Whe │ │ │ │ -0001de70: 6e0a 7573 6564 2074 6869 7320 6f70 7469 n.used this opti │ │ │ │ -0001de80: 6f6e 206d 6179 2061 6c6c 6f77 2074 6865 on may allow the │ │ │ │ -0001de90: 2075 7365 7220 746f 2073 7065 6564 2075 user to speed u │ │ │ │ -0001dea0: 7020 7468 6520 636f 6d70 7574 6174 696f p the computatio │ │ │ │ -0001deb0: 6e20 6279 2074 656c 6c69 6e67 0a67 6976 n by telling.giv │ │ │ │ -0001dec0: 696e 6720 7468 6520 6d65 7468 6f64 2061 ing the method a │ │ │ │ -0001ded0: 206c 6973 7420 6f66 2069 6e64 6963 6573 list of indices │ │ │ │ -0001dee0: 2066 6f72 2074 6865 2067 656e 6572 6174 for the generat │ │ │ │ -0001def0: 6f72 7320 6f66 2074 6865 2069 6e70 7574 ors of the input │ │ │ │ -0001df00: 2069 6465 616c 2074 6861 742c 0a77 6865 ideal that,.whe │ │ │ │ -0001df10: 6e20 7461 6b65 6e20 746f 6765 7468 6572 n taken together │ │ │ │ -0001df20: 2c20 6465 6669 6e65 2061 2073 6d6f 6f74 , define a smoot │ │ │ │ -0001df30: 6820 7375 6273 6368 656d 6520 6f66 2074 h subscheme of t │ │ │ │ -0001df40: 6865 2061 6d62 6965 6e74 2073 7061 6365 he ambient space │ │ │ │ -0001df50: 2e20 5468 6973 0a6f 7074 696f 6e20 7769 . This.option wi │ │ │ │ -0001df60: 6c6c 2062 6520 6967 6e6f 7265 6420 6f74 ll be ignored ot │ │ │ │ -0001df70: 6865 7277 6973 652e 0a0a 2b2d 2d2d 2d2d herwise...+----- │ │ │ │ -0001df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dcc0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0001dcd0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0001dce0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0001dcf0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0001dd00: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0001dd10: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0001dd20: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ +0001dd30: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +0001dd40: 7365 732e 6d32 3a32 3534 313a 302e 0a1f ses.m2:2541:0... │ │ │ │ +0001dd50: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ +0001dd60: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ +0001dd70: 6f2c 204e 6f64 653a 2049 6e64 734f 6653 o, Node: IndsOfS │ │ │ │ +0001dd80: 6d6f 6f74 682c 204e 6578 743a 2049 6e70 mooth, Next: Inp │ │ │ │ +0001dd90: 7574 4973 536d 6f6f 7468 2c20 5072 6576 utIsSmooth, Prev │ │ │ │ +0001dda0: 3a20 4575 6c65 7241 6666 696e 652c 2055 : EulerAffine, U │ │ │ │ +0001ddb0: 703a 2054 6f70 0a0a 496e 6473 4f66 536d p: Top..IndsOfSm │ │ │ │ +0001ddc0: 6f6f 7468 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ooth.*********** │ │ │ │ +0001ddd0: 2a0a 0a44 6573 6372 6970 7469 6f6e 0a3d *..Description.= │ │ │ │ +0001dde0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0001ddf0: 6f70 7469 6f6e 2049 6e64 734f 6653 6d6f option IndsOfSmo │ │ │ │ +0001de00: 6f74 6820 6973 206f 6e6c 7920 7573 6564 oth is only used │ │ │ │ +0001de10: 2062 7920 7468 6520 636f 6d6d 616e 6473 by the commands │ │ │ │ +0001de20: 202a 6e6f 7465 2043 534d 3a20 4353 4d2c *note CSM: CSM, │ │ │ │ +0001de30: 2c20 616e 6420 2a6e 6f74 650a 4575 6c65 , and *note.Eule │ │ │ │ +0001de40: 723a 2045 756c 6572 2c20 696e 2063 6f6d r: Euler, in com │ │ │ │ +0001de50: 6269 6e61 7469 6f6e 2077 6974 6820 7468 bination with th │ │ │ │ +0001de60: 6520 6f70 7469 6f6e 204d 6574 686f 643d e option Method= │ │ │ │ +0001de70: 3e44 6972 6563 7443 6f6d 706c 6574 496e >DirectCompletIn │ │ │ │ +0001de80: 742e 2057 6865 6e0a 7573 6564 2074 6869 t. When.used thi │ │ │ │ +0001de90: 7320 6f70 7469 6f6e 206d 6179 2061 6c6c s option may all │ │ │ │ +0001dea0: 6f77 2074 6865 2075 7365 7220 746f 2073 ow the user to s │ │ │ │ +0001deb0: 7065 6564 2075 7020 7468 6520 636f 6d70 peed up the comp │ │ │ │ +0001dec0: 7574 6174 696f 6e20 6279 2074 656c 6c69 utation by telli │ │ │ │ +0001ded0: 6e67 0a67 6976 696e 6720 7468 6520 6d65 ng.giving the me │ │ │ │ +0001dee0: 7468 6f64 2061 206c 6973 7420 6f66 2069 thod a list of i │ │ │ │ +0001def0: 6e64 6963 6573 2066 6f72 2074 6865 2067 ndices for the g │ │ │ │ +0001df00: 656e 6572 6174 6f72 7320 6f66 2074 6865 enerators of the │ │ │ │ +0001df10: 2069 6e70 7574 2069 6465 616c 2074 6861 input ideal tha │ │ │ │ +0001df20: 742c 0a77 6865 6e20 7461 6b65 6e20 746f t,.when taken to │ │ │ │ +0001df30: 6765 7468 6572 2c20 6465 6669 6e65 2061 gether, define a │ │ │ │ +0001df40: 2073 6d6f 6f74 6820 7375 6273 6368 656d smooth subschem │ │ │ │ +0001df50: 6520 6f66 2074 6865 2061 6d62 6965 6e74 e of the ambient │ │ │ │ +0001df60: 2073 7061 6365 2e20 5468 6973 0a6f 7074 space. This.opt │ │ │ │ +0001df70: 696f 6e20 7769 6c6c 2062 6520 6967 6e6f ion will be igno │ │ │ │ +0001df80: 7265 6420 6f74 6865 7277 6973 652e 0a0a red otherwise... │ │ │ │ +0001df90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0001dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001dfc0: 0a7c 6931 203a 2052 203d 204d 756c 7469 .|i1 : R = Multi │ │ │ │ -0001dfd0: 5072 6f6a 436f 6f72 6452 696e 6728 7b32 ProjCoordRing({2 │ │ │ │ -0001dfe0: 2c32 7d29 2020 2020 2020 2020 2020 2020 ,2}) │ │ │ │ -0001dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e000: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001dfd0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2052 203d -----+.|i1 : R = │ │ │ │ +0001dfe0: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ +0001dff0: 696e 6728 7b32 2c32 7d29 2020 2020 2020 ing({2,2}) │ │ │ │ +0001e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e010: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e040: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e050: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ -0001e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ 0001e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e090: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001e0a0: 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 207c 0a7c 6f31 |.|o1 │ │ │ │ -0001e0e0: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -0001e0f0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -0001e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e0f0: 207c 0a7c 6f31 203a 2050 6f6c 796e 6f6d |.|o1 : Polynom │ │ │ │ +0001e100: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ 0001e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e120: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0001e130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0001e170: 2049 3d69 6465 616c 2852 5f30 2a52 5f31 I=ideal(R_0*R_1 │ │ │ │ -0001e180: 2a52 5f33 2d52 5f30 5e32 2a52 5f33 2c72 *R_3-R_0^2*R_3,r │ │ │ │ -0001e190: 616e 646f 6d28 7b30 2c31 7d2c 5229 2c72 andom({0,1},R),r │ │ │ │ -0001e1a0: 616e 646f 6d28 7b31 2c32 7d2c 5229 293b andom({1,2},R)); │ │ │ │ -0001e1b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ +0001e190: 5f30 2a52 5f31 2a52 5f33 2d52 5f30 5e32 _0*R_1*R_3-R_0^2 │ │ │ │ +0001e1a0: 2a52 5f33 2c72 616e 646f 6d28 7b30 2c31 *R_3,random({0,1 │ │ │ │ +0001e1b0: 7d2c 5229 2c72 616e 646f 6d28 7b31 2c32 },R),random({1,2 │ │ │ │ +0001e1c0: 7d2c 5229 293b 7c0a 7c20 2020 2020 2020 },R));|.| │ │ │ │ 0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e1f0: 2020 2020 2020 207c 0a7c 6f32 203a 2049 |.|o2 : I │ │ │ │ -0001e200: 6465 616c 206f 6620 5220 2020 2020 2020 deal of R │ │ │ │ -0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a |. │ │ │ │ -0001e240: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0001e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 2d2b 0a7c 6933 203a 2074 696d -----+.|i3 : tim │ │ │ │ -0001e290: 6520 4353 4d28 492c 4d65 7468 6f64 3d3e e CSM(I,Method=> │ │ │ │ -0001e2a0: 4469 7265 6374 436f 6d70 6c65 7449 6e74 DirectCompletInt │ │ │ │ -0001e2b0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001e2d0: 2d2d 2075 7365 6420 312e 3732 3139 3673 -- used 1.72196s │ │ │ │ -0001e2e0: 2028 6370 7529 3b20 312e 3136 3739 3273 (cpu); 1.16792s │ │ │ │ -0001e2f0: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0001e300: 6329 2020 2020 2020 2020 2020 2020 2020 c) │ │ │ │ -0001e310: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 332e |.| -- used 3. │ │ │ │ +0001e2f0: 3837 3235 3773 2028 6370 7529 3b20 312e 87257s (cpu); 1. │ │ │ │ +0001e300: 3039 3031 3473 2028 7468 7265 6164 293b 09014s (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 7c0a 7c20 2020 |.| │ │ │ │ -0001e360: 2020 2020 3220 3220 2020 2020 3220 2020 2 2 2 │ │ │ │ -0001e370: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -0001e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e370: 7c0a 7c20 2020 2020 2020 3220 3220 2020 |.| 2 2 │ │ │ │ +0001e380: 2020 3220 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ 0001e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3a0: 207c 0a7c 6f33 203d 2032 6820 6820 202b |.|o3 = 2h h + │ │ │ │ -0001e3b0: 2032 6820 6820 202b 2035 6820 6820 2020 2h h + 5h h │ │ │ │ -0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001e3f0: 2020 3120 3220 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ -0001e400: 2020 3120 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ -0001e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e420: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001e430: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0001e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ +0001e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e440: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0001e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e470: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ -0001e480: 5b68 202e 2e68 205d 2020 2020 2020 2020 [h ..h ] │ │ │ │ -0001e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 ] │ │ │ │ 0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e4c0: 2020 2020 2020 2020 2031 2020 2032 2020 1 2 │ │ │ │ -0001e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e4d0: 2020 207c 0a7c 2020 2020 2020 2020 2031 |.| 1 │ │ │ │ +0001e4e0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e500: 2020 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d |.|o3 : ---- │ │ │ │ -0001e510: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ -0001e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e510: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0001e520: 3a20 2d2d 2d2d 2d2d 2d2d 2d2d 2020 2020 : ---------- │ │ │ │ 0001e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e540: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001e550: 2020 2020 2020 3320 2020 3320 2020 2020 3 3 │ │ │ │ -0001e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e560: 207c 0a7c 2020 2020 2020 2020 3320 2020 |.| 3 │ │ │ │ +0001e570: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e590: 2020 7c0a 7c20 2020 2020 2028 6820 2c20 |.| (h , │ │ │ │ -0001e5a0: 6820 2920 2020 2020 2020 2020 2020 2020 h ) │ │ │ │ -0001e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 ) │ │ │ │ 0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001e5e0: 2020 2020 3120 2020 3220 2020 2020 2020 1 2 │ │ │ │ -0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e5e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e5f0: 0a7c 2020 2020 2020 2020 3120 2020 3220 .| 1 2 │ │ │ │ 0001e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e620: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0001e630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e630: 2020 2020 2020 7c0a 2b2d 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 2d2b 0a7c 6934 203a 2074 -------+.|i4 : t │ │ │ │ -0001e670: 696d 6520 4353 4d28 492c 4d65 7468 6f64 ime CSM(I,Method │ │ │ │ -0001e680: 3d3e 4469 7265 6374 436f 6d70 6c65 7449 =>DirectCompletI │ │ │ │ -0001e690: 6e74 2c49 6e64 734f 6653 6d6f 6f74 683d nt,IndsOfSmooth= │ │ │ │ -0001e6a0: 3e7b 312c 327d 2920 2020 2020 2020 7c0a >{1,2}) |. │ │ │ │ -0001e6b0: 7c20 2d2d 2075 7365 6420 312e 3430 3632 | -- used 1.4062 │ │ │ │ -0001e6c0: 3273 2028 6370 7529 3b20 312e 3035 3636 2s (cpu); 1.0566 │ │ │ │ -0001e6d0: 3773 2028 7468 7265 6164 293b 2030 7320 7s (thread); 0s │ │ │ │ -0001e6e0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ -0001e6f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0001e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001e670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001e680: 6934 203a 2074 696d 6520 4353 4d28 492c i4 : time CSM(I, │ │ │ │ +0001e690: 4d65 7468 6f64 3d3e 4469 7265 6374 436f Method=>DirectCo │ │ │ │ +0001e6a0: 6d70 6c65 7449 6e74 2c49 6e64 734f 6653 mpletInt,IndsOfS │ │ │ │ +0001e6b0: 6d6f 6f74 683d 3e7b 312c 327d 2920 2020 mooth=>{1,2}) │ │ │ │ +0001e6c0: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +0001e6d0: 332e 3937 3538 3473 2028 6370 7529 3b20 3.97584s (cpu); │ │ │ │ +0001e6e0: 312e 3036 3834 3773 2028 7468 7265 6164 1.06847s (thread │ │ │ │ +0001e6f0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0001e700: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 0001e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001e740: 2020 2020 2020 3220 3220 2020 2020 3220 2 2 2 │ │ │ │ -0001e750: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -0001e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e750: 2020 7c0a 7c20 2020 2020 2020 3220 3220 |.| 2 2 │ │ │ │ +0001e760: 2020 2020 3220 2020 2020 2020 2020 3220 2 2 │ │ │ │ 0001e770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e780: 2020 207c 0a7c 6f34 203d 2032 6820 6820 |.|o4 = 2h h │ │ │ │ -0001e790: 202b 2032 6820 6820 202b 2035 6820 6820 + 2h h + 5h h │ │ │ │ -0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001e7d0: 2020 2020 3120 3220 2020 2020 3120 3220 1 2 1 2 │ │ │ │ -0001e7e0: 2020 2020 3120 3220 2020 2020 2020 2020 1 2 │ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e790: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ +0001e7a0: 2032 6820 6820 202b 2032 6820 6820 202b 2h h + 2h h + │ │ │ │ +0001e7b0: 2035 6820 6820 2020 2020 2020 2020 2020 5h h │ │ │ │ +0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ 0001e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e810: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e820: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0001e830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e850: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001e860: 5a5a 5b68 202e 2e68 205d 2020 2020 2020 ZZ[h ..h ] │ │ │ │ -0001e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 207c | │ │ │ │ -0001e8a0: 0a7c 2020 2020 2020 2020 2031 2020 2032 .| 1 2 │ │ │ │ -0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001e890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ -0001e8e0: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d |.|o4 : -- │ │ │ │ -0001e8f0: 2d2d 2d2d 2d2d 2d2d 2020 2020 2020 2020 -------- │ │ │ │ -0001e900: 2020 2020 2020 2020 2020 2020 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 │ │ │ │ -0001e920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001e930: 2020 2020 2020 2020 3320 2020 3320 2020 3 3 │ │ │ │ -0001e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ -0001e970: 2020 2020 7c0a 7c20 2020 2020 2028 6820 |.| (h │ │ │ │ -0001e980: 2c20 6820 2920 2020 2020 2020 2020 2020 , h ) │ │ │ │ -0001e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ -0001e9b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001e9c0: 2020 2020 2020 3120 2020 3220 2020 2020 1 2 │ │ │ │ -0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ -0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 ---------------- │ │ │ │ 0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ea40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 6e63 ---------+..Func │ │ │ │ -0001ea50: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -0001ea60: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -0001ea70: 6564 2049 6e64 734f 6653 6d6f 6f74 683a ed IndsOfSmooth: │ │ │ │ -0001ea80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0001ea90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +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 │ │ │ │ +0001ea70: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ +0001ea80: 6e74 206e 616d 6564 2049 6e64 734f 6653 nt named IndsOfS │ │ │ │ +0001ea90: 6d6f 6f74 683a 0a3d 3d3d 3d3d 3d3d 3d3d mooth:.========= │ │ │ │ 0001eaa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001eab0: 3d3d 3d3d 3d0a 0a20 202a 2022 4353 4d28 =====.. * "CSM( │ │ │ │ -0001eac0: 2e2e 2e2c 496e 6473 4f66 536d 6f6f 7468 ...,IndsOfSmooth │ │ │ │ -0001ead0: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ -0001eae0: 6e6f 7465 2043 534d 3a20 4353 4d2c 202d note CSM: CSM, - │ │ │ │ -0001eaf0: 2d20 5468 650a 2020 2020 4368 6572 6e2d - The. Chern- │ │ │ │ -0001eb00: 5363 6877 6172 747a 2d4d 6163 5068 6572 Schwartz-MacPher │ │ │ │ -0001eb10: 736f 6e20 636c 6173 730a 2020 2a20 4575 son class. * Eu │ │ │ │ -0001eb20: 6c65 7228 2e2e 2e2c 496e 6473 4f66 536d ler(...,IndsOfSm │ │ │ │ -0001eb30: 6f6f 7468 3d3e 2e2e 2e29 2028 6d69 7373 ooth=>...) (miss │ │ │ │ -0001eb40: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ -0001eb50: 6e29 0a0a 466f 7220 7468 6520 7072 6f67 n)..For the prog │ │ │ │ -0001eb60: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0001eb70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0001eb80: 626a 6563 7420 2a6e 6f74 6520 496e 6473 bject *note Inds │ │ │ │ -0001eb90: 4f66 536d 6f6f 7468 3a20 496e 6473 4f66 OfSmooth: IndsOf │ │ │ │ -0001eba0: 536d 6f6f 7468 2c20 6973 2061 202a 6e6f Smooth, is a *no │ │ │ │ -0001ebb0: 7465 2073 796d 626f 6c3a 0a28 4d61 6361 te symbol:.(Maca │ │ │ │ -0001ebc0: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -0001ebd0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ -0001ebe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001eab0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001eac0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +0001ead0: 2022 4353 4d28 2e2e 2e2c 496e 6473 4f66 "CSM(...,IndsOf │ │ │ │ +0001eae0: 536d 6f6f 7468 3d3e 2e2e 2e29 2220 2d2d Smooth=>...)" -- │ │ │ │ +0001eaf0: 2073 6565 202a 6e6f 7465 2043 534d 3a20 see *note CSM: │ │ │ │ +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 │ │ │ │ +0001eb60: 6e74 6174 696f 6e29 0a0a 466f 7220 7468 ntation)..For th │ │ │ │ +0001eb70: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0001eb80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0001eb90: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0001eba0: 6520 496e 6473 4f66 536d 6f6f 7468 3a20 e IndsOfSmooth: │ │ │ │ +0001ebb0: 496e 6473 4f66 536d 6f6f 7468 2c20 6973 IndsOfSmooth, is │ │ │ │ +0001ebc0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ +0001ebd0: 0a28 4d61 6361 756c 6179 3244 6f63 2953 .(Macaulay2Doc)S │ │ │ │ +0001ebe0: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ 0001ebf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec20: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -0001ec30: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -0001ec40: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -0001ec50: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -0001ec60: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -0001ec70: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0001ec80: 2f70 6163 6b61 6765 732f 0a43 6861 7261 /packages/.Chara │ │ │ │ -0001ec90: 6374 6572 6973 7469 6343 6c61 7373 6573 cteristicClasses │ │ │ │ -0001eca0: 2e6d 323a 3234 3832 3a30 2e0a 1f0a 4669 .m2:2482:0....Fi │ │ │ │ -0001ecb0: 6c65 3a20 4368 6172 6163 7465 7269 7374 le: Characterist │ │ │ │ -0001ecc0: 6963 436c 6173 7365 732e 696e 666f 2c20 icClasses.info, │ │ │ │ -0001ecd0: 4e6f 6465 3a20 496e 7075 7449 7353 6d6f Node: InputIsSmo │ │ │ │ -0001ece0: 6f74 682c 204e 6578 743a 2069 734d 756c oth, Next: isMul │ │ │ │ -0001ecf0: 7469 486f 6d6f 6765 6e65 6f75 732c 2050 tiHomogeneous, P │ │ │ │ -0001ed00: 7265 763a 2049 6e64 734f 6653 6d6f 6f74 rev: IndsOfSmoot │ │ │ │ -0001ed10: 682c 2055 703a 2054 6f70 0a0a 496e 7075 h, Up: Top..Inpu │ │ │ │ -0001ed20: 7449 7353 6d6f 6f74 680a 2a2a 2a2a 2a2a tIsSmooth.****** │ │ │ │ -0001ed30: 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 6970 *******..Descrip │ │ │ │ -0001ed40: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0001ed50: 0a0a 5468 6520 6f70 7469 6f6e 2049 6e70 ..The option Inp │ │ │ │ -0001ed60: 7574 4973 536d 6f6f 7468 2069 7320 6f6e utIsSmooth is on │ │ │ │ -0001ed70: 6c79 2075 7365 6420 6279 2074 6865 2063 ly used by the c │ │ │ │ -0001ed80: 6f6d 6d61 6e64 7320 2a6e 6f74 6520 4353 ommands *note CS │ │ │ │ -0001ed90: 4d3a 2043 534d 2c2c 2061 6e64 0a2a 6e6f M: CSM,, and.*no │ │ │ │ -0001eda0: 7465 2045 756c 6572 3a20 4575 6c65 722c te Euler: Euler, │ │ │ │ -0001edb0: 2e20 4966 2074 6865 2069 6e70 7574 2069 . If the input i │ │ │ │ -0001edc0: 6465 616c 2069 7320 6b6e 6f77 6e20 746f deal is known to │ │ │ │ -0001edd0: 2064 6566 696e 6520 6120 736d 6f6f 7468 define a smooth │ │ │ │ -0001ede0: 2073 7562 7363 6865 6d65 0a73 6574 7469 subscheme.setti │ │ │ │ -0001edf0: 6e67 2074 6869 7320 6f70 7469 6f6e 2074 ng this option t │ │ │ │ -0001ee00: 6f20 7472 7565 2077 696c 6c20 7370 6565 o true will spee │ │ │ │ -0001ee10: 6420 7570 2063 6f6d 7075 7461 7469 6f6e d up computation │ │ │ │ -0001ee20: 7320 2869 7420 6973 2073 6574 2074 6f20 s (it is set to │ │ │ │ -0001ee30: 6661 6c73 6520 6279 0a64 6566 6175 6c74 false by.default │ │ │ │ -0001ee40: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d )...+----------- │ │ │ │ -0001ee50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001ec30: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0001ec40: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0001ec50: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0001ec60: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0001ec70: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0001ec80: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0001ec90: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0001eca0: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +0001ecb0: 6c61 7373 6573 2e6d 323a 3234 3832 3a30 lasses.m2:2482:0 │ │ │ │ +0001ecc0: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +0001ecd0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +0001ece0: 696e 666f 2c20 4e6f 6465 3a20 496e 7075 info, Node: Inpu │ │ │ │ +0001ecf0: 7449 7353 6d6f 6f74 682c 204e 6578 743a tIsSmooth, Next: │ │ │ │ +0001ed00: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ +0001ed10: 6f75 732c 2050 7265 763a 2049 6e64 734f ous, Prev: IndsO │ │ │ │ +0001ed20: 6653 6d6f 6f74 682c 2055 703a 2054 6f70 fSmooth, Up: Top │ │ │ │ +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 │ │ │ │ +0001ed70: 6f6e 2049 6e70 7574 4973 536d 6f6f 7468 on InputIsSmooth │ │ │ │ +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 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0001ee80: 203a 2052 203d 205a 5a2f 3332 3734 395b : R = ZZ/32749[ │ │ │ │ -0001ee90: 785f 302e 2e78 5f34 5d3b 2020 2020 2020 x_0..x_4]; │ │ │ │ -0001eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eeb0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -0001eec0: 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 2d2b 0a7c -------------+.| │ │ │ │ -0001eef0: 6932 203a 2049 3d69 6465 616c 2872 616e i2 : I=ideal(ran │ │ │ │ -0001ef00: 646f 6d28 322c 5229 2c72 616e 646f 6d28 dom(2,R),random( │ │ │ │ -0001ef10: 322c 5229 2c72 616e 646f 6d28 312c 5229 2,R),random(1,R) │ │ │ │ -0001ef20: 293b 2020 2020 7c0a 7c20 2020 2020 2020 ); |.| │ │ │ │ -0001ef30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0001ef60: 0a7c 6f32 203a 2049 6465 616c 206f 6620 .|o2 : Ideal of │ │ │ │ -0001ef70: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -0001efa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001efb0: 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 │ │ │ │ +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: 2d2b 0a7c 6933 203a 2074 696d 6520 4353 -+.|i3 : time CS │ │ │ │ -0001efe0: 4d20 4920 2020 2020 2020 2020 2020 2020 M I │ │ │ │ -0001eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f000: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0001f010: 2075 7365 6420 302e 3631 3837 3534 7320 used 0.618754s │ │ │ │ -0001f020: 2863 7075 293b 2030 2e34 3636 3337 3673 (cpu); 0.466376s │ │ │ │ -0001f030: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -0001f040: 6329 207c 0a7c 2020 2020 2020 2020 2020 c) |.| │ │ │ │ -0001f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f070: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f080: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ -0001f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0b0: 2020 2020 207c 0a7c 6f33 203d 2034 6820 |.|o3 = 4h │ │ │ │ +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 3739 3935 3633 7320 2863 sed 0.799563s (c │ │ │ │ +0001f040: 7075 293b 2030 2e34 3138 3831 3273 2028 pu); 0.418812s ( │ │ │ │ +0001f050: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +0001f060: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f090: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f0a0: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +0001f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f0f0: 7c20 2020 2020 2020 3120 2020 2020 2020 | 1 │ │ │ │ +0001f0d0: 2020 2020 2020 7c0a 7c6f 3320 3d20 3468 |.|o3 = 4h │ │ │ │ +0001f0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f120: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f110: 7c0a 7c20 2020 2020 2020 3120 2020 2020 |.| 1 │ │ │ │ +0001f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f140: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0001f150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f160: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020 |.| ZZ[h ] │ │ │ │ +0001f160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f190: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001f1a0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ -0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1d0: 2020 7c0a 7c6f 3320 3a20 2d2d 2d2d 2d2d |.|o3 : ------ │ │ │ │ +0001f180: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68 |.| ZZ[h │ │ │ │ +0001f190: 205d 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +0001f1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f1b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0001f1c0: 7c20 2020 2020 2020 2020 3120 2020 2020 | 1 │ │ │ │ +0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f200: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0001f210: 2020 2020 2020 3520 2020 2020 2020 2020 5 │ │ │ │ +0001f1f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ +0001f200: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f240: 2020 2020 7c0a 7c20 2020 2020 2020 6820 |.| h │ │ │ │ +0001f230: 2020 7c0a 7c20 2020 2020 2020 2035 2020 |.| 5 │ │ │ │ +0001f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f270: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0001f280: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ +0001f260: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f270: 2020 2020 2020 6820 2020 2020 2020 2020 h │ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2b0: 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0001f2f0: 0a7c 6934 203a 2074 696d 6520 4353 4d28 .|i4 : time CSM( │ │ │ │ -0001f300: 492c 496e 7075 7449 7353 6d6f 6f74 683d I,InputIsSmooth= │ │ │ │ -0001f310: 3e74 7275 6529 2020 2020 2020 2020 2020 >true) │ │ │ │ -0001f320: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -0001f330: 7365 6420 302e 3033 3339 3837 7320 2863 sed 0.033987s (c │ │ │ │ -0001f340: 7075 293b 2030 2e30 3333 3432 3133 7320 pu); 0.0334213s │ │ │ │ -0001f350: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0001f360: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f390: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001f3a0: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ +0001f2a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001f2b0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f2e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f310: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +0001f320: 3a20 7469 6d65 2043 534d 2849 2c49 6e70 : time CSM(I,Inp │ │ │ │ +0001f330: 7574 4973 536d 6f6f 7468 3d3e 7472 7565 utIsSmooth=>true │ │ │ │ +0001f340: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +0001f350: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +0001f360: 302e 3035 3636 3030 3573 2028 6370 7529 0.0566005s (cpu) │ │ │ │ +0001f370: 3b20 302e 3033 3336 3233 3173 2028 7468 ; 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0. │ │ │ │ -0001f700: 3033 3137 3730 3973 2028 7468 7265 6164 0317709s (thread │ │ │ │ -0001f710: 293b 2030 7320 2867 6329 7c0a 7c20 2020 ); 0s (gc)|.| │ │ │ │ -0001f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f750: 2020 2020 7c0a 7c20 2020 2020 2020 3320 |.| 3 │ │ │ │ +0001f5d0: 2020 7c0a 7c20 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ +0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f600: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001f610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f640: 2d2d 2d2d 2d2d 2b0a 0a4e 6f74 6520 7468 ------+..Note th │ │ │ │ +0001f650: 6174 206f 6e65 2063 6f75 6c64 2c20 6571 at one could, eq │ │ │ │ +0001f660: 7569 7661 6c65 6e74 6c79 2c20 7573 6520 uivalently, use │ │ │ │ +0001f670: 7468 6520 636f 6d6d 616e 6420 2a6e 6f74 the command *not │ │ │ │ +0001f680: 6520 4368 6572 6e3a 2043 6865 726e 2c20 e Chern: Chern, │ │ │ │ +0001f690: 696e 7374 6561 640a 696e 2074 6869 7320 instead.in this │ │ │ │ +0001f6a0: 6361 7365 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 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 3534 3632 3036 7320 used 0.0546206s │ │ │ │ +0001f730: 2863 7075 293b 2030 2e30 3332 3334 3039 (cpu); 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(missing │ │ │ │ -0001fad0: 2064 6f63 756d 656e 7461 7469 6f6e 290a documentation). │ │ │ │ -0001fae0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -0001faf0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -0001fb00: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -0001fb10: 6374 202a 6e6f 7465 2049 6e70 7574 4973 ct *note InputIs │ │ │ │ -0001fb20: 536d 6f6f 7468 3a20 496e 7075 7449 7353 Smooth: InputIsS │ │ │ │ -0001fb30: 6d6f 6f74 682c 2069 7320 6120 2a6e 6f74 mooth, is a *not │ │ │ │ -0001fb40: 6520 7379 6d62 6f6c 3a0a 284d 6163 6175 e symbol:.(Macau │ │ │ │ -0001fb50: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e lay2Doc)Symbol,. │ │ │ │ -0001fb60: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ -0001fb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f990: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f9a0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0001f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f9d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0001f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 -----------+..Fu │ │ │ │ +0001fa10: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +0001fa20: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +0001fa30: 616d 6564 2049 6e70 7574 4973 536d 6f6f amed InputIsSmoo │ │ │ │ +0001fa40: 7468 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d th:.============ │ │ │ │ +0001fa50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001fa60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001fa70: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +0001fa80: 4353 4d28 2e2e 2e2c 496e 7075 7449 7353 CSM(...,InputIsS │ │ │ │ +0001fa90: 6d6f 6f74 683d 3e2e 2e2e 2922 202d 2d20 mooth=>...)" -- │ │ │ │ +0001faa0: 7365 6520 2a6e 6f74 6520 4353 4d3a 2043 see *note CSM: C │ │ │ │ +0001fab0: 534d 2c20 2d2d 2054 6865 0a20 2020 2043 SM, -- The. C │ │ │ │ +0001fac0: 6865 726e 2d53 6368 7761 7274 7a2d 4d61 hern-Schwartz-Ma │ │ │ │ +0001fad0: 6350 6865 7273 6f6e 2063 6c61 7373 0a20 cPherson class. │ │ │ │ +0001fae0: 202a 2045 756c 6572 282e 2e2e 2c49 6e70 * Euler(...,Inp │ │ │ │ +0001faf0: 7574 4973 536d 6f6f 7468 3d3e 2e2e 2e29 utIsSmooth=>...) │ │ │ │ +0001fb00: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ +0001fb10: 6e74 6174 696f 6e29 0a0a 466f 7220 7468 ntation)..For th │ │ │ │ +0001fb20: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0001fb30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0001fb40: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0001fb50: 6520 496e 7075 7449 7353 6d6f 6f74 683a e InputIsSmooth: │ │ │ │ +0001fb60: 2049 6e70 7574 4973 536d 6f6f 7468 2c20 InputIsSmooth, │ │ │ │ +0001fb70: 6973 2061 202a 6e6f 7465 2073 796d 626f is a *note symbo │ │ │ │ +0001fb80: 6c3a 0a28 4d61 6361 756c 6179 3244 6f63 l:.(Macaulay2Doc │ │ │ │ +0001fb90: 2953 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d )Symbol,...----- │ │ │ │ 0001fba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fbb0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0001fbc0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0001fbd0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0001fbe0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0001fbf0: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -0001fc00: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0001fc10: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ -0001fc20: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -0001fc30: 6d32 3a32 3530 303a 302e 0a1f 0a46 696c m2:2500:0....Fil │ │ │ │ -0001fc40: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ -0001fc50: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ -0001fc60: 6f64 653a 2069 734d 756c 7469 486f 6d6f ode: isMultiHomo │ │ │ │ -0001fc70: 6765 6e65 6f75 732c 204e 6578 743a 204d geneous, Next: M │ │ │ │ -0001fc80: 6574 686f 642c 2050 7265 763a 2049 6e70 ethod, Prev: Inp │ │ │ │ -0001fc90: 7574 4973 536d 6f6f 7468 2c20 5570 3a20 utIsSmooth, Up: │ │ │ │ -0001fca0: 546f 700a 0a69 734d 756c 7469 486f 6d6f Top..isMultiHomo │ │ │ │ -0001fcb0: 6765 6e65 6f75 7320 2d2d 2043 6865 636b geneous -- Check │ │ │ │ -0001fcc0: 7320 6966 2061 6e20 6964 6561 6c20 6973 s if an ideal is │ │ │ │ -0001fcd0: 2068 6f6d 6f67 656e 656f 7573 2077 6974 homogeneous wit │ │ │ │ -0001fce0: 6820 7265 7370 6563 7420 746f 2074 6865 h respect to the │ │ │ │ -0001fcf0: 2067 7261 6469 6e67 206f 6e20 6974 7320 grading on its │ │ │ │ -0001fd00: 7269 6e67 2028 692e 652e 206d 756c 7469 ring (i.e. multi │ │ │ │ -0001fd10: 2d68 6f6d 6f67 656e 656f 7573 2069 6e20 -homogeneous in │ │ │ │ -0001fd20: 7468 6520 6d75 6c74 692d 6772 6164 6564 the multi-graded │ │ │ │ -0001fd30: 2063 6173 6529 0a2a 2a2a 2a2a 2a2a 2a2a case).********* │ │ │ │ -0001fd40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fd50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fd60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0001fbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0001fbf0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0001fc00: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0001fc10: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0001fc20: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0001fc30: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +0001fc40: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0001fc50: 732f 0a43 6861 7261 6374 6572 6973 7469 s/.Characteristi │ │ │ │ +0001fc60: 6343 6c61 7373 6573 2e6d 323a 3235 3030 cClasses.m2:2500 │ │ │ │ +0001fc70: 3a30 2e0a 1f0a 4669 6c65 3a20 4368 6172 :0....File: Char │ │ │ │ +0001fc80: 6163 7465 7269 7374 6963 436c 6173 7365 acteristicClasse │ │ │ │ +0001fc90: 732e 696e 666f 2c20 4e6f 6465 3a20 6973 s.info, Node: is │ │ │ │ +0001fca0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ +0001fcb0: 2c20 4e65 7874 3a20 4d65 7468 6f64 2c20 , Next: Method, │ │ │ │ +0001fcc0: 5072 6576 3a20 496e 7075 7449 7353 6d6f Prev: InputIsSmo │ │ │ │ +0001fcd0: 6f74 682c 2055 703a 2054 6f70 0a0a 6973 oth, Up: Top..is │ │ │ │ +0001fce0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ +0001fcf0: 202d 2d20 4368 6563 6b73 2069 6620 616e -- Checks if an │ │ │ │ +0001fd00: 2069 6465 616c 2069 7320 686f 6d6f 6765 ideal is homoge │ │ │ │ +0001fd10: 6e65 6f75 7320 7769 7468 2072 6573 7065 neous with respe │ │ │ │ +0001fd20: 6374 2074 6f20 7468 6520 6772 6164 696e ct to the gradin │ │ │ │ +0001fd30: 6720 6f6e 2069 7473 2072 696e 6720 2869 g on its ring (i │ │ │ │ +0001fd40: 2e65 2e20 6d75 6c74 692d 686f 6d6f 6765 .e. multi-homoge │ │ │ │ +0001fd50: 6e65 6f75 7320 696e 2074 6865 206d 756c neous in the mul │ │ │ │ +0001fd60: 7469 2d67 7261 6465 6420 6361 7365 290a ti-graded case). │ │ │ │ 0001fd70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fd90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fda0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001fdb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001fdc0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ -0001fdd0: 6167 653a 200a 2020 2020 2020 2020 6973 age: . is │ │ │ │ -0001fde0: 4d75 6c74 6948 6f6d 6f67 656e 656f 7573 MultiHomogeneous │ │ │ │ -0001fdf0: 2049 0a20 2020 2020 2020 2069 734d 756c I. isMul │ │ │ │ -0001fe00: 7469 486f 6d6f 6765 6e65 6f75 7320 660a tiHomogeneous f. │ │ │ │ -0001fe10: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -0001fe20: 2020 2a20 492c 2061 6e20 2a6e 6f74 6520 * I, an *note │ │ │ │ -0001fe30: 6964 6561 6c3a 2028 4d61 6361 756c 6179 ideal: (Macaulay │ │ │ │ -0001fe40: 3244 6f63 2949 6465 616c 2c2c 2061 6e20 2Doc)Ideal,, an │ │ │ │ -0001fe50: 6964 6561 6c20 696e 2061 2067 7261 6465 ideal in a grade │ │ │ │ -0001fe60: 6420 6f72 0a20 2020 2020 2020 206d 756c d or. mul │ │ │ │ -0001fe70: 7469 2d67 7261 6465 6420 7269 6e67 0a20 ti-graded ring. │ │ │ │ -0001fe80: 2020 2020 202a 2066 2c20 6120 2a6e 6f74 * f, a *not │ │ │ │ -0001fe90: 6520 7269 6e67 2065 6c65 6d65 6e74 3a20 e ring element: │ │ │ │ -0001fea0: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ -0001feb0: 6e67 456c 656d 656e 742c 2c20 6120 656c ngElement,, a el │ │ │ │ -0001fec0: 656d 656e 7420 696e 2061 0a20 2020 2020 ement in a. │ │ │ │ -0001fed0: 2020 2067 7261 6465 6420 6f72 206d 756c graded or mul │ │ │ │ -0001fee0: 7469 2d67 7261 6465 6420 7269 6e67 0a20 ti-graded ring. │ │ │ │ -0001fef0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0001ff00: 2020 2a20 6120 2a6e 6f74 6520 426f 6f6c * a *note Bool │ │ │ │ -0001ff10: 6561 6e20 7661 6c75 653a 2028 4d61 6361 ean value: (Maca │ │ │ │ -0001ff20: 756c 6179 3244 6f63 2942 6f6f 6c65 616e ulay2Doc)Boolean │ │ │ │ -0001ff30: 2c2c 200a 0a44 6573 6372 6970 7469 6f6e ,, ..Description │ │ │ │ -0001ff40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5465 .===========..Te │ │ │ │ -0001ff50: 7374 7320 6966 2074 6865 2069 6e70 7574 sts if the input │ │ │ │ -0001ff60: 2049 6465 616c 206f 7220 5269 6e67 456c Ideal or RingEl │ │ │ │ -0001ff70: 656d 656e 7420 6973 2048 6f6d 6f67 656e ement is Homogen │ │ │ │ -0001ff80: 656f 7573 2077 6974 6820 7265 7370 6563 eous with respec │ │ │ │ -0001ff90: 7420 746f 2074 6865 0a67 7261 6469 6e67 t to the.grading │ │ │ │ -0001ffa0: 206f 6e20 7468 6520 7269 6e67 2e20 486f on the ring. Ho │ │ │ │ -0001ffb0: 6d6f 6765 6e65 6f75 7320 696e 7075 7420 mogeneous input │ │ │ │ -0001ffc0: 6973 2072 6571 7569 7265 6420 666f 7220 is required for │ │ │ │ -0001ffd0: 616c 6c20 6d65 7468 6f64 7320 746f 2063 all methods to c │ │ │ │ -0001ffe0: 6f6d 7075 7465 0a63 6861 7261 6374 6572 ompute.character │ │ │ │ -0001fff0: 6973 7469 6320 636c 6173 7365 732e 0a0a istic classes... │ │ │ │ -00020000: 5468 6973 206d 6574 686f 6420 776f 726b This method work │ │ │ │ -00020010: 7320 666f 7220 6964 6561 6c73 2069 6e20 s for ideals in │ │ │ │ -00020020: 7468 6520 6772 6164 6564 2063 6f6f 7264 the graded coord │ │ │ │ -00020030: 696e 6174 6520 7269 6e67 7320 6f66 2074 inate rings of t │ │ │ │ -00020040: 6f72 6963 2076 6172 6965 7469 6573 2c0a oric varieties,. │ │ │ │ -00020050: 616e 6420 6865 6e63 6520 666f 7220 7072 and hence for pr │ │ │ │ -00020060: 6f64 7563 7473 206f 6620 7072 6f6a 6563 oducts of projec │ │ │ │ -00020070: 7469 7665 2073 7061 6365 732e 2054 6865 tive spaces. The │ │ │ │ -00020080: 7365 2063 616e 2062 6520 6372 6561 7465 se can be create │ │ │ │ -00020090: 6420 6469 7265 6374 6c79 2c20 6f72 0a75 d directly, or.u │ │ │ │ -000200a0: 7369 6e67 206d 6574 686f 6473 2074 6865 sing methods the │ │ │ │ -000200b0: 202a 6e6f 7465 204d 756c 7469 5072 6f6a *note MultiProj │ │ │ │ -000200c0: 436f 6f72 6452 696e 673a 204d 756c 7469 CoordRing: Multi │ │ │ │ -000200d0: 5072 6f6a 436f 6f72 6452 696e 672c 206d ProjCoordRing, m │ │ │ │ -000200e0: 6574 686f 6420 6f66 2074 6869 730a 7061 ethod of this.pa │ │ │ │ -000200f0: 636b 6167 652c 206f 7220 7769 7468 206d ckage, or with m │ │ │ │ -00020100: 6574 686f 6473 2066 726f 6d20 7468 6520 ethods from the │ │ │ │ -00020110: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ -00020120: 7469 6573 2050 6163 6b61 6765 2e0a 0a2b ties Package...+ │ │ │ │ -00020130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00020170: 0a7c 6931 203a 2052 3d4d 756c 7469 5072 .|i1 : R=MultiPr │ │ │ │ -00020180: 6f6a 436f 6f72 6452 696e 6728 7b31 2c32 ojCoordRing({1,2 │ │ │ │ -00020190: 2c31 7d29 2020 2020 2020 2020 2020 2020 ,1}) │ │ │ │ -000201a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000201c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fdc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fdd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fde0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fdf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0001fe00: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0001fe10: 2020 2020 2020 2069 734d 756c 7469 486f isMultiHo │ │ │ │ +0001fe20: 6d6f 6765 6e65 6f75 7320 490a 2020 2020 mogeneous I. │ │ │ │ +0001fe30: 2020 2020 6973 4d75 6c74 6948 6f6d 6f67 isMultiHomog │ │ │ │ +0001fe40: 656e 656f 7573 2066 0a20 202a 2049 6e70 eneous f. * Inp │ │ │ │ +0001fe50: 7574 733a 0a20 2020 2020 202a 2049 2c20 uts:. * I, │ │ │ │ +0001fe60: 616e 202a 6e6f 7465 2069 6465 616c 3a20 an *note ideal: │ │ │ │ +0001fe70: 284d 6163 6175 6c61 7932 446f 6329 4964 (Macaulay2Doc)Id │ │ │ │ +0001fe80: 6561 6c2c 2c20 616e 2069 6465 616c 2069 eal,, an ideal i │ │ │ │ +0001fe90: 6e20 6120 6772 6164 6564 206f 720a 2020 n a graded or. │ │ │ │ +0001fea0: 2020 2020 2020 6d75 6c74 692d 6772 6164 multi-grad │ │ │ │ +0001feb0: 6564 2072 696e 670a 2020 2020 2020 2a20 ed ring. * │ │ │ │ +0001fec0: 662c 2061 202a 6e6f 7465 2072 696e 6720 f, a *note ring │ │ │ │ +0001fed0: 656c 656d 656e 743a 2028 4d61 6361 756c element: (Macaul │ │ │ │ +0001fee0: 6179 3244 6f63 2952 696e 6745 6c65 6d65 ay2Doc)RingEleme │ │ │ │ +0001fef0: 6e74 2c2c 2061 2065 6c65 6d65 6e74 2069 nt,, a element i │ │ │ │ +0001ff00: 6e20 610a 2020 2020 2020 2020 6772 6164 n a. grad │ │ │ │ +0001ff10: 6564 206f 7220 6d75 6c74 692d 6772 6164 ed or multi-grad │ │ │ │ +0001ff20: 6564 2072 696e 670a 2020 2a20 4f75 7470 ed ring. * Outp │ │ │ │ +0001ff30: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +0001ff40: 6e6f 7465 2042 6f6f 6c65 616e 2076 616c note Boolean val │ │ │ │ +0001ff50: 7565 3a20 284d 6163 6175 6c61 7932 446f ue: (Macaulay2Do │ │ │ │ +0001ff60: 6329 426f 6f6c 6561 6e2c 2c20 0a0a 4465 c)Boolean,, ..De │ │ │ │ +0001ff70: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0001ff80: 3d3d 3d3d 3d0a 0a54 6573 7473 2069 6620 =====..Tests if │ │ │ │ +0001ff90: 7468 6520 696e 7075 7420 4964 6561 6c20 the input Ideal │ │ │ │ +0001ffa0: 6f72 2052 696e 6745 6c65 6d65 6e74 2069 or RingElement i │ │ │ │ +0001ffb0: 7320 486f 6d6f 6765 6e65 6f75 7320 7769 s Homogeneous wi │ │ │ │ +0001ffc0: 7468 2072 6573 7065 6374 2074 6f20 7468 th respect to th │ │ │ │ +0001ffd0: 650a 6772 6164 696e 6720 6f6e 2074 6865 e.grading on the │ │ │ │ +0001ffe0: 2072 696e 672e 2048 6f6d 6f67 656e 656f ring. Homogeneo │ │ │ │ +0001fff0: 7573 2069 6e70 7574 2069 7320 7265 7175 us input is requ │ │ │ │ +00020000: 6972 6564 2066 6f72 2061 6c6c 206d 6574 ired for all met │ │ │ │ +00020010: 686f 6473 2074 6f20 636f 6d70 7574 650a hods to compute. │ │ │ │ +00020020: 6368 6172 6163 7465 7269 7374 6963 2063 characteristic c │ │ │ │ +00020030: 6c61 7373 6573 2e0a 0a54 6869 7320 6d65 lasses...This me │ │ │ │ +00020040: 7468 6f64 2077 6f72 6b73 2066 6f72 2069 thod works for i │ │ │ │ +00020050: 6465 616c 7320 696e 2074 6865 2067 7261 deals in the gra │ │ │ │ +00020060: 6465 6420 636f 6f72 6469 6e61 7465 2072 ded coordinate r │ │ │ │ +00020070: 696e 6773 206f 6620 746f 7269 6320 7661 ings of toric va │ │ │ │ +00020080: 7269 6574 6965 732c 0a61 6e64 2068 656e rieties,.and hen │ │ │ │ +00020090: 6365 2066 6f72 2070 726f 6475 6374 7320 ce for products │ │ │ │ +000200a0: 6f66 2070 726f 6a65 6374 6976 6520 7370 of projective sp │ │ │ │ +000200b0: 6163 6573 2e20 5468 6573 6520 6361 6e20 aces. These can │ │ │ │ +000200c0: 6265 2063 7265 6174 6564 2064 6972 6563 be created direc │ │ │ │ +000200d0: 746c 792c 206f 720a 7573 696e 6720 6d65 tly, or.using me │ │ │ │ +000200e0: 7468 6f64 7320 7468 6520 2a6e 6f74 6520 thods the *note │ │ │ │ +000200f0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ +00020100: 6e67 3a20 4d75 6c74 6950 726f 6a43 6f6f ng: MultiProjCoo │ │ │ │ +00020110: 7264 5269 6e67 2c20 6d65 7468 6f64 206f rdRing, method o │ │ │ │ +00020120: 6620 7468 6973 0a70 6163 6b61 6765 2c20 f this.package, │ │ │ │ +00020130: 6f72 2077 6974 6820 6d65 7468 6f64 7320 or with methods │ │ │ │ +00020140: 6672 6f6d 2074 6865 204e 6f72 6d61 6c54 from the NormalT │ │ │ │ +00020150: 6f72 6963 5661 7269 6574 6965 7320 5061 oricVarieties Pa │ │ │ │ +00020160: 636b 6167 652e 0a0a 2b2d 2d2d 2d2d 2d2d ckage...+------- │ │ │ │ +00020170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000201a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +000201b0: 523d 4d75 6c74 6950 726f 6a43 6f6f 7264 R=MultiProjCoord │ │ │ │ +000201c0: 5269 6e67 287b 312c 322c 317d 2920 2020 Ring({1,2,1}) │ │ │ │ 000201d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000201f0: 2020 207c 0a7c 6f31 203d 2052 2020 2020 |.|o1 = R │ │ │ │ +000201e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000201f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020230: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00020220: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00020230: 3120 3d20 5220 2020 2020 2020 2020 2020 1 = R │ │ │ │ 00020240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020270: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ -00020280: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00020260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020270: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00020280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000202a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000202b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000202c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00020300: 203a 2078 3d67 656e 7328 5229 2020 2020 : x=gens(R) │ │ │ │ -00020310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202b0: 7c0a 7c6f 3120 3a20 506f 6c79 6e6f 6d69 |.|o1 : Polynomi │ │ │ │ +000202c0: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +000202d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000202f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020330: 2d2d 2d2d 2b0a 7c69 3220 3a20 783d 6765 ----+.|i2 : x=ge │ │ │ │ +00020340: 6e73 2852 2920 2020 2020 2020 2020 2020 ns(R) │ │ │ │ 00020350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020380: 0a7c 6f32 203d 207b 7820 2c20 7820 2c20 .|o2 = {x , x , │ │ │ │ -00020390: 7820 2c20 7820 2c20 7820 2c20 7820 2c20 x , x , x , x , │ │ │ │ -000203a0: 7820 7d20 2020 2020 2020 2020 2020 2020 x } │ │ │ │ -000203b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000203c0: 207c 0a7c 2020 2020 2020 2030 2020 2031 |.| 0 1 │ │ │ │ -000203d0: 2020 2032 2020 2033 2020 2034 2020 2035 2 3 4 5 │ │ │ │ -000203e0: 2020 2036 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -000203f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020370: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203b0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +000203c0: 7b78 202c 2078 202c 2078 202c 2078 202c {x , x , x , x , │ │ │ │ +000203d0: 2078 202c 2078 202c 2078 207d 2020 2020 x , x , x } │ │ │ │ +000203e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000203f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00020400: 2020 2020 3020 2020 3120 2020 3220 2020 0 1 2 │ │ │ │ +00020410: 3320 2020 3420 2020 3520 2020 3620 2020 3 4 5 6 │ │ │ │ 00020420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020440: 2020 2020 207c 0a7c 6f32 203a 204c 6973 |.|o2 : Lis │ │ │ │ -00020450: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00020430: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00020440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -00020490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000204c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -000204d0: 2049 3d69 6465 616c 2878 5f30 5e32 2a78 I=ideal(x_0^2*x │ │ │ │ -000204e0: 5f33 2d78 5f31 2a78 5f30 2a78 5f34 2c78 _3-x_1*x_0*x_4,x │ │ │ │ -000204f0: 5f36 5e33 2920 2020 2020 2020 2020 2020 _6^3) │ │ │ │ -00020500: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00020510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020480: 7c6f 3220 3a20 4c69 7374 2020 2020 2020 |o2 : List │ │ │ │ +00020490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000204c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000204d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000204e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000204f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020500: 2d2d 2b0a 7c69 3320 3a20 493d 6964 6561 --+.|i3 : I=idea │ │ │ │ +00020510: 6c28 785f 305e 322a 785f 332d 785f 312a l(x_0^2*x_3-x_1* │ │ │ │ +00020520: 785f 302a 785f 342c 785f 365e 3329 2020 x_0*x_4,x_6^3) │ │ │ │ 00020530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020550: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ -00020560: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ +00020540: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00020550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020580: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00020590: 0a7c 6f33 203d 2069 6465 616c 2028 7820 .|o3 = ideal (x │ │ │ │ -000205a0: 7820 202d 2078 2078 2078 202c 2078 2029 x - x x x , x ) │ │ │ │ +00020580: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020590: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000205a0: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ 000205b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000205d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000205e0: 2030 2033 2020 2020 3020 3120 3420 2020 0 3 0 1 4 │ │ │ │ -000205f0: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -00020600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020610: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00020620: 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 │ │ │ │ 00020630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020650: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ -00020660: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +00020640: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00020650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -000206a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000206d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -000206e0: 2069 734d 756c 7469 486f 6d6f 6765 6e65 isMultiHomogene │ │ │ │ -000206f0: 6f75 7320 4920 2020 2020 2020 2020 2020 ous I │ │ │ │ -00020700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00020720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020680: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020690: 7c6f 3320 3a20 4964 6561 6c20 6f66 2052 |o3 : Ideal of R │ │ │ │ +000206a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000206d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000206e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000206f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020710: 2d2d 2b0a 7c69 3420 3a20 6973 4d75 6c74 --+.|i4 : isMult │ │ │ │ +00020720: 6948 6f6d 6f67 656e 656f 7573 2049 2020 iHomogeneous I │ │ │ │ 00020730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020760: 6f34 203d 2074 7275 6520 2020 2020 2020 o4 = true │ │ │ │ +00020750: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00020760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020790: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000207a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -000207b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000207e0: 2d2b 0a7c 6935 203a 2069 734d 756c 7469 -+.|i5 : isMulti │ │ │ │ -000207f0: 486f 6d6f 6765 6e65 6f75 7320 6964 6561 Homogeneous idea │ │ │ │ -00020800: 6c28 785f 302a 785f 332d 785f 312a 785f l(x_0*x_3-x_1*x_ │ │ │ │ -00020810: 302a 785f 342c 785f 365e 3329 2020 2020 0*x_4,x_6^3) │ │ │ │ -00020820: 2020 207c 0a7c 496e 7075 7420 7465 726d |.|Input term │ │ │ │ -00020830: 2062 656c 6f77 2069 7320 6e6f 7420 686f below is not ho │ │ │ │ -00020840: 6d6f 6765 6e65 6f75 7320 7769 7468 2072 mogeneous with r │ │ │ │ -00020850: 6573 7065 6374 2074 6f20 7468 6520 6772 espect to the gr │ │ │ │ -00020860: 6164 696e 677c 0a7c 2d20 7820 7820 7820 ading|.|- x x x │ │ │ │ -00020870: 202b 2078 2078 2020 2020 2020 2020 2020 + x x │ │ │ │ -00020880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208a0: 2020 2020 2020 207c 0a7c 2020 2030 2031 |.| 0 1 │ │ │ │ -000208b0: 2034 2020 2020 3020 3320 2020 2020 2020 4 0 3 │ │ │ │ +00020790: 2020 2020 2020 7c0a 7c6f 3420 3d20 7472 |.|o4 = tr │ │ │ │ +000207a0: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +000207b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000207c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000207d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000207e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000207f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020810: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +00020820: 3a20 6973 4d75 6c74 6948 6f6d 6f67 656e : isMultiHomogen │ │ │ │ +00020830: 656f 7573 2069 6465 616c 2878 5f30 2a78 eous ideal(x_0*x │ │ │ │ +00020840: 5f33 2d78 5f31 2a78 5f30 2a78 5f34 2c78 _3-x_1*x_0*x_4,x │ │ │ │ +00020850: 5f36 5e33 2920 2020 2020 2020 7c0a 7c49 _6^3) |.|I │ │ │ │ +00020860: 6e70 7574 2074 6572 6d20 6265 6c6f 7720 nput term below │ │ │ │ +00020870: 6973 206e 6f74 2068 6f6d 6f67 656e 656f is not homogeneo │ │ │ │ +00020880: 7573 2077 6974 6820 7265 7370 6563 7420 us with respect │ │ │ │ +00020890: 746f 2074 6865 2067 7261 6469 6e67 7c0a to the grading|. │ │ │ │ +000208a0: 7c2d 2078 2078 2078 2020 2b20 7820 7820 |- x x x + x x │ │ │ │ +000208b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000208c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000208d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000208e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000208f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000208e0: 7c0a 7c20 2020 3020 3120 3420 2020 2030 |.| 0 1 4 0 │ │ │ │ +000208f0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00020900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020920: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00020930: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ +00020920: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020960: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00020970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000209a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000209b0: 0a0a 4e6f 7465 2074 6861 7420 666f 7220 ..Note that for │ │ │ │ -000209c0: 616e 2069 6465 616c 2074 6f20 6265 206d an ideal to be m │ │ │ │ -000209d0: 756c 7469 2d68 6f6d 6f67 656e 656f 7573 ulti-homogeneous │ │ │ │ -000209e0: 2074 6865 2064 6567 7265 6520 7665 6374 the degree vect │ │ │ │ -000209f0: 6f72 206f 6620 616c 6c0a 6d6f 6e6f 6d69 or of all.monomi │ │ │ │ -00020a00: 616c 7320 696e 2061 2067 6976 656e 2067 als in a given g │ │ │ │ -00020a10: 656e 6572 6174 6f72 206d 7573 7420 6265 enerator must be │ │ │ │ -00020a20: 2074 6865 2073 616d 652e 0a0a 5761 7973 the same...Ways │ │ │ │ -00020a30: 2074 6f20 7573 6520 6973 4d75 6c74 6948 to use isMultiH │ │ │ │ -00020a40: 6f6d 6f67 656e 656f 7573 3a0a 3d3d 3d3d omogeneous:.==== │ │ │ │ -00020a50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00020a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00020a70: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e "isMultiHomogen │ │ │ │ -00020a80: 656f 7573 2849 6465 616c 2922 0a20 202a eous(Ideal)". * │ │ │ │ -00020a90: 2022 6973 4d75 6c74 6948 6f6d 6f67 656e "isMultiHomogen │ │ │ │ -00020aa0: 656f 7573 2852 696e 6745 6c65 6d65 6e74 eous(RingElement │ │ │ │ -00020ab0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -00020ac0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -00020ad0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -00020ae0: 626a 6563 7420 2a6e 6f74 6520 6973 4d75 bject *note isMu │ │ │ │ -00020af0: 6c74 6948 6f6d 6f67 656e 656f 7573 3a20 ltiHomogeneous: │ │ │ │ -00020b00: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f isMultiHomogeneo │ │ │ │ -00020b10: 7573 2c20 6973 2061 202a 6e6f 7465 206d us, is a *note m │ │ │ │ -00020b20: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ -00020b30: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00020b40: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ -00020b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -00020ba0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -00020bb0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -00020bc0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -00020bd0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00020be0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -00020bf0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00020c00: 636b 6167 6573 2f0a 4368 6172 6163 7465 ckages/.Characte │ │ │ │ -00020c10: 7269 7374 6963 436c 6173 7365 732e 6d32 risticClasses.m2 │ │ │ │ -00020c20: 3a32 3031 323a 302e 0a1f 0a46 696c 653a :2012:0....File: │ │ │ │ -00020c30: 2043 6861 7261 6374 6572 6973 7469 6343 CharacteristicC │ │ │ │ -00020c40: 6c61 7373 6573 2e69 6e66 6f2c 204e 6f64 lasses.info, Nod │ │ │ │ -00020c50: 653a 204d 6574 686f 642c 204e 6578 743a e: Method, Next: │ │ │ │ -00020c60: 204d 756c 7469 5072 6f6a 436f 6f72 6452 MultiProjCoordR │ │ │ │ -00020c70: 696e 672c 2050 7265 763a 2069 734d 756c ing, Prev: isMul │ │ │ │ -00020c80: 7469 486f 6d6f 6765 6e65 6f75 732c 2055 tiHomogeneous, U │ │ │ │ -00020c90: 703a 2054 6f70 0a0a 4d65 7468 6f64 0a2a p: Top..Method.* │ │ │ │ -00020ca0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ -00020cb0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00020cc0: 5468 6520 6f70 7469 6f6e 204d 6574 686f The option Metho │ │ │ │ -00020cd0: 6420 6973 206f 6e6c 7920 7573 6564 2062 d is only used b │ │ │ │ -00020ce0: 7920 7468 6520 636f 6d6d 616e 6473 202a y the commands * │ │ │ │ -00020cf0: 6e6f 7465 2043 534d 3a20 4353 4d2c 2061 note CSM: CSM, a │ │ │ │ -00020d00: 6e64 202a 6e6f 7465 2045 756c 6572 3a0a nd *note Euler:. │ │ │ │ -00020d10: 4575 6c65 722c 2061 6e64 206f 6e6c 7920 Euler, and only │ │ │ │ -00020d20: 696e 2063 6f6d 6269 6e61 7469 6f6e 2077 in combination w │ │ │ │ -00020d30: 6974 6820 2a6e 6f74 6520 436f 6d70 4d65 ith *note CompMe │ │ │ │ -00020d40: 7468 6f64 3a0a 436f 6d70 4d65 7468 6f64 thod:.CompMethod │ │ │ │ -00020d50: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ -00020d60: 7265 652e 2054 6865 204d 6574 686f 6420 ree. The Method │ │ │ │ -00020d70: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ -00020d80: 6f6e 2077 696c 6c20 616c 7761 7973 2062 on will always b │ │ │ │ -00020d90: 650a 7573 6564 2077 6974 6820 2a6e 6f74 e.used with *not │ │ │ │ -00020da0: 6520 436f 6d70 4d65 7468 6f64 3a20 436f e CompMethod: Co │ │ │ │ -00020db0: 6d70 4d65 7468 6f64 2c20 506e 5265 7369 mpMethod, PnResi │ │ │ │ -00020dc0: 6475 616c 206f 7220 6265 7274 696e 692e dual or bertini. │ │ │ │ -00020dd0: 2057 6865 6e20 7468 6520 696e 7075 740a When the input. │ │ │ │ -00020de0: 6964 6561 6c20 6973 2061 2063 6f6d 706c ideal is a compl │ │ │ │ -00020df0: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ -00020e00: 206f 6e65 206d 6179 2c20 706f 7465 6e74 one may, potent │ │ │ │ -00020e10: 6961 6c6c 792c 2073 7065 6564 2075 7020 ially, speed up │ │ │ │ -00020e20: 7468 6520 636f 6d70 7574 6174 696f 6e0a the computation. │ │ │ │ -00020e30: 6279 2073 6574 7469 6e67 204d 6574 686f by setting Metho │ │ │ │ -00020e40: 643d 3e20 4469 7265 6374 436f 6d70 6c65 d=> DirectComple │ │ │ │ -00020e50: 7465 496e 742e 2054 6865 206f 7074 696f teInt. The optio │ │ │ │ -00020e60: 6e20 4d65 7468 6f64 2069 7320 6f6e 6c79 n Method is only │ │ │ │ -00020e70: 2075 7365 6420 6279 2074 6865 0a63 6f6d used by the.com │ │ │ │ -00020e80: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a mands *note CSM: │ │ │ │ -00020e90: 2043 534d 2c20 616e 6420 2a6e 6f74 6520 CSM, and *note │ │ │ │ -00020ea0: 4575 6c65 723a 2045 756c 6572 2c20 616e Euler: Euler, an │ │ │ │ -00020eb0: 6420 6f6e 6c79 2069 6e20 636f 6d62 696e d only in combin │ │ │ │ -00020ec0: 6174 696f 6e20 7769 7468 0a2a 6e6f 7465 ation with.*note │ │ │ │ -00020ed0: 2043 6f6d 704d 6574 686f 643a 2043 6f6d CompMethod: Com │ │ │ │ -00020ee0: 704d 6574 686f 642c 3d3e 5072 6f6a 6563 pMethod,=>Projec │ │ │ │ -00020ef0: 7469 7665 4465 6772 6565 2e20 5468 6520 tiveDegree. The │ │ │ │ -00020f00: 4d65 7468 6f64 2049 6e63 6c75 7369 6f6e Method Inclusion │ │ │ │ -00020f10: 4578 636c 7573 696f 6e0a 7769 6c6c 2061 Exclusion.will a │ │ │ │ -00020f20: 6c77 6179 7320 6265 2075 7365 6420 7769 lways be used wi │ │ │ │ -00020f30: 7468 202a 6e6f 7465 2043 6f6d 704d 6574 th *note CompMet │ │ │ │ -00020f40: 686f 643a 2043 6f6d 704d 6574 686f 642c hod: CompMethod, │ │ │ │ -00020f50: 2050 6e52 6573 6964 7561 6c20 6f72 2062 PnResidual or b │ │ │ │ -00020f60: 6572 7469 6e69 2e0a 0a2b 2d2d 2d2d 2d2d ertini...+------ │ │ │ │ -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 2b0a 7c69 3120 3a20 5220 3d20 5a5a --+.|i1 : R = ZZ │ │ │ │ -00020fb0: 2f33 3237 3439 5b78 5f30 2e2e 785f 365d /32749[x_0..x_6] │ │ │ │ -00020fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020fd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ff0: 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,...------- │ │ │ │ +00020b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020bd0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +00020be0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +00020bf0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +00020c00: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +00020c10: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +00020c20: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +00020c30: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00020c40: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ +00020c50: 6c61 7373 6573 2e6d 323a 3230 3132 3a30 lasses.m2:2012:0 │ │ │ │ +00020c60: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ +00020c70: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +00020c80: 696e 666f 2c20 4e6f 6465 3a20 4d65 7468 info, Node: Meth │ │ │ │ +00020c90: 6f64 2c20 4e65 7874 3a20 4d75 6c74 6950 od, Next: MultiP │ │ │ │ +00020ca0: 726f 6a43 6f6f 7264 5269 6e67 2c20 5072 rojCoordRing, Pr │ │ │ │ +00020cb0: 6576 3a20 6973 4d75 6c74 6948 6f6d 6f67 ev: isMultiHomog │ │ │ │ +00020cc0: 656e 656f 7573 2c20 5570 3a20 546f 700a eneous, Up: Top. │ │ │ │ +00020cd0: 0a4d 6574 686f 640a 2a2a 2a2a 2a2a 0a0a .Method.******.. │ │ │ │ +00020ce0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00020cf0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 7074 =======..The opt │ │ │ │ +00020d00: 696f 6e20 4d65 7468 6f64 2069 7320 6f6e ion Method is on │ │ │ │ +00020d10: 6c79 2075 7365 6420 6279 2074 6865 2063 ly used by the c │ │ │ │ +00020d20: 6f6d 6d61 6e64 7320 2a6e 6f74 6520 4353 ommands *note CS │ │ │ │ +00020d30: 4d3a 2043 534d 2c20 616e 6420 2a6e 6f74 M: CSM, and *not │ │ │ │ +00020d40: 6520 4575 6c65 723a 0a45 756c 6572 2c20 e Euler:.Euler, │ │ │ │ +00020d50: 616e 6420 6f6e 6c79 2069 6e20 636f 6d62 and only in comb │ │ │ │ +00020d60: 696e 6174 696f 6e20 7769 7468 202a 6e6f ination with *no │ │ │ │ +00020d70: 7465 2043 6f6d 704d 6574 686f 643a 0a43 te CompMethod:.C │ │ │ │ +00020d80: 6f6d 704d 6574 686f 642c 3d3e 5072 6f6a ompMethod,=>Proj │ │ │ │ +00020d90: 6563 7469 7665 4465 6772 6565 2e20 5468 ectiveDegree. Th │ │ │ │ +00020da0: 6520 4d65 7468 6f64 2049 6e63 6c75 7369 e Method Inclusi │ │ │ │ +00020db0: 6f6e 4578 636c 7573 696f 6e20 7769 6c6c onExclusion will │ │ │ │ +00020dc0: 2061 6c77 6179 7320 6265 0a75 7365 6420 always be.used │ │ │ │ +00020dd0: 7769 7468 202a 6e6f 7465 2043 6f6d 704d with *note CompM │ │ │ │ +00020de0: 6574 686f 643a 2043 6f6d 704d 6574 686f ethod: CompMetho │ │ │ │ +00020df0: 642c 2050 6e52 6573 6964 7561 6c20 6f72 d, PnResidual or │ │ │ │ +00020e00: 2062 6572 7469 6e69 2e20 5768 656e 2074 bertini. When t │ │ │ │ +00020e10: 6865 2069 6e70 7574 0a69 6465 616c 2069 he input.ideal i │ │ │ │ +00020e20: 7320 6120 636f 6d70 6c65 7465 2069 6e74 s a complete int │ │ │ │ +00020e30: 6572 7365 6374 696f 6e20 6f6e 6520 6d61 ersection one ma │ │ │ │ +00020e40: 792c 2070 6f74 656e 7469 616c 6c79 2c20 y, potentially, │ │ │ │ +00020e50: 7370 6565 6420 7570 2074 6865 2063 6f6d speed up the com │ │ │ │ +00020e60: 7075 7461 7469 6f6e 0a62 7920 7365 7474 putation.by sett │ │ │ │ +00020e70: 696e 6720 4d65 7468 6f64 3d3e 2044 6972 ing Method=> Dir │ │ │ │ +00020e80: 6563 7443 6f6d 706c 6574 6549 6e74 2e20 ectCompleteInt. │ │ │ │ +00020e90: 5468 6520 6f70 7469 6f6e 204d 6574 686f The option Metho │ │ │ │ +00020ea0: 6420 6973 206f 6e6c 7920 7573 6564 2062 d is only used b │ │ │ │ +00020eb0: 7920 7468 650a 636f 6d6d 616e 6473 202a y the.commands * │ │ │ │ +00020ec0: 6e6f 7465 2043 534d 3a20 4353 4d2c 2061 note CSM: CSM, a │ │ │ │ +00020ed0: 6e64 202a 6e6f 7465 2045 756c 6572 3a20 nd *note Euler: │ │ │ │ +00020ee0: 4575 6c65 722c 2061 6e64 206f 6e6c 7920 Euler, and only │ │ │ │ +00020ef0: 696e 2063 6f6d 6269 6e61 7469 6f6e 2077 in combination w │ │ │ │ +00020f00: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65 ith.*note CompMe │ │ │ │ +00020f10: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ +00020f20: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ +00020f30: 7265 652e 2054 6865 204d 6574 686f 6420 ree. The Method │ │ │ │ +00020f40: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ +00020f50: 6f6e 0a77 696c 6c20 616c 7761 7973 2062 on.will always b │ │ │ │ +00020f60: 6520 7573 6564 2077 6974 6820 2a6e 6f74 e used with *not │ │ │ │ +00020f70: 6520 436f 6d70 4d65 7468 6f64 3a20 436f e CompMethod: Co │ │ │ │ +00020f80: 6d70 4d65 7468 6f64 2c20 506e 5265 7369 mpMethod, PnResi │ │ │ │ +00020f90: 6475 616c 206f 7220 6265 7274 696e 692e dual or bertini. │ │ │ │ +00020fa0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00020fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00020fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00020fe0: 203a 2052 203d 205a 5a2f 3332 3734 395b : R = ZZ/32749[ │ │ │ │ +00020ff0: 785f 302e 2e78 5f36 5d20 2020 2020 2020 x_0..x_6] │ │ │ │ 00021000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021010: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00021020: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00021010: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00021020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021050: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00021050: 207c 0a7c 6f31 203d 2052 2020 2020 2020 |.|o1 = R │ │ │ │ 00021060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021080: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00021090: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ -000210a0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00021080: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00021090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000210a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000210b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000210c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000210d0: 2d2d 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 2b0a 7c69 3220 3a20 493d 6964 ----+.|i2 : I=id │ │ │ │ -00021110: 6561 6c28 7261 6e64 6f6d 2832 2c52 292c eal(random(2,R), │ │ │ │ -00021120: 7261 6e64 6f6d 2831 2c52 292c 525f 302a random(1,R),R_0* │ │ │ │ -00021130: 525f 312a 525f 362d 525f 305e 3329 3b7c R_1*R_6-R_0^3);| │ │ │ │ -00021140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00021150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021170: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ -00021180: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +000210c0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +000210d0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +000210e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000210f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021100: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00021110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00021140: 6932 203a 2049 3d69 6465 616c 2872 616e i2 : I=ideal(ran │ │ │ │ +00021150: 646f 6d28 322c 5229 2c72 616e 646f 6d28 dom(2,R),random( │ │ │ │ +00021160: 312c 5229 2c52 5f30 2a52 5f31 2a52 5f36 1,R),R_0*R_1*R_6 │ │ │ │ +00021170: 2d52 5f30 5e33 293b 7c0a 7c20 2020 2020 -R_0^3);|.| │ │ │ │ +00021180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000211a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000211b0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 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: 2b0a 7c69 3320 3a20 7469 6d65 2043 534d +.|i3 : time CSM │ │ │ │ -00021200: 2049 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ -00021210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021220: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00021230: 2d20 7573 6564 2031 2e31 3939 3131 7320 - used 1.19911s │ │ │ │ -00021240: 2863 7075 293b 2030 2e38 3737 3230 3973 (cpu); 0.877209s │ │ │ │ -00021250: 2028 7468 7265 6164 293b 2030 7320 2867 (thread); 0s (g │ │ │ │ -00021260: 6329 2020 2020 7c0a 7c20 2020 2020 2020 c) |.| │ │ │ │ -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: 207c 0a7c 2020 2020 2020 2020 3520 2020 |.| 5 │ │ │ │ -000212b0: 2020 2034 2020 2020 2033 2020 2020 2020 4 3 │ │ │ │ +000211b0: 2020 207c 0a7c 6f32 203a 2049 6465 616c |.|o2 : Ideal │ │ │ │ +000211c0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +000211d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000211e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000211f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00021200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021220: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00021230: 2074 696d 6520 4353 4d20 4920 2020 2020 time CSM I │ │ │ │ +00021240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021260: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +00021270: 322e 3139 3339 3873 2028 6370 7529 3b20 2.19398s (cpu); │ │ │ │ +00021280: 302e 3933 3034 3973 2028 7468 7265 6164 0.93049s (thread │ │ │ │ +00021290: 293b 2030 7320 2867 6329 2020 2020 207c ); 0s (gc) | │ │ │ │ +000212a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000212b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000212c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000212d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000212e0: 3320 3d20 3132 6820 202b 2031 3068 2020 3 = 12h + 10h │ │ │ │ -000212f0: 2b20 3668 2020 2020 2020 2020 2020 2020 + 6h │ │ │ │ +000212d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000212e0: 2020 2020 2035 2020 2020 2020 3420 2020 5 4 │ │ │ │ +000212f0: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00021300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021310: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00021320: 2020 3120 2020 2020 2031 2020 2020 2031 1 1 1 │ │ │ │ +00021310: 2020 2020 207c 0a7c 6f33 203d 2031 3268 |.|o3 = 12h │ │ │ │ +00021320: 2020 2b20 3130 6820 202b 2036 6820 2020 + 10h + 6h │ │ │ │ 00021330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021350: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021350: 7c0a 7c20 2020 2020 2020 2031 2020 2020 |.| 1 │ │ │ │ +00021360: 2020 3120 2020 2020 3120 2020 2020 2020 1 1 │ │ │ │ 00021370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021380: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00021390: 2020 2020 205a 5a5b 6820 5d20 2020 2020 ZZ[h ] │ │ │ │ +00021380: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00021390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000213a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000213b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000213c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000213d0: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ +000213c0: 2020 2020 2020 7c0a 7c20 2020 2020 5a5a |.| ZZ │ │ │ │ +000213d0: 5b68 205d 2020 2020 2020 2020 2020 2020 [h ] │ │ │ │ 000213e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000213f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021400: 2020 207c 0a7c 6f33 203a 202d 2d2d 2d2d |.|o3 : ----- │ │ │ │ -00021410: 2d20 2020 2020 2020 2020 2020 2020 2020 - │ │ │ │ +00021400: 207c 0a7c 2020 2020 2020 2020 2031 2020 |.| 1 │ │ │ │ +00021410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00021440: 7c20 2020 2020 2020 2037 2020 2020 2020 | 7 │ │ │ │ +00021430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00021440: 3320 3a20 2d2d 2d2d 2d2d 2020 2020 2020 3 : ------ │ │ │ │ 00021450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021470: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00021480: 2020 2068 2020 2020 2020 2020 2020 2020 h │ │ │ │ +00021470: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00021480: 2020 3720 2020 2020 2020 2020 2020 2020 7 │ │ │ │ 00021490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000214a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214b0: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ +000214b0: 2020 7c0a 7c20 2020 2020 2020 6820 2020 |.| h │ │ │ │ 000214c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000214d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000214e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000214f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ -00021500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021520: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ -00021530: 3a20 7469 6d65 2043 534d 2849 2c4d 6574 : time CSM(I,Met │ │ │ │ -00021540: 686f 643d 3e44 6972 6563 7443 6f6d 706c hod=>DirectCompl │ │ │ │ -00021550: 6574 6549 6e74 2920 2020 2020 2020 2020 eteInt) │ │ │ │ -00021560: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00021570: 2030 2e33 3130 3536 3273 2028 6370 7529 0.310562s (cpu) │ │ │ │ -00021580: 3b20 302e 3233 3333 3431 7320 2874 6872 ; 0.233341s (thr │ │ │ │ -00021590: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ -000215a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000215b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000215e0: 2020 2020 2020 3520 2020 2020 2034 2020 5 4 │ │ │ │ -000215f0: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +000214e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000214f0: 2020 2020 2020 2020 3120 2020 2020 2020 1 │ │ │ │ +00021500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021520: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00021530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021560: 2d2d 2d2b 0a7c 6934 203a 2074 696d 6520 ---+.|i4 : time │ │ │ │ +00021570: 4353 4d28 492c 4d65 7468 6f64 3d3e 4469 CSM(I,Method=>Di │ │ │ │ +00021580: 7265 6374 436f 6d70 6c65 7465 496e 7429 rectCompleteInt) │ │ │ │ +00021590: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000215a0: 7c20 2d2d 2075 7365 6420 302e 3536 3836 | -- used 0.5686 │ │ │ │ +000215b0: 3132 7320 2863 7075 293b 2030 2e32 3131 12s (cpu); 0.211 │ │ │ │ +000215c0: 3531 3773 2028 7468 7265 6164 293b 2030 517s (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 │ │ │ │ 00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021610: 2020 2020 2020 7c0a 7c6f 3420 3d20 3132 |.|o4 = 12 │ │ │ │ -00021620: 6820 202b 2031 3068 2020 2b20 3668 2020 h + 10h + 6h │ │ │ │ +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 2020 │ │ │ │ -00021650: 207c 0a7c 2020 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ -00021660: 2020 2031 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +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 │ │ │ │ 00021670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00021690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216a0: 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 │ │ │ │ +000216a0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 000216b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216c0: 2020 2020 2020 207c 0a7c 2020 2020 205a |.| Z │ │ │ │ -000216d0: 5a5b 6820 5d20 2020 2020 2020 2020 2020 Z[h ] │ │ │ │ +000216c0: 2020 2020 207c 0a7c 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 7c0a 7c20 2020 2020 2020 2020 3120 |.| 1 │ │ │ │ +00021700: 7c0a 7c20 2020 2020 5a5a 5b68 205d 2020 |.| ZZ[h ] │ │ │ │ 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 207c 0a7c |.| │ │ │ │ -00021740: 6f34 203a 202d 2d2d 2d2d 2d20 2020 2020 o4 : ------ │ │ │ │ +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 │ │ │ │ 00021760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021770: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00021780: 2020 2037 2020 2020 2020 2020 2020 2020 7 │ │ │ │ +00021770: 2020 2020 2020 7c0a 7c6f 3420 3a20 2d2d |.|o4 : -- │ │ │ │ +00021780: 2d2d 2d2d 2020 2020 2020 2020 2020 2020 ---- │ │ │ │ 00021790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000217a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000217b0: 2020 207c 0a7c 2020 2020 2020 2068 2020 |.| h │ │ │ │ +000217b0: 207c 0a7c 2020 2020 2020 2020 3720 2020 |.| 7 │ │ │ │ 000217c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000217d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000217e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000217f0: 7c20 2020 2020 2020 2031 2020 2020 2020 | 1 │ │ │ │ +000217e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000217f0: 2020 2020 2020 6820 2020 2020 2020 2020 h │ │ │ │ 00021800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021820: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00021830: 2d2d 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 ---------------- │ │ │ │ -00021860: 2d2d 2d2d 2b0a 0a57 6865 6e20 7573 696e ----+..When usin │ │ │ │ -00021870: 6720 7468 6520 4469 7265 6374 436f 6d70 g the DirectComp │ │ │ │ -00021880: 6c65 7465 496e 7420 6d65 7468 6f64 206f leteInt method o │ │ │ │ -00021890: 6e65 206d 6179 2070 6f74 656e 7469 616c ne may potential │ │ │ │ -000218a0: 6c79 2066 7572 7468 6572 2073 7065 6564 ly further speed │ │ │ │ -000218b0: 2075 700a 636f 6d70 7574 6174 696f 6e20 up.computation │ │ │ │ -000218c0: 7469 6d65 2062 7920 7370 6563 6966 7969 time by specifyi │ │ │ │ -000218d0: 6e67 2077 6861 7420 7375 6273 6574 206f ng what subset o │ │ │ │ -000218e0: 6620 7468 6520 6765 6e65 7261 746f 7273 f the generators │ │ │ │ -000218f0: 206f 6620 7468 6520 696e 7075 7420 6964 of the input id │ │ │ │ -00021900: 6561 6c0a 6465 6669 6e65 2061 2073 6d6f eal.define a smo │ │ │ │ -00021910: 6f74 6820 7375 6273 6368 656d 6520 2869 oth subscheme (i │ │ │ │ -00021920: 6620 7468 6973 2069 7320 6b6e 6f77 6e29 f this is known) │ │ │ │ -00021930: 2c20 7365 6520 2a6e 6f74 6520 496e 6473 , see *note Inds │ │ │ │ -00021940: 4f66 536d 6f6f 7468 3a0a 496e 6473 4f66 OfSmooth:.IndsOf │ │ │ │ -00021950: 536d 6f6f 7468 2c2e 0a0a 4675 6e63 7469 Smooth,...Functi │ │ │ │ -00021960: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00021970: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ -00021980: 204d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d Method:.======= │ │ │ │ -00021990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000219b0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 4353 =======.. * "CS │ │ │ │ -000219c0: 4d28 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e M(...,Method=>.. │ │ │ │ -000219d0: 2e29 2220 2d2d 2073 6565 202a 6e6f 7465 .)" -- see *note │ │ │ │ -000219e0: 2043 534d 3a20 4353 4d2c 202d 2d20 5468 CSM: CSM, -- Th │ │ │ │ -000219f0: 650a 2020 2020 4368 6572 6e2d 5363 6877 e. Chern-Schw │ │ │ │ -00021a00: 6172 747a 2d4d 6163 5068 6572 736f 6e20 artz-MacPherson │ │ │ │ -00021a10: 636c 6173 730a 2020 2a20 4575 6c65 7228 class. * Euler( │ │ │ │ -00021a20: 2e2e 2e2c 4d65 7468 6f64 3d3e 2e2e 2e29 ...,Method=>...) │ │ │ │ -00021a30: 2028 6d69 7373 696e 6720 646f 6375 6d65 (missing docume │ │ │ │ -00021a40: 6e74 6174 696f 6e29 0a0a 466f 7220 7468 ntation)..For th │ │ │ │ -00021a50: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00021a60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00021a70: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00021a80: 6520 4d65 7468 6f64 3a20 4d65 7468 6f64 e Method: Method │ │ │ │ -00021a90: 2c20 6973 2061 202a 6e6f 7465 2073 796d , is a *note sym │ │ │ │ -00021aa0: 626f 6c3a 2028 4d61 6361 756c 6179 3244 bol: (Macaulay2D │ │ │ │ -00021ab0: 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2d2d oc)Symbol,...--- │ │ │ │ -00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ -00021b10: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -00021b20: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -00021b30: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -00021b40: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -00021b50: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -00021b60: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -00021b70: 6765 732f 0a43 6861 7261 6374 6572 6973 ges/.Characteris │ │ │ │ -00021b80: 7469 6343 6c61 7373 6573 2e6d 323a 3234 ticClasses.m2:24 │ │ │ │ -00021b90: 3332 3a30 2e0a 1f0a 4669 6c65 3a20 4368 32:0....File: Ch │ │ │ │ -00021ba0: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -00021bb0: 7365 732e 696e 666f 2c20 4e6f 6465 3a20 ses.info, Node: │ │ │ │ -00021bc0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -00021bd0: 6e67 2c20 4e65 7874 3a20 4f75 7470 7574 ng, Next: Output │ │ │ │ -00021be0: 2c20 5072 6576 3a20 4d65 7468 6f64 2c20 , Prev: Method, │ │ │ │ -00021bf0: 5570 3a20 546f 700a 0a4d 756c 7469 5072 Up: Top..MultiPr │ │ │ │ -00021c00: 6f6a 436f 6f72 6452 696e 6720 2d2d 2041 ojCoordRing -- A │ │ │ │ -00021c10: 2071 7569 636b 2077 6179 2074 6f20 6275 quick way to bu │ │ │ │ -00021c20: 696c 6420 7468 6520 636f 6f72 6469 6e61 ild the coordina │ │ │ │ -00021c30: 7465 2072 696e 6720 6f66 2061 2070 726f te ring of a pro │ │ │ │ -00021c40: 6475 6374 206f 6620 7072 6f6a 6563 7469 duct of projecti │ │ │ │ -00021c50: 7665 2073 7061 6365 730a 2a2a 2a2a 2a2a ve spaces.****** │ │ │ │ -00021c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021820: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00021830: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00021840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021860: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00021870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000218a0: 5768 656e 2075 7369 6e67 2074 6865 2044 When using the D │ │ │ │ +000218b0: 6972 6563 7443 6f6d 706c 6574 6549 6e74 irectCompleteInt │ │ │ │ +000218c0: 206d 6574 686f 6420 6f6e 6520 6d61 7920 method one may │ │ │ │ +000218d0: 706f 7465 6e74 6961 6c6c 7920 6675 7274 potentially furt │ │ │ │ +000218e0: 6865 7220 7370 6565 6420 7570 0a63 6f6d her speed up.com │ │ │ │ +000218f0: 7075 7461 7469 6f6e 2074 696d 6520 6279 putation time by │ │ │ │ +00021900: 2073 7065 6369 6679 696e 6720 7768 6174 specifying what │ │ │ │ +00021910: 2073 7562 7365 7420 6f66 2074 6865 2067 subset of the g │ │ │ │ +00021920: 656e 6572 6174 6f72 7320 6f66 2074 6865 enerators of the │ │ │ │ +00021930: 2069 6e70 7574 2069 6465 616c 0a64 6566 input ideal.def │ │ │ │ +00021940: 696e 6520 6120 736d 6f6f 7468 2073 7562 ine a smooth sub │ │ │ │ +00021950: 7363 6865 6d65 2028 6966 2074 6869 7320 scheme (if this │ │ │ │ +00021960: 6973 206b 6e6f 776e 292c 2073 6565 202a is known), see * │ │ │ │ +00021970: 6e6f 7465 2049 6e64 734f 6653 6d6f 6f74 note IndsOfSmoot │ │ │ │ +00021980: 683a 0a49 6e64 734f 6653 6d6f 6f74 682c h:.IndsOfSmooth, │ │ │ │ +00021990: 2e0a 0a46 756e 6374 696f 6e73 2077 6974 ...Functions wit │ │ │ │ +000219a0: 6820 6f70 7469 6f6e 616c 2061 7267 756d h optional argum │ │ │ │ +000219b0: 656e 7420 6e61 6d65 6420 4d65 7468 6f64 ent named Method │ │ │ │ +000219c0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +000219d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000219e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000219f0: 0a0a 2020 2a20 2243 534d 282e 2e2e 2c4d .. * "CSM(...,M │ │ │ │ +00021a00: 6574 686f 643d 3e2e 2e2e 2922 202d 2d20 ethod=>...)" -- │ │ │ │ +00021a10: 7365 6520 2a6e 6f74 6520 4353 4d3a 2043 see *note CSM: C │ │ │ │ +00021a20: 534d 2c20 2d2d 2054 6865 0a20 2020 2043 SM, -- The. C │ │ │ │ +00021a30: 6865 726e 2d53 6368 7761 7274 7a2d 4d61 hern-Schwartz-Ma │ │ │ │ +00021a40: 6350 6865 7273 6f6e 2063 6c61 7373 0a20 cPherson class. │ │ │ │ +00021a50: 202a 2045 756c 6572 282e 2e2e 2c4d 6574 * Euler(...,Met │ │ │ │ +00021a60: 686f 643d 3e2e 2e2e 2920 286d 6973 7369 hod=>...) (missi │ │ │ │ +00021a70: 6e67 2064 6f63 756d 656e 7461 7469 6f6e ng documentation │ │ │ │ +00021a80: 290a 0a46 6f72 2074 6865 2070 726f 6772 )..For the progr │ │ │ │ +00021a90: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00021aa0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00021ab0: 6a65 6374 202a 6e6f 7465 204d 6574 686f ject *note Metho │ │ │ │ +00021ac0: 643a 204d 6574 686f 642c 2069 7320 6120 d: Method, is a │ │ │ │ +00021ad0: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d *note symbol: (M │ │ │ │ +00021ae0: 6163 6175 6c61 7932 446f 6329 5379 6d62 acaulay2Doc)Symb │ │ │ │ +00021af0: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ +00021b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021b40: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +00021b50: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +00021b60: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +00021b70: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +00021b80: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +00021b90: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +00021ba0: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ +00021bb0: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ +00021bc0: 7365 732e 6d32 3a32 3433 323a 302e 0a1f ses.m2:2432:0... │ │ │ │ +00021bd0: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ +00021be0: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ +00021bf0: 6f2c 204e 6f64 653a 204d 756c 7469 5072 o, Node: MultiPr │ │ │ │ +00021c00: 6f6a 436f 6f72 6452 696e 672c 204e 6578 ojCoordRing, Nex │ │ │ │ +00021c10: 743a 204f 7574 7075 742c 2050 7265 763a t: Output, Prev: │ │ │ │ +00021c20: 204d 6574 686f 642c 2055 703a 2054 6f70 Method, Up: Top │ │ │ │ +00021c30: 0a0a 4d75 6c74 6950 726f 6a43 6f6f 7264 ..MultiProjCoord │ │ │ │ +00021c40: 5269 6e67 202d 2d20 4120 7175 6963 6b20 Ring -- A quick │ │ │ │ +00021c50: 7761 7920 746f 2062 7569 6c64 2074 6865 way to build the │ │ │ │ +00021c60: 2063 6f6f 7264 696e 6174 6520 7269 6e67 coordinate ring │ │ │ │ +00021c70: 206f 6620 6120 7072 6f64 7563 7420 6f66 of a product of │ │ │ │ +00021c80: 2070 726f 6a65 6374 6976 6520 7370 6163 projective spac │ │ │ │ +00021c90: 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a es.************* │ │ │ │ 00021ca0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -00021cc0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -00021cd0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -00021ce0: 6e67 2044 696d 730a 2020 2020 2020 2020 ng Dims. │ │ │ │ -00021cf0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -00021d00: 6e67 2028 436f 6566 6652 696e 672c 4469 ng (CoeffRing,Di │ │ │ │ -00021d10: 6d73 290a 2020 2020 2020 2020 4d75 6c74 ms). Mult │ │ │ │ -00021d20: 6950 726f 6a43 6f6f 7264 5269 6e67 2028 iProjCoordRing ( │ │ │ │ -00021d30: 7661 722c 4469 6d73 290a 2020 2020 2020 var,Dims). │ │ │ │ -00021d40: 2020 4d75 6c74 6950 726f 6a43 6f6f 7264 MultiProjCoord │ │ │ │ -00021d50: 5269 6e67 2028 436f 6566 6652 696e 672c Ring (CoeffRing, │ │ │ │ -00021d60: 7661 722c 4469 6d73 290a 2020 2a20 496e var,Dims). * In │ │ │ │ -00021d70: 7075 7473 3a0a 2020 2020 2020 2a20 4469 puts:. * Di │ │ │ │ -00021d80: 6d73 2c20 6120 2a6e 6f74 6520 6c69 7374 ms, a *note list │ │ │ │ -00021d90: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00021da0: 4c69 7374 2c2c 2072 6570 7265 7365 6e74 List,, represent │ │ │ │ -00021db0: 696e 6720 7468 6520 6469 6d65 6e73 696f ing the dimensio │ │ │ │ -00021dc0: 6e73 206f 660a 2020 2020 2020 2020 7468 ns of. th │ │ │ │ -00021dd0: 6520 7072 6f6a 6563 7469 7665 2073 7061 e projective spa │ │ │ │ -00021de0: 6365 732c 2069 2e65 2e20 7b6e 5f31 2c2e ces, i.e. {n_1,. │ │ │ │ -00021df0: 2e2e 2c6e 5f6d 7d20 636f 7272 6573 706f ..,n_m} correspo │ │ │ │ -00021e00: 6e64 7320 746f 205c 5050 5e7b 6e5f 317d nds to \PP^{n_1} │ │ │ │ -00021e10: 0a20 2020 2020 2020 2078 2e2e 2e2e 2078 . x.... x │ │ │ │ -00021e20: 205c 5050 5e7b 6e5f 6d7d 0a20 2020 2020 \PP^{n_m}. │ │ │ │ -00021e30: 202a 2043 6f65 6666 5269 6e67 2c20 6120 * CoeffRing, a │ │ │ │ -00021e40: 2a6e 6f74 6520 7269 6e67 3a20 284d 6163 *note ring: (Mac │ │ │ │ -00021e50: 6175 6c61 7932 446f 6329 5269 6e67 2c2c aulay2Doc)Ring,, │ │ │ │ -00021e60: 2074 6865 2063 6f65 6666 6963 6965 6e74 the coefficient │ │ │ │ -00021e70: 2072 696e 6720 6f66 0a20 2020 2020 2020 ring of. │ │ │ │ -00021e80: 2074 6865 2067 7261 6465 6420 706f 6c79 the graded poly │ │ │ │ -00021e90: 6e6f 6d69 616c 2072 696e 6720 746f 2062 nomial ring to b │ │ │ │ -00021ea0: 6520 6275 696c 7420 6279 2074 6865 206d e built by the m │ │ │ │ -00021eb0: 6574 686f 642c 2062 7920 6465 6661 756c ethod, by defaul │ │ │ │ -00021ec0: 7420 7468 6973 0a20 2020 2020 2020 2069 t this. i │ │ │ │ -00021ed0: 7320 5c5a 5a2f 3332 3734 390a 2020 2020 s \ZZ/32749. │ │ │ │ -00021ee0: 2020 2a20 7661 722c 2061 202a 6e6f 7465 * var, a *note │ │ │ │ -00021ef0: 2073 796d 626f 6c3a 2028 4d61 6361 756c symbol: (Macaul │ │ │ │ -00021f00: 6179 3244 6f63 2953 796d 626f 6c2c 2c20 ay2Doc)Symbol,, │ │ │ │ -00021f10: 746f 2062 6520 7573 6564 2066 6f72 2074 to be used for t │ │ │ │ -00021f20: 6865 0a20 2020 2020 2020 2069 6e74 6572 he. inter │ │ │ │ -00021f30: 6d65 6469 6174 6573 206f 6620 7468 6520 mediates of the │ │ │ │ -00021f40: 6772 6164 6564 2070 6f6c 796e 6f6d 6961 graded polynomia │ │ │ │ -00021f50: 6c20 7269 6e67 2074 6f20 6265 2062 7569 l ring to be bui │ │ │ │ -00021f60: 6c74 2062 7920 7468 6520 6d65 7468 6f64 lt by the method │ │ │ │ -00021f70: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -00021f80: 2020 2020 2a20 6120 2a6e 6f74 6520 7269 * a *note ri │ │ │ │ -00021f90: 6e67 3a20 284d 6163 6175 6c61 7932 446f ng: (Macaulay2Do │ │ │ │ -00021fa0: 6329 5269 6e67 2c2c 2074 6865 2067 7261 c)Ring,, the gra │ │ │ │ -00021fb0: 6465 6420 636f 6f72 6469 6e61 7465 2072 ded coordinate r │ │ │ │ -00021fc0: 696e 6720 6f66 2074 6865 0a20 2020 2020 ing of the. │ │ │ │ -00021fd0: 2020 205c 5050 5e7b 6e5f 317d 2078 2e2e \PP^{n_1} x.. │ │ │ │ -00021fe0: 2e2e 2078 205c 5050 5e7b 6e5f 6d7d 2077 .. x \PP^{n_m} w │ │ │ │ -00021ff0: 6865 7265 207b 6e5f 312c 2e2e 2e2c 6e5f here {n_1,...,n_ │ │ │ │ -00022000: 6d7d 2069 7320 7468 6520 696e 7075 7420 m} is the input │ │ │ │ -00022010: 6c69 7374 206f 660a 2020 2020 2020 2020 list of. │ │ │ │ -00022020: 6469 6d65 6e73 696f 6e73 0a0a 4465 7363 dimensions..Desc │ │ │ │ -00022030: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -00022040: 3d3d 3d0a 0a43 6f6d 7075 7465 7320 7468 ===..Computes th │ │ │ │ -00022050: 6520 6772 6164 6564 2063 6f6f 7264 696e e graded coordin │ │ │ │ -00022060: 6174 6520 7269 6e67 206f 6620 7468 6520 ate ring of the │ │ │ │ -00022070: 5c50 505e 7b6e 5f31 7d20 782e 2e2e 2e20 \PP^{n_1} x.... │ │ │ │ -00022080: 7820 5c50 505e 7b6e 5f6d 7d20 7768 6572 x \PP^{n_m} wher │ │ │ │ -00022090: 650a 7b6e 5f31 2c2e 2e2e 2c6e 5f6d 7d20 e.{n_1,...,n_m} │ │ │ │ -000220a0: 6973 2074 6865 2069 6e70 7574 206c 6973 is the input lis │ │ │ │ -000220b0: 7420 6f66 2064 696d 656e 7369 6f6e 732e t of dimensions. │ │ │ │ -000220c0: 2054 6869 7320 6d65 7468 6f64 2069 7320 This method is │ │ │ │ -000220d0: 7573 6564 2074 6f20 7175 6963 6b6c 790a used to quickly. │ │ │ │ -000220e0: 6275 696c 6420 7468 6520 636f 6f72 6469 build the coordi │ │ │ │ -000220f0: 6e61 7465 2072 696e 6720 6f66 2061 2070 nate ring of a p │ │ │ │ -00022100: 726f 6475 6374 206f 6620 7072 6f6a 6563 roduct of projec │ │ │ │ -00022110: 7469 7665 2073 7061 6365 7320 666f 7220 tive spaces for │ │ │ │ -00022120: 7573 6520 696e 0a63 6f6d 7075 7461 7469 use in.computati │ │ │ │ -00022130: 6f6e 732e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ons...+--------- │ │ │ │ -00022140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00021cb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021ce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00021cf0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00021d00: 0a20 2020 2020 2020 204d 756c 7469 5072 . MultiPr │ │ │ │ +00021d10: 6f6a 436f 6f72 6452 696e 6720 4469 6d73 ojCoordRing Dims │ │ │ │ +00021d20: 0a20 2020 2020 2020 204d 756c 7469 5072 . MultiPr │ │ │ │ +00021d30: 6f6a 436f 6f72 6452 696e 6720 2843 6f65 ojCoordRing (Coe │ │ │ │ +00021d40: 6666 5269 6e67 2c44 696d 7329 0a20 2020 ffRing,Dims). │ │ │ │ +00021d50: 2020 2020 204d 756c 7469 5072 6f6a 436f MultiProjCo │ │ │ │ +00021d60: 6f72 6452 696e 6720 2876 6172 2c44 696d ordRing (var,Dim │ │ │ │ +00021d70: 7329 0a20 2020 2020 2020 204d 756c 7469 s). Multi │ │ │ │ +00021d80: 5072 6f6a 436f 6f72 6452 696e 6720 2843 ProjCoordRing (C │ │ │ │ +00021d90: 6f65 6666 5269 6e67 2c76 6172 2c44 696d oeffRing,var,Dim │ │ │ │ +00021da0: 7329 0a20 202a 2049 6e70 7574 733a 0a20 s). * Inputs:. │ │ │ │ +00021db0: 2020 2020 202a 2044 696d 732c 2061 202a * Dims, a * │ │ │ │ +00021dc0: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +00021dd0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +00021de0: 7265 7072 6573 656e 7469 6e67 2074 6865 representing the │ │ │ │ +00021df0: 2064 696d 656e 7369 6f6e 7320 6f66 0a20 dimensions of. │ │ │ │ +00021e00: 2020 2020 2020 2074 6865 2070 726f 6a65 the proje │ │ │ │ +00021e10: 6374 6976 6520 7370 6163 6573 2c20 692e ctive spaces, i. │ │ │ │ +00021e20: 652e 207b 6e5f 312c 2e2e 2e2c 6e5f 6d7d e. {n_1,...,n_m} │ │ │ │ +00021e30: 2063 6f72 7265 7370 6f6e 6473 2074 6f20 corresponds to │ │ │ │ +00021e40: 5c50 505e 7b6e 5f31 7d0a 2020 2020 2020 \PP^{n_1}. │ │ │ │ +00021e50: 2020 782e 2e2e 2e20 7820 5c50 505e 7b6e x.... x \PP^{n │ │ │ │ +00021e60: 5f6d 7d0a 2020 2020 2020 2a20 436f 6566 _m}. * Coef │ │ │ │ +00021e70: 6652 696e 672c 2061 202a 6e6f 7465 2072 fRing, a *note r │ │ │ │ +00021e80: 696e 673a 2028 4d61 6361 756c 6179 3244 ing: (Macaulay2D │ │ │ │ +00021e90: 6f63 2952 696e 672c 2c20 7468 6520 636f oc)Ring,, the co │ │ │ │ +00021ea0: 6566 6669 6369 656e 7420 7269 6e67 206f efficient ring o │ │ │ │ +00021eb0: 660a 2020 2020 2020 2020 7468 6520 6772 f. the gr │ │ │ │ +00021ec0: 6164 6564 2070 6f6c 796e 6f6d 6961 6c20 aded polynomial │ │ │ │ +00021ed0: 7269 6e67 2074 6f20 6265 2062 7569 6c74 ring to be built │ │ │ │ +00021ee0: 2062 7920 7468 6520 6d65 7468 6f64 2c20 by the method, │ │ │ │ +00021ef0: 6279 2064 6566 6175 6c74 2074 6869 730a by default this. │ │ │ │ +00021f00: 2020 2020 2020 2020 6973 205c 5a5a 2f33 is \ZZ/3 │ │ │ │ +00021f10: 3237 3439 0a20 2020 2020 202a 2076 6172 2749. * var │ │ │ │ +00021f20: 2c20 6120 2a6e 6f74 6520 7379 6d62 6f6c , a *note symbol │ │ │ │ +00021f30: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021f40: 5379 6d62 6f6c 2c2c 2074 6f20 6265 2075 Symbol,, to be u │ │ │ │ +00021f50: 7365 6420 666f 7220 7468 650a 2020 2020 sed for the. │ │ │ │ +00021f60: 2020 2020 696e 7465 726d 6564 6961 7465 intermediate │ │ │ │ +00021f70: 7320 6f66 2074 6865 2067 7261 6465 6420 s of the graded │ │ │ │ +00021f80: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +00021f90: 746f 2062 6520 6275 696c 7420 6279 2074 to be built by t │ │ │ │ +00021fa0: 6865 206d 6574 686f 640a 2020 2a20 4f75 he method. * Ou │ │ │ │ +00021fb0: 7470 7574 733a 0a20 2020 2020 202a 2061 tputs:. * a │ │ │ │ +00021fc0: 202a 6e6f 7465 2072 696e 673a 2028 4d61 *note ring: (Ma │ │ │ │ +00021fd0: 6361 756c 6179 3244 6f63 2952 696e 672c caulay2Doc)Ring, │ │ │ │ +00021fe0: 2c20 7468 6520 6772 6164 6564 2063 6f6f , the graded coo │ │ │ │ +00021ff0: 7264 696e 6174 6520 7269 6e67 206f 6620 rdinate ring of │ │ │ │ +00022000: 7468 650a 2020 2020 2020 2020 5c50 505e the. \PP^ │ │ │ │ +00022010: 7b6e 5f31 7d20 782e 2e2e 2e20 7820 5c50 {n_1} x.... x \P │ │ │ │ +00022020: 505e 7b6e 5f6d 7d20 7768 6572 6520 7b6e P^{n_m} where {n │ │ │ │ +00022030: 5f31 2c2e 2e2e 2c6e 5f6d 7d20 6973 2074 _1,...,n_m} is t │ │ │ │ +00022040: 6865 2069 6e70 7574 206c 6973 7420 6f66 he input list of │ │ │ │ +00022050: 0a20 2020 2020 2020 2064 696d 656e 7369 . dimensi │ │ │ │ +00022060: 6f6e 730a 0a44 6573 6372 6970 7469 6f6e ons..Description │ │ │ │ +00022070: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f .===========..Co │ │ │ │ +00022080: 6d70 7574 6573 2074 6865 2067 7261 6465 mputes the grade │ │ │ │ +00022090: 6420 636f 6f72 6469 6e61 7465 2072 696e d coordinate rin │ │ │ │ +000220a0: 6720 6f66 2074 6865 205c 5050 5e7b 6e5f g of the \PP^{n_ │ │ │ │ +000220b0: 317d 2078 2e2e 2e2e 2078 205c 5050 5e7b 1} x.... x \PP^{ │ │ │ │ +000220c0: 6e5f 6d7d 2077 6865 7265 0a7b 6e5f 312c n_m} where.{n_1, │ │ │ │ +000220d0: 2e2e 2e2c 6e5f 6d7d 2069 7320 7468 6520 ...,n_m} is the │ │ │ │ +000220e0: 696e 7075 7420 6c69 7374 206f 6620 6469 input list of di │ │ │ │ +000220f0: 6d65 6e73 696f 6e73 2e20 5468 6973 206d mensions. This m │ │ │ │ +00022100: 6574 686f 6420 6973 2075 7365 6420 746f ethod is used to │ │ │ │ +00022110: 2071 7569 636b 6c79 0a62 7569 6c64 2074 quickly.build t │ │ │ │ +00022120: 6865 2063 6f6f 7264 696e 6174 6520 7269 he coordinate ri │ │ │ │ +00022130: 6e67 206f 6620 6120 7072 6f64 7563 7420 ng of a product │ │ │ │ +00022140: 6f66 2070 726f 6a65 6374 6976 6520 7370 of projective sp │ │ │ │ +00022150: 6163 6573 2066 6f72 2075 7365 2069 6e0a aces for use in. │ │ │ │ +00022160: 636f 6d70 7574 6174 696f 6e73 2e0a 0a2b computations...+ │ │ │ │ 00022170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022180: 2d2d 2d2d 2b0a 7c69 3120 3a20 533d 4d75 ----+.|i1 : S=Mu │ │ │ │ -00022190: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -000221a0: 2851 512c 7379 6d62 6f6c 207a 2c7b 312c (QQ,symbol z,{1, │ │ │ │ -000221b0: 332c 337d 2920 2020 2020 2020 2020 2020 3,3}) │ │ │ │ -000221c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000221d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000221e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000221b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000221c0: 6931 203a 2053 3d4d 756c 7469 5072 6f6a i1 : S=MultiProj │ │ │ │ +000221d0: 436f 6f72 6452 696e 6728 5151 2c73 796d CoordRing(QQ,sym │ │ │ │ +000221e0: 626f 6c20 7a2c 7b31 2c33 2c33 7d29 2020 bol z,{1,3,3}) │ │ │ │ 000221f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022220: 2020 2020 7c0a 7c6f 3120 3d20 5320 2020 |.|o1 = S │ │ │ │ +00022220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022270: 2020 2020 7c0a 7c20 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 │ │ │ │ 00022280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000222a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000222b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222c0: 2020 2020 7c0a 7c6f 3120 3a20 506f 6c79 |.|o1 : Poly │ │ │ │ -000222d0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000222d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022310: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00022320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000222f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022300: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00022310: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00022320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022340: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 00022350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022360: 2d2d 2d2d 2b0a 7c69 3220 3a20 6465 6772 ----+.|i2 : degr │ │ │ │ -00022370: 6565 7320 5320 2020 2020 2020 2020 2020 ees 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000223a0: 6932 203a 2064 6567 7265 6573 2053 2020 i2 : degrees S │ │ │ │ +000223b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000223e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 000223f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022400: 2020 2020 7c0a 7c6f 3220 3d20 7b7b 312c |.|o2 = {{1, │ │ │ │ -00022410: 2030 2c20 307d 2c20 7b31 2c20 302c 2030 0, 0}, {1, 0, 0 │ │ │ │ -00022420: 7d2c 207b 302c 2031 2c20 307d 2c20 7b30 }, {0, 1, 0}, {0 │ │ │ │ -00022430: 2c20 312c 2030 7d2c 207b 302c 2031 2c20 , 1, 0}, {0, 1, │ │ │ │ -00022440: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b 0}, {0, 1, 0}, { │ │ │ │ -00022450: 302c 2020 7c0a 7c20 2020 2020 2d2d 2d2d 0, |.| ---- │ │ │ │ -00022460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000224a0: 2d2d 2d2d 7c0a 7c20 2020 2020 302c 2031 ----|.| 0, 1 │ │ │ │ -000224b0: 7d2c 207b 302c 2030 2c20 317d 2c20 7b30 }, {0, 0, 1}, {0 │ │ │ │ -000224c0: 2c20 302c 2031 7d2c 207b 302c 2030 2c20 , 0, 1}, {0, 0, │ │ │ │ -000224d0: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ -000224e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00022500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022430: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022440: 6f32 203d 207b 7b31 2c20 302c 2030 7d2c o2 = {{1, 0, 0}, │ │ │ │ +00022450: 207b 312c 2030 2c20 307d 2c20 7b30 2c20 {1, 0, 0}, {0, │ │ │ │ +00022460: 312c 2030 7d2c 207b 302c 2031 2c20 307d 1, 0}, {0, 1, 0} │ │ │ │ +00022470: 2c20 7b30 2c20 312c 2030 7d2c 207b 302c , {0, 1, 0}, {0, │ │ │ │ +00022480: 2031 2c20 307d 2c20 7b30 2c20 207c 0a7c 1, 0}, {0, |.| │ │ │ │ +00022490: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +000224a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000224b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000224c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000224d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +000224e0: 2020 2020 2030 2c20 317d 2c20 7b30 2c20 0, 1}, {0, │ │ │ │ +000224f0: 302c 2031 7d2c 207b 302c 2030 2c20 317d 0, 1}, {0, 0, 1} │ │ │ │ +00022500: 2c20 7b30 2c20 302c 2031 7d7d 2020 2020 , {0, 0, 1}} │ │ │ │ 00022510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022520: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022540: 2020 2020 7c0a 7c6f 3220 3a20 4c69 7374 |.|o2 : List │ │ │ │ +00022540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022590: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -000225a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022570: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022580: 6f32 203a 204c 6973 7420 2020 2020 2020 o2 : List │ │ │ │ +00022590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000225c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 000225d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000225e0: 2d2d 2d2d 2b0a 7c69 3320 3a20 523d 4d75 ----+.|i3 : R=Mu │ │ │ │ -000225f0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ -00022600: 207b 322c 337d 2020 2020 2020 2020 2020 {2,3} │ │ │ │ -00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022630: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000225e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000225f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00022610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00022620: 6933 203a 2052 3d4d 756c 7469 5072 6f6a i3 : R=MultiProj │ │ │ │ +00022630: 436f 6f72 6452 696e 6720 7b32 2c33 7d20 CoordRing {2,3} │ │ │ │ 00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022660: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022680: 2020 2020 7c0a 7c6f 3320 3d20 5220 2020 |.|o3 = R │ │ │ │ +00022680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000226a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000226d0: 2020 2020 7c0a 7c20 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 │ │ │ │ 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 2020 2020 │ │ │ │ +00022700: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022720: 2020 2020 7c0a 7c6f 3320 3a20 506f 6c79 |.|o3 : Poly 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229a0: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74 |.|o4 : Quot │ │ │ │ -000229b0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +000229a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000229b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 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 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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 |.+ │ │ │ │ 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2020 2020 │ │ │ │ -00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022ca0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cc0: 2020 2020 7c0a 7c6f 3620 3d20 4120 2020 |.|o6 = A │ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d10: 2020 2020 7c0a 7c20 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 │ │ │ │ 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 2020 2020 │ │ │ │ +00022d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d60: 2020 2020 7c0a 7c6f 3620 3a20 5175 6f74 |.|o6 : Quot │ │ │ │ -00022d70: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00022d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022db0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 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 ---------------- │ │ │ │ +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 |.+ │ │ │ │ 00022df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022e00: 2d2d 2d2d 2b0a 7c69 3720 3a20 6465 7363 ----+.|i7 : desc │ │ │ │ -00022e10: 7269 6265 2041 2020 2020 2020 2020 2020 ribe 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +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 │ │ │ │ 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 2020 2020 │ │ │ │ +00022e80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00022e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ea0: 2020 2020 7c0a 7c20 2020 2020 5a5a 5b68 |.| ZZ[h │ │ │ │ -00022eb0: 202e 2e68 205d 2020 2020 2020 2020 2020 ..h ] │ │ │ │ +00022ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ec0: 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 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00022f00: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +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 │ │ │ │ 00022f10: 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 7c0a 7c6f 3720 3d20 2d2d 2d2d |.|o7 = ---- │ │ │ │ -00022f50: 2d2d 2d2d 2d2d 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 │ │ │ │ 00022f60: 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 7c0a 7c20 2020 2020 2020 2033 |.| 3 │ │ │ │ -00022fa0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +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 │ │ │ │ 00022fb0: 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 7c0a 7c20 2020 2020 2028 6820 |.| (h │ │ │ │ -00022ff0: 2c20 6820 2920 2020 2020 2020 2020 2020 , h ) │ │ │ │ +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 │ │ │ │ 00023000: 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 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ -00023040: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +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 │ │ │ │ 00023050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023080: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 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 ---------------- │ │ │ │ +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 |.+ │ │ │ │ 000230c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000230d0: 2d2d 2d2d 2b0a 7c69 3820 3a20 5365 6772 ----+.|i8 : Segr │ │ │ │ -000230e0: 6528 412c 6964 6561 6c20 7261 6e64 6f6d e(A,ideal random │ │ │ │ -000230f0: 287b 312c 317d 2c52 2929 2020 2020 2020 ({1,1},R)) │ │ │ │ -00023100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023120: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00023130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000230d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000230e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000230f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00023110: 6938 203a 2053 6567 7265 2841 2c69 6465 i8 : Segre(A,ide │ │ │ │ +00023120: 616c 2072 616e 646f 6d28 7b31 2c31 7d2c al random({1,1}, │ │ │ │ +00023130: 5229 2920 2020 2020 2020 2020 2020 2020 R)) │ │ │ │ 00023140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023150: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 00023160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023170: 2020 2020 7c0a 7c20 2020 2020 2020 2032 |.| 2 │ │ │ │ -00023180: 2033 2020 2020 2032 2032 2020 2020 2020 3 2 2 │ │ │ │ -00023190: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ -000231a0: 2032 2020 2020 3320 2020 2032 2020 2020 2 3 2 │ │ │ │ -000231b0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ -000231c0: 2020 2020 7c0a 7c6f 3820 3d20 3130 6820 |.|o8 = 10h │ │ │ │ -000231d0: 6820 202d 2036 6820 6820 202d 2034 6820 h - 6h h - 4h │ │ │ │ -000231e0: 6820 202b 2033 6820 6820 202b 2033 6820 h + 3h h + 3h │ │ │ │ -000231f0: 6820 202b 2068 2020 2d20 6820 202d 2032 h + h - h - 2 │ │ │ │ -00023200: 6820 6820 202d 2068 2020 2b20 6820 202b h h - h + h + │ │ │ │ -00023210: 2068 2020 7c0a 7c20 2020 2020 2020 2031 h |.| 1 │ │ │ │ -00023220: 2032 2020 2020 2031 2032 2020 2020 2031 2 1 2 1 │ │ │ │ -00023230: 2032 2020 2020 2031 2032 2020 2020 2031 2 1 2 1 │ │ │ │ -00023240: 2032 2020 2020 3220 2020 2031 2020 2020 2 2 1 │ │ │ │ -00023250: 2031 2032 2020 2020 3220 2020 2031 2020 1 2 2 1 │ │ │ │ -00023260: 2020 3220 7c0a 7c20 2020 2020 2020 2020 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 │ │ │ │ +00023170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000231b0: 2020 2020 2020 2020 3220 3320 2020 2020 2 3 │ │ │ │ +000231c0: 3220 3220 2020 2020 2020 3320 2020 2020 2 2 3 │ │ │ │ +000231d0: 3220 2020 2020 2020 2020 3220 2020 2033 2 2 3 │ │ │ │ +000231e0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +000231f0: 2032 2020 2020 2020 2020 2020 207c 0a7c 2 |.| │ │ │ │ +00023200: 6f38 203d 2031 3068 2068 2020 2d20 3668 o8 = 10h h - 6h │ │ │ │ +00023210: 2068 2020 2d20 3468 2068 2020 2b20 3368 h - 4h h + 3h │ │ │ │ +00023220: 2068 2020 2b20 3368 2068 2020 2b20 6820 h + 3h h + h │ │ │ │ +00023230: 202d 2068 2020 2d20 3268 2068 2020 2d20 - h - 2h h - │ │ │ │ +00023240: 6820 202b 2068 2020 2b20 6820 207c 0a7c h + h + h |.| │ │ │ │ +00023250: 2020 2020 2020 2020 3120 3220 2020 2020 1 2 │ │ │ │ +00023260: 3120 3220 2020 2020 3120 3220 2020 2020 1 2 1 2 │ │ │ │ +00023270: 3120 3220 2020 2020 3120 3220 2020 2032 1 2 1 2 2 │ │ │ │ +00023280: 2020 2020 3120 2020 2020 3120 3220 2020 1 1 2 │ │ │ │ +00023290: 2032 2020 2020 3120 2020 2032 207c 0a7c 2 1 2 |.| │ │ │ │ 000232a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232b0: 2020 2020 7c0a 7c6f 3820 3a20 4120 2020 |.|o8 : A │ │ │ │ +000232b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000232d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000232f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ -00023310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000232e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000232f0: 6f38 203a 2041 2020 2020 2020 2020 2020 o8 : A │ │ │ │ +00023300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023330: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 00023340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023350: 2d2d 2d2d 2b0a 0a57 6179 7320 746f 2075 ----+..Ways to u │ │ │ │ -00023360: 7365 204d 756c 7469 5072 6f6a 436f 6f72 se MultiProjCoor │ │ │ │ -00023370: 6452 696e 673a 0a3d 3d3d 3d3d 3d3d 3d3d dRing:.========= │ │ │ │ -00023380: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023390: 3d3d 3d3d 3d3d 0a0a 2020 2a20 224d 756c ======.. * "Mul │ │ │ │ -000233a0: 7469 5072 6f6a 436f 6f72 6452 696e 6728 tiProjCoordRing( │ │ │ │ -000233b0: 4c69 7374 2922 0a20 202a 2022 4d75 6c74 List)". * "Mult │ │ │ │ -000233c0: 6950 726f 6a43 6f6f 7264 5269 6e67 2852 iProjCoordRing(R │ │ │ │ -000233d0: 696e 672c 4c69 7374 2922 0a20 202a 2022 ing,List)". * " │ │ │ │ -000233e0: 4d75 6c74 6950 726f 6a43 6f6f 7264 5269 MultiProjCoordRi │ │ │ │ -000233f0: 6e67 2852 696e 672c 5379 6d62 6f6c 2c4c ng(Ring,Symbol,L │ │ │ │ -00023400: 6973 7429 220a 2020 2a20 224d 756c 7469 ist)". * "Multi │ │ │ │ -00023410: 5072 6f6a 436f 6f72 6452 696e 6728 5379 ProjCoordRing(Sy │ │ │ │ -00023420: 6d62 6f6c 2c4c 6973 7429 220a 0a46 6f72 mbol,List)"..For │ │ │ │ -00023430: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -00023440: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00023450: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -00023460: 6e6f 7465 204d 756c 7469 5072 6f6a 436f note MultiProjCo │ │ │ │ -00023470: 6f72 6452 696e 673a 204d 756c 7469 5072 ordRing: MultiPr │ │ │ │ -00023480: 6f6a 436f 6f72 6452 696e 672c 2069 7320 ojCoordRing, is │ │ │ │ -00023490: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ -000234a0: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ -000234b0: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -000234c0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ -000234d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000234e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000234f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023510: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00023520: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00023530: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00023540: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00023550: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00023560: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00023570: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00023580: 0a43 6861 7261 6374 6572 6973 7469 6343 .CharacteristicC │ │ │ │ -00023590: 6c61 7373 6573 2e6d 323a 3230 3530 3a30 lasses.m2:2050:0 │ │ │ │ -000235a0: 2e0a 1f0a 4669 6c65 3a20 4368 6172 6163 ....File: Charac │ │ │ │ -000235b0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ -000235c0: 696e 666f 2c20 4e6f 6465 3a20 4f75 7470 info, Node: Outp │ │ │ │ -000235d0: 7574 2c20 4e65 7874 3a20 7072 6f62 6162 ut, Next: probab │ │ │ │ -000235e0: 696c 6973 7469 6320 616c 676f 7269 7468 ilistic algorith │ │ │ │ -000235f0: 6d2c 2050 7265 763a 204d 756c 7469 5072 m, Prev: MultiPr │ │ │ │ -00023600: 6f6a 436f 6f72 6452 696e 672c 2055 703a ojCoordRing, Up: │ │ │ │ -00023610: 2054 6f70 0a0a 4f75 7470 7574 0a2a 2a2a Top..Output.*** │ │ │ │ -00023620: 2a2a 2a0a 0a44 6573 6372 6970 7469 6f6e ***..Description │ │ │ │ -00023630: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -00023640: 6520 6f70 7469 6f6e 204f 7574 7075 7420 e option Output │ │ │ │ -00023650: 6973 206f 6e6c 7920 7573 6564 2062 7920 is only used by │ │ │ │ -00023660: 7468 6520 636f 6d6d 616e 6473 202a 6e6f the commands *no │ │ │ │ -00023670: 7465 2043 534d 3a20 4353 4d2c 2c20 2a6e te CSM: CSM,, *n │ │ │ │ -00023680: 6f74 6520 5365 6772 653a 0a53 6567 7265 ote Segre:.Segre │ │ │ │ -00023690: 2c2c 202a 6e6f 7465 2043 6865 726e 3a20 ,, *note Chern: │ │ │ │ -000236a0: 4368 6572 6e2c 2061 6e64 202a 6e6f 7465 Chern, and *note │ │ │ │ -000236b0: 2045 756c 6572 3a20 4575 6c65 722c 2074 Euler: Euler, t │ │ │ │ -000236c0: 6f20 7370 6563 6966 7920 7468 6520 7479 o specify the ty │ │ │ │ -000236d0: 7065 206f 660a 6f75 7470 7574 2074 6f20 pe of.output to │ │ │ │ -000236e0: 6265 2072 6574 7572 6e65 6420 746f 2074 be returned to t │ │ │ │ -000236f0: 6865 2075 7365 642e 2054 6869 7320 6f70 he used. This op │ │ │ │ -00023700: 7469 6f6e 2077 696c 6c20 6265 2069 676e tion will be ign │ │ │ │ -00023710: 6f72 6564 2077 6865 6e20 7573 6564 2077 ored when used w │ │ │ │ -00023720: 6974 680a 2a6e 6f74 6520 436f 6d70 4d65 ith.*note CompMe │ │ │ │ -00023730: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ -00023740: 2c20 506e 5265 7369 6475 616c 206f 7220 , PnResidual or │ │ │ │ -00023750: 6265 7274 696e 692e 2054 6865 206f 7074 bertini. The opt │ │ │ │ -00023760: 696f 6e20 7769 6c6c 2061 6c73 6f20 6265 ion will also be │ │ │ │ -00023770: 0a69 676e 6f72 6520 7768 656e 202a 6e6f .ignore when *no │ │ │ │ -00023780: 7465 204d 6574 686f 643a 204d 6574 686f te Method: Metho │ │ │ │ -00023790: 642c 3d3e 4469 7265 6374 436f 6d70 6c65 d,=>DirectComple │ │ │ │ -000237a0: 7465 496e 7420 6973 2075 7365 642e 2054 teInt is used. T │ │ │ │ -000237b0: 6865 2064 6566 6175 6c74 0a6f 7574 7075 he default.outpu │ │ │ │ -000237c0: 7420 666f 7220 616c 6c20 7468 6573 6520 t for all these │ │ │ │ -000237d0: 6d65 7468 6f64 7320 6973 2043 686f 7752 methods is ChowR │ │ │ │ -000237e0: 696e 6745 6c65 6c6d 656e 7420 7768 6963 ingElelment whic │ │ │ │ -000237f0: 6820 7769 6c6c 2072 6574 7572 6e20 616e h will return an │ │ │ │ -00023800: 2065 6c65 6d65 6e74 0a6f 6620 7468 6520 element.of the │ │ │ │ -00023810: 6170 7072 6f70 7269 6174 6520 4368 6f77 appropriate Chow │ │ │ │ -00023820: 2072 696e 672e 2041 6c6c 206d 6574 686f ring. All metho │ │ │ │ -00023830: 6473 2061 6c73 6f20 6861 7665 2061 6e20 ds also have an │ │ │ │ -00023840: 6f70 7469 6f6e 2048 6173 6846 6f72 6d20 option HashForm │ │ │ │ -00023850: 7768 6963 680a 7265 7475 726e 7320 6164 which.returns ad │ │ │ │ -00023860: 6469 7469 6f6e 616c 2069 6e66 6f72 6d61 ditional informa │ │ │ │ -00023870: 7469 6f6e 2063 6f6d 7075 7465 6420 6279 tion computed by │ │ │ │ -00023880: 2074 6865 206d 6574 686f 6473 2064 7572 the methods dur │ │ │ │ -00023890: 696e 6720 7468 6569 7220 7374 616e 6461 ing their standa │ │ │ │ -000238a0: 7264 0a6f 7065 7261 7469 6f6e 2e0a 0a2b rd.operation...+ │ │ │ │ -000238b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000238f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023900: 6931 203a 2052 203d 205a 5a2f 3332 3734 i1 : R = ZZ/3274 │ │ │ │ -00023910: 395b 785f 302e 2e78 5f36 5d20 2020 2020 9[x_0..x_6] │ │ │ │ -00023920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023940: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00023390: 5761 7973 2074 6f20 7573 6520 4d75 6c74 Ways to use Mult │ │ │ │ +000233a0: 6950 726f 6a43 6f6f 7264 5269 6e67 3a0a iProjCoordRing:. │ │ │ │ +000233b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000233c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +000233d0: 0a20 202a 2022 4d75 6c74 6950 726f 6a43 . * "MultiProjC │ │ │ │ +000233e0: 6f6f 7264 5269 6e67 284c 6973 7429 220a oordRing(List)". │ │ │ │ +000233f0: 2020 2a20 224d 756c 7469 5072 6f6a 436f * "MultiProjCo │ │ │ │ +00023400: 6f72 6452 696e 6728 5269 6e67 2c4c 6973 ordRing(Ring,Lis │ │ │ │ +00023410: 7429 220a 2020 2a20 224d 756c 7469 5072 t)". * "MultiPr │ │ │ │ +00023420: 6f6a 436f 6f72 6452 696e 6728 5269 6e67 ojCoordRing(Ring │ │ │ │ +00023430: 2c53 796d 626f 6c2c 4c69 7374 2922 0a20 ,Symbol,List)". │ │ │ │ +00023440: 202a 2022 4d75 6c74 6950 726f 6a43 6f6f * "MultiProjCoo │ │ │ │ +00023450: 7264 5269 6e67 2853 796d 626f 6c2c 4c69 rdRing(Symbol,Li │ │ │ │ +00023460: 7374 2922 0a0a 466f 7220 7468 6520 7072 st)"..For the pr │ │ │ │ +00023470: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00023480: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00023490: 206f 626a 6563 7420 2a6e 6f74 6520 4d75 object *note Mu │ │ │ │ +000234a0: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +000234b0: 3a20 4d75 6c74 6950 726f 6a43 6f6f 7264 : MultiProjCoord │ │ │ │ +000234c0: 5269 6e67 2c20 6973 2061 202a 6e6f 7465 Ring, is a *note │ │ │ │ +000234d0: 206d 6574 686f 640a 6675 6e63 7469 6f6e method.function │ │ │ │ +000234e0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +000234f0: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ +00023500: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00023510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023550: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00023560: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00023570: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00023580: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00023590: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +000235a0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +000235b0: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ +000235c0: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +000235d0: 6d32 3a32 3035 303a 302e 0a1f 0a46 696c m2:2050:0....Fil │ │ │ │ +000235e0: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ +000235f0: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ +00023600: 6f64 653a 204f 7574 7075 742c 204e 6578 ode: Output, Nex │ │ │ │ +00023610: 743a 2070 726f 6261 6269 6c69 7374 6963 t: probabilistic │ │ │ │ +00023620: 2061 6c67 6f72 6974 686d 2c20 5072 6576 algorithm, Prev │ │ │ │ +00023630: 3a20 4d75 6c74 6950 726f 6a43 6f6f 7264 : MultiProjCoord │ │ │ │ +00023640: 5269 6e67 2c20 5570 3a20 546f 700a 0a4f Ring, Up: Top..O │ │ │ │ +00023650: 7574 7075 740a 2a2a 2a2a 2a2a 0a0a 4465 utput.******..De │ │ │ │ +00023660: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00023670: 3d3d 3d3d 3d0a 0a54 6865 206f 7074 696f =====..The optio │ │ │ │ +00023680: 6e20 4f75 7470 7574 2069 7320 6f6e 6c79 n Output is only │ │ │ │ +00023690: 2075 7365 6420 6279 2074 6865 2063 6f6d used by the com │ │ │ │ +000236a0: 6d61 6e64 7320 2a6e 6f74 6520 4353 4d3a mands *note CSM: │ │ │ │ +000236b0: 2043 534d 2c2c 202a 6e6f 7465 2053 6567 CSM,, *note Seg │ │ │ │ +000236c0: 7265 3a0a 5365 6772 652c 2c20 2a6e 6f74 re:.Segre,, *not │ │ │ │ +000236d0: 6520 4368 6572 6e3a 2043 6865 726e 2c20 e Chern: Chern, │ │ │ │ +000236e0: 616e 6420 2a6e 6f74 6520 4575 6c65 723a and *note Euler: │ │ │ │ +000236f0: 2045 756c 6572 2c20 746f 2073 7065 6369 Euler, to speci │ │ │ │ +00023700: 6679 2074 6865 2074 7970 6520 6f66 0a6f fy the type of.o │ │ │ │ +00023710: 7574 7075 7420 746f 2062 6520 7265 7475 utput to be retu │ │ │ │ +00023720: 726e 6564 2074 6f20 7468 6520 7573 6564 rned to the used │ │ │ │ +00023730: 2e20 5468 6973 206f 7074 696f 6e20 7769 . This option wi │ │ │ │ +00023740: 6c6c 2062 6520 6967 6e6f 7265 6420 7768 ll be ignored wh │ │ │ │ +00023750: 656e 2075 7365 6420 7769 7468 0a2a 6e6f en used with.*no │ │ │ │ +00023760: 7465 2043 6f6d 704d 6574 686f 643a 2043 te CompMethod: C │ │ │ │ +00023770: 6f6d 704d 6574 686f 642c 2050 6e52 6573 ompMethod, PnRes │ │ │ │ +00023780: 6964 7561 6c20 6f72 2062 6572 7469 6e69 idual or bertini │ │ │ │ +00023790: 2e20 5468 6520 6f70 7469 6f6e 2077 696c . The option wil │ │ │ │ +000237a0: 6c20 616c 736f 2062 650a 6967 6e6f 7265 l also be.ignore │ │ │ │ +000237b0: 2077 6865 6e20 2a6e 6f74 6520 4d65 7468 when *note Meth │ │ │ │ +000237c0: 6f64 3a20 4d65 7468 6f64 2c3d 3e44 6972 od: Method,=>Dir │ │ │ │ +000237d0: 6563 7443 6f6d 706c 6574 6549 6e74 2069 ectCompleteInt i │ │ │ │ +000237e0: 7320 7573 6564 2e20 5468 6520 6465 6661 s used. The defa │ │ │ │ +000237f0: 756c 740a 6f75 7470 7574 2066 6f72 2061 ult.output for a │ │ │ │ +00023800: 6c6c 2074 6865 7365 206d 6574 686f 6473 ll these methods │ │ │ │ +00023810: 2069 7320 4368 6f77 5269 6e67 456c 656c is ChowRingElel │ │ │ │ +00023820: 6d65 6e74 2077 6869 6368 2077 696c 6c20 ment which will │ │ │ │ +00023830: 7265 7475 726e 2061 6e20 656c 656d 656e return an elemen │ │ │ │ +00023840: 740a 6f66 2074 6865 2061 7070 726f 7072 t.of the appropr │ │ │ │ +00023850: 6961 7465 2043 686f 7720 7269 6e67 2e20 iate Chow ring. │ │ │ │ +00023860: 416c 6c20 6d65 7468 6f64 7320 616c 736f All methods also │ │ │ │ +00023870: 2068 6176 6520 616e 206f 7074 696f 6e20 have an option │ │ │ │ +00023880: 4861 7368 466f 726d 2077 6869 6368 0a72 HashForm which.r │ │ │ │ +00023890: 6574 7572 6e73 2061 6464 6974 696f 6e61 eturns additiona │ │ │ │ +000238a0: 6c20 696e 666f 726d 6174 696f 6e20 636f l information co │ │ │ │ +000238b0: 6d70 7574 6564 2062 7920 7468 6520 6d65 mputed by the me │ │ │ │ +000238c0: 7468 6f64 7320 6475 7269 6e67 2074 6865 thods during the │ │ │ │ +000238d0: 6972 2073 7461 6e64 6172 640a 6f70 6572 ir standard.oper │ │ │ │ +000238e0: 6174 696f 6e2e 0a0a 2b2d 2d2d 2d2d 2d2d ation...+------- │ │ │ │ +000238f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023930: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5220 ------+.|i1 : R │ │ │ │ +00023940: 3d20 5a5a 2f33 3237 3439 5b78 5f30 2e2e = ZZ/32749[x_0.. │ │ │ │ +00023950: 785f 365d 2020 2020 2020 2020 2020 2020 x_6] │ │ │ │ 00023960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023990: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000239a0: 6f31 203d 2052 2020 2020 2020 2020 2020 o1 = R │ │ │ │ +00023980: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000239a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000239d0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5220 |.|o1 = R │ │ │ │ +000239e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023a40: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ -00023a50: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00023a20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023ae0: 6932 203a 2041 3d43 686f 7752 696e 6728 i2 : A=ChowRing( │ │ │ │ -00023af0: 5229 2020 2020 2020 2020 2020 2020 2020 R) │ │ │ │ -00023b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023a70: 2020 2020 2020 7c0a 7c6f 3120 3a20 506f |.|o1 : Po │ │ │ │ +00023a80: 6c79 6e6f 6d69 616c 5269 6e67 2020 2020 lynomialRing │ │ │ │ +00023a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ac0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023b10: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 413d ------+.|i2 : A= │ │ │ │ +00023b20: 4368 6f77 5269 6e67 2852 2920 2020 2020 ChowRing(R) │ │ │ │ 00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023b80: 6f32 203d 2041 2020 2020 2020 2020 2020 o2 = A │ │ │ │ +00023b60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023bb0: 2020 2020 2020 7c0a 7c6f 3220 3d20 4120 |.|o2 = A │ │ │ │ +00023bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023c20: 6f32 203a 2051 756f 7469 656e 7452 696e o2 : QuotientRin │ │ │ │ -00023c30: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00023c00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c60: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00023c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023cc0: 6933 203a 2049 3d69 6465 616c 2872 616e i3 : I=ideal(ran │ │ │ │ -00023cd0: 646f 6d28 322c 5229 2c52 5f30 2a52 5f31 dom(2,R),R_0*R_1 │ │ │ │ -00023ce0: 2a52 5f36 2d52 5f30 5e33 293b 2020 2020 *R_6-R_0^3); │ │ │ │ -00023cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c50: 2020 2020 2020 7c0a 7c6f 3220 3a20 5175 |.|o2 : Qu │ │ │ │ +00023c60: 6f74 6965 6e74 5269 6e67 2020 2020 2020 otientRing │ │ │ │ +00023c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ca0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023cf0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 493d ------+.|i3 : I= │ │ │ │ +00023d00: 6964 6561 6c28 7261 6e64 6f6d 2832 2c52 ideal(random(2,R │ │ │ │ +00023d10: 292c 525f 302a 525f 312a 525f 362d 525f ),R_0*R_1*R_6-R_ │ │ │ │ +00023d20: 305e 3329 3b20 2020 2020 2020 2020 2020 0^3); │ │ │ │ 00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023d60: 6f33 203a 2049 6465 616c 206f 6620 5220 o3 : Ideal of R │ │ │ │ +00023d40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023da0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00023db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023e00: 6934 203a 2063 736d 3d43 534d 2841 2c49 i4 : csm=CSM(A,I │ │ │ │ -00023e10: 2c4f 7574 7075 743d 3e48 6173 6846 6f72 ,Output=>HashFor │ │ │ │ -00023e20: 6d29 2020 2020 2020 2020 2020 2020 2020 m) │ │ │ │ -00023e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023d90: 2020 2020 2020 7c0a 7c6f 3320 3a20 4964 |.|o3 : Id │ │ │ │ +00023da0: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ +00023db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023de0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00023e30: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 6373 ------+.|i4 : cs │ │ │ │ +00023e40: 6d3d 4353 4d28 412c 492c 4f75 7470 7574 m=CSM(A,I,Output │ │ │ │ +00023e50: 3d3e 4861 7368 466f 726d 2920 2020 2020 =>HashForm) │ │ │ │ 00023e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023ea0: 6f34 203d 204d 7574 6162 6c65 4861 7368 o4 = MutableHash │ │ │ │ -00023eb0: 5461 626c 657b 2e2e 2e34 2e2e 2e7d 2020 Table{...4...} │ │ │ │ +00023e80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ee0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023ed0: 2020 2020 2020 7c0a 7c6f 3420 3d20 4d75 |.|o4 = Mu │ │ │ │ +00023ee0: 7461 626c 6548 6173 6854 6162 6c65 7b2e tableHashTable{. │ │ │ │ +00023ef0: 2e2e 342e 2e2e 7d20 2020 2020 2020 2020 ..4...} │ │ │ │ 00023f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023f40: 6f34 203a 204d 7574 6162 6c65 4861 7368 o4 : MutableHash │ │ │ │ -00023f50: 5461 626c 6520 2020 2020 2020 2020 2020 Table │ │ │ │ +00023f20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00023f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023f80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00023f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00023fe0: 6935 203a 2070 6565 6b20 6373 6d20 2020 i5 : peek csm │ │ │ │ -00023ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023f70: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d75 |.|o4 : Mu │ │ │ │ +00023f80: 7461 626c 6548 6173 6854 6162 6c65 2020 tableHashTable │ │ │ │ +00023f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023fc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 │ │ │ │ -000246b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -000246c0: 2d2d 2d2d 2d2d 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 2d2b 0a7c -------------+.| │ │ │ │ -00024710: 6938 203a 2063 3d43 6865 726e 2820 492c i8 : c=Chern( I, │ │ │ │ -00024720: 204f 7574 7075 743d 3e48 6173 6846 6f72 Output=>HashFor │ │ │ │ -00024730: 6d29 2020 2020 2020 2020 2020 2020 2020 m) │ │ │ │ -00024740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024750: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024760: 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 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2020 3120 1 1 1 │ │ │ │ -00025cd0: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ -00025ce0: 2020 3120 2020 2020 2020 2020 207c 0a7c 1 |.| │ │ │ │ -00025cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d10: 3620 2020 2020 2035 2020 2020 2020 3420 6 5 4 │ │ │ │ -00025d20: 2020 2020 2033 2020 2020 2020 3220 2020 3 2 │ │ │ │ -00025d30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00025d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025d50: 2020 2020 2020 207b 317d 203d 3e20 3668 {1} => 6h │ │ │ │ -00025d60: 2020 2b20 3137 6820 202b 2032 3868 2020 + 17h + 28h │ │ │ │ -00025d70: 2b20 3237 6820 202b 2031 3468 2020 2b20 + 27h + 14h + │ │ │ │ -00025d80: 3368 2020 2020 2020 2020 2020 207c 0a7c 3h |.| │ │ │ │ -00025d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025db0: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ -00025dc0: 2020 2020 2031 2020 2020 2020 3120 2020 1 1 │ │ │ │ -00025dd0: 2020 3120 2020 2020 2020 2020 207c 0a2b 1 |.+ │ │ │ │ -00025de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00025e30: 5468 6520 4d75 7461 626c 6548 6173 6854 The MutableHashT │ │ │ │ -00025e40: 6162 6c65 2072 6574 7572 6e65 6420 7769 able returned wi │ │ │ │ -00025e50: 7468 2074 6865 206f 7074 696f 6e20 4f75 th the option Ou │ │ │ │ -00025e60: 7470 7574 3d3e 4861 7368 466f 726d 2063 tput=>HashForm c │ │ │ │ -00025e70: 6f6e 7461 696e 730a 6469 6666 6572 656e ontains.differen │ │ │ │ -00025e80: 7420 696e 666f 726d 6174 696f 6e20 6465 t information de │ │ │ │ -00025e90: 7065 6e64 696e 6720 6f6e 2074 6865 206d pending on the m │ │ │ │ -00025ea0: 6574 686f 6420 7769 7468 2077 6869 6368 ethod with which │ │ │ │ -00025eb0: 2069 7420 6973 2075 7365 642e 0a41 6464 it is used..Add │ │ │ │ -00025ec0: 6974 696f 6e61 6c6c 7920 6966 2074 6865 itionally if the │ │ │ │ -00025ed0: 206f 7074 696f 6e20 2a6e 6f74 6520 496e option *note In │ │ │ │ -00025ee0: 7075 7449 7353 6d6f 6f74 683a 2049 6e70 putIsSmooth: Inp │ │ │ │ -00025ef0: 7574 4973 536d 6f6f 7468 2c20 6973 2075 utIsSmooth, is u │ │ │ │ -00025f00: 7365 6420 7468 656e 2074 6865 0a68 6173 sed then the.has │ │ │ │ -00025f10: 6820 7461 626c 6520 7265 7475 726e 6564 h table returned │ │ │ │ -00025f20: 2062 7920 7468 6520 6d65 7468 6f64 7320 by the methods │ │ │ │ -00025f30: 4575 6c65 7220 616e 6420 4353 4d20 7769 Euler and CSM wi │ │ │ │ -00025f40: 6c6c 2062 6520 7468 6520 7361 6d65 2061 ll be the same a │ │ │ │ -00025f50: 7320 7468 6174 0a72 6574 7572 6e65 6420 s that.returned │ │ │ │ -00025f60: 6279 2043 6865 726e 2e20 5768 656e 2075 by Chern. When u │ │ │ │ -00025f70: 7369 6e67 2074 6865 202a 6e6f 7465 2043 sing the *note C │ │ │ │ -00025f80: 534d 3a20 4353 4d2c 2020 636f 6d6d 616e SM: CSM, comman │ │ │ │ -00025f90: 6420 696e 2074 6865 2064 6566 6175 6c74 d in the default │ │ │ │ -00025fa0: 0a63 6f6e 6669 6775 7261 7469 6f6e 7320 .configurations │ │ │ │ -00025fb0: 2874 6861 7420 6973 202a 6e6f 7465 204d (that is *note M │ │ │ │ -00025fc0: 6574 686f 643a 204d 6574 686f 642c 3d3e ethod: Method,=> │ │ │ │ -00025fd0: 496e 636c 7573 696f 6e45 7863 6c75 7369 InclusionExclusi │ │ │ │ -00025fe0: 6f6e 2c20 2a6e 6f74 650a 436f 6d70 4d65 on, *note.CompMe │ │ │ │ -00025ff0: 7468 6f64 3a20 436f 6d70 4d65 7468 6f64 thod: CompMethod │ │ │ │ -00026000: 2c3d 3e50 726f 6a65 6374 6976 6544 6567 ,=>ProjectiveDeg │ │ │ │ -00026010: 7265 6529 2074 6865 7265 2069 7320 7468 ree) there is th │ │ │ │ -00026020: 6520 6164 6469 7469 6f6e 616c 206f 7074 e additional opt │ │ │ │ -00026030: 696f 6e20 746f 0a73 6574 204f 7574 7075 ion to.set Outpu │ │ │ │ -00026040: 743d 3e48 6173 6846 6f72 6d58 4c2e 2054 t=>HashFormXL. T │ │ │ │ -00026050: 6869 7320 7265 7475 726e 7320 616c 6c20 his returns all │ │ │ │ -00026060: 7468 6520 7573 7561 6c20 696e 666f 726d the usual inform │ │ │ │ -00026070: 6174 696f 6e20 7468 6174 0a4f 7574 7075 ation that.Outpu │ │ │ │ -00026080: 743d 3e48 6173 6846 6f72 6d20 776f 756c t=>HashForm woul │ │ │ │ -00026090: 6420 666f 7220 7468 6973 2063 6f6e 6669 d for this confi │ │ │ │ -000260a0: 6775 7261 7469 6f6e 2077 6974 6820 7468 guration with th │ │ │ │ -000260b0: 6520 6164 6469 7469 6f6e 206f 6620 7468 e addition of th │ │ │ │ -000260c0: 650a 7072 6f6a 6563 7469 7665 2064 6567 e.projective deg │ │ │ │ -000260d0: 7265 6573 2061 6e64 2053 6567 7265 2063 rees and Segre c │ │ │ │ -000260e0: 6c61 7373 6573 206f 6620 7369 6e67 756c lasses of singul │ │ │ │ -000260f0: 6172 6974 7920 7375 6273 6368 656d 6573 arity subschemes │ │ │ │ -00026100: 2067 656e 6572 6174 6564 2062 7920 7468 generated by th │ │ │ │ -00026110: 650a 6879 7065 7273 7572 6661 6365 7320 e.hypersurfaces │ │ │ │ -00026120: 636f 6e73 6964 6572 6564 2069 6e20 7468 considered in th │ │ │ │ -00026130: 6520 696e 636c 7573 696f 6e2f 6578 636c e inclusion/excl │ │ │ │ -00026140: 7573 696f 6e20 7072 6f63 6564 7572 652c usion procedure, │ │ │ │ -00026150: 2074 6861 7420 6973 2069 6e0a 6669 6e64 that is in.find │ │ │ │ -00026160: 696e 6720 7468 6520 4353 4d20 636c 6173 ing the CSM clas │ │ │ │ -00026170: 7320 6f66 2061 6c6c 2068 7970 6572 7375 s of all hypersu │ │ │ │ -00026180: 7266 6163 6573 2067 656e 6572 6174 6564 rfaces generated │ │ │ │ -00026190: 2062 7920 7461 6b69 6e67 2061 2070 726f by taking a pro │ │ │ │ -000261a0: 6475 6374 206f 660a 736f 6d65 2073 7562 duct of.some sub │ │ │ │ -000261b0: 7365 7473 206f 6620 6765 6e65 7261 746f sets of generato │ │ │ │ -000261c0: 7273 206f 6620 7468 6520 696e 7075 7420 rs of the input │ │ │ │ -000261d0: 6964 6561 6c2e 204e 6f74 6520 7468 6174 ideal. Note that │ │ │ │ -000261e0: 2c20 7369 6e63 6520 7468 6520 4353 4d20 , since the CSM │ │ │ │ -000261f0: 636c 6173 730a 6f66 2061 2073 7562 7363 class.of a subsc │ │ │ │ -00026200: 6865 6d65 2065 7175 616c 7320 7468 6520 heme equals the │ │ │ │ -00026210: 4353 4d20 636c 6173 7320 6f66 2069 7473 CSM class of its │ │ │ │ -00026220: 2072 6564 7563 6564 2073 6368 656d 652c reduced scheme, │ │ │ │ -00026230: 206f 7220 6571 7569 7661 6c65 6e74 6c79 or equivalently │ │ │ │ -00026240: 2066 6f72 0a75 7320 7468 6520 4353 4d20 for.us the CSM │ │ │ │ -00026250: 636c 6173 7320 636f 7272 6573 706f 6e64 class correspond │ │ │ │ -00026260: 696e 6720 746f 2061 6e20 6964 6561 6c20 ing to an ideal │ │ │ │ -00026270: 4920 6571 7561 6c73 2074 6865 2043 534d I equals the CSM │ │ │ │ -00026280: 2063 6c61 7373 206f 6620 7468 650a 7261 class of the.ra │ │ │ │ -00026290: 6469 6361 6c20 6f66 2049 2c20 7468 656e dical of I, then │ │ │ │ -000262a0: 2069 6e74 6572 6e61 6c6c 7920 7765 2061 internally we a │ │ │ │ -000262b0: 6c77 6179 7320 776f 726b 2077 6974 6820 lways work with │ │ │ │ -000262c0: 7261 6469 6361 6c20 6964 6561 6c73 2028 radical ideals ( │ │ │ │ -000262d0: 666f 720a 6566 6669 6369 656e 6379 2072 for.efficiency r │ │ │ │ -000262e0: 6561 736f 6e73 292e 2048 656e 6365 2074 easons). Hence t │ │ │ │ -000262f0: 6865 2070 726f 6a65 6374 6976 6520 6465 he projective de │ │ │ │ -00026300: 6772 6565 7320 616e 6420 5365 6772 6520 grees and Segre │ │ │ │ -00026310: 636c 6173 7365 7320 636f 6d70 7574 6564 classes computed │ │ │ │ -00026320: 0a69 6e74 6572 6e61 6c6c 7920 7769 6c6c .internally will │ │ │ │ -00026330: 2062 6520 7468 6f73 6520 6f66 2074 6865 be those of the │ │ │ │ -00026340: 2072 6164 6963 616c 206f 6620 616e 2069 radical of an i │ │ │ │ -00026350: 6465 616c 2064 6566 696e 6564 2062 7920 deal defined by │ │ │ │ -00026360: 6120 706f 6c79 6e6f 6d69 616c 0a77 6869 a polynomial.whi │ │ │ │ -00026370: 6368 2069 7320 6120 7072 6f64 7563 7420 ch is a product │ │ │ │ -00026380: 6f66 2073 6f6d 6520 7375 6273 6574 206f of some subset o │ │ │ │ -00026390: 6620 7468 6520 6765 6e65 7261 746f 7273 f the generators │ │ │ │ -000263a0: 2e20 5765 2069 6c6c 7573 7472 6174 6520 . We illustrate │ │ │ │ -000263b0: 7468 6973 2077 6974 6820 616e 0a65 7861 this with an.exa │ │ │ │ -000263c0: 6d70 6c65 2062 656c 6f77 2e0a 0a2b 2d2d mple below...+-- │ │ │ │ -000263d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000263f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00026420: 3420 3a20 6373 6d58 4c68 6173 683d 4353 4 : csmXLhash=CS │ │ │ │ -00026430: 4d28 412c 492c 4f75 7470 7574 3d3e 4861 M(A,I,Output=>Ha │ │ │ │ -00026440: 7368 466f 726d 584c 2920 2020 2020 2020 shFormXL) │ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026460: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00026470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a40: 2020 2020 2020 2020 2020 2020 2020 207d } │ │ │ │ +00025a50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025a70: 2020 2020 2020 2020 2020 2020 3620 2020 6 │ │ │ │ +00025a80: 2020 2035 2020 2020 2020 3420 2020 2020 5 4 │ │ │ │ +00025a90: 2033 2020 2020 2020 3220 2020 2020 2020 3 2 │ │ │ │ +00025aa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ac0: 7b30 2c20 317d 203d 3e20 3268 2020 2b20 {0, 1} => 2h + │ │ │ │ +00025ad0: 3233 6820 202b 2033 3268 2020 2b20 3333 23h + 32h + 33 │ │ │ │ +00025ae0: 6820 202b 2031 3868 2020 2b20 3568 2020 h + 18h + 5h │ │ │ │ +00025af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b10: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00025b20: 2020 2031 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ +00025b30: 2031 2020 2020 2020 3120 2020 2020 3120 1 1 1 │ │ │ │ +00025b40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025b60: 2020 2020 2020 2020 2020 3620 2020 2020 6 │ │ │ │ +00025b70: 2035 2020 2020 2020 3420 2020 2020 2033 5 4 3 │ │ │ │ +00025b80: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025b90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025bb0: 4353 4d20 3d3e 2031 3068 2020 2b20 3132 CSM => 10h + 12 │ │ │ │ +00025bc0: 6820 202b 2032 3268 2020 2b20 3136 6820 h + 22h + 16h │ │ │ │ +00025bd0: 202b 2036 6820 2020 2020 2020 2020 2020 + 6h │ │ │ │ +00025be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c00: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00025c10: 2031 2020 2020 2020 3120 2020 2020 2031 1 1 1 │ │ │ │ +00025c20: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00025c30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025c50: 2020 2020 2020 2020 2036 2020 2020 2020 6 │ │ │ │ +00025c60: 3520 2020 2020 2034 2020 2020 2020 3320 5 4 3 │ │ │ │ +00025c70: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025c80: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025ca0: 7b30 7d20 3d3e 2036 6820 202b 2031 3868 {0} => 6h + 18h │ │ │ │ +00025cb0: 2020 2b20 3236 6820 202b 2032 3268 2020 + 26h + 22h │ │ │ │ +00025cc0: 2b20 3130 6820 202b 2032 6820 2020 2020 + 10h + 2h │ │ │ │ +00025cd0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025cf0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +00025d00: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00025d10: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00025d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d40: 2020 2020 2020 2020 2036 2020 2020 2020 6 │ │ │ │ +00025d50: 3520 2020 2020 2034 2020 2020 2020 3320 5 4 3 │ │ │ │ +00025d60: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00025d70: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025d90: 7b31 7d20 3d3e 2036 6820 202b 2031 3768 {1} => 6h + 17h │ │ │ │ +00025da0: 2020 2b20 3238 6820 202b 2032 3768 2020 + 28h + 27h │ │ │ │ +00025db0: 2b20 3134 6820 202b 2033 6820 2020 2020 + 14h + 3h │ │ │ │ +00025dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025de0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +00025df0: 3120 2020 2020 2031 2020 2020 2020 3120 1 1 1 │ │ │ │ +00025e00: 2020 2020 2031 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00025e10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00025e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00025e60: 2d2d 2d2d 2d2d 2b0a 0a54 6865 204d 7574 ------+..The Mut │ │ │ │ +00025e70: 6162 6c65 4861 7368 5461 626c 6520 7265 ableHashTable re │ │ │ │ +00025e80: 7475 726e 6564 2077 6974 6820 7468 6520 turned with the │ │ │ │ +00025e90: 6f70 7469 6f6e 204f 7574 7075 743d 3e48 option Output=>H │ │ │ │ +00025ea0: 6173 6846 6f72 6d20 636f 6e74 6169 6e73 ashForm contains │ │ │ │ +00025eb0: 0a64 6966 6665 7265 6e74 2069 6e66 6f72 .different infor │ │ │ │ +00025ec0: 6d61 7469 6f6e 2064 6570 656e 6469 6e67 mation depending │ │ │ │ +00025ed0: 206f 6e20 7468 6520 6d65 7468 6f64 2077 on the method w │ │ │ │ +00025ee0: 6974 6820 7768 6963 6820 6974 2069 7320 ith which it is │ │ │ │ +00025ef0: 7573 6564 2e0a 4164 6469 7469 6f6e 616c used..Additional │ │ │ │ +00025f00: 6c79 2069 6620 7468 6520 6f70 7469 6f6e ly if the option │ │ │ │ +00025f10: 202a 6e6f 7465 2049 6e70 7574 4973 536d *note InputIsSm │ │ │ │ +00025f20: 6f6f 7468 3a20 496e 7075 7449 7353 6d6f ooth: InputIsSmo │ │ │ │ +00025f30: 6f74 682c 2069 7320 7573 6564 2074 6865 oth, is used the │ │ │ │ +00025f40: 6e20 7468 650a 6861 7368 2074 6162 6c65 n the.hash table │ │ │ │ +00025f50: 2072 6574 7572 6e65 6420 6279 2074 6865 returned by the │ │ │ │ +00025f60: 206d 6574 686f 6473 2045 756c 6572 2061 methods Euler a │ │ │ │ +00025f70: 6e64 2043 534d 2077 696c 6c20 6265 2074 nd CSM will be t │ │ │ │ +00025f80: 6865 2073 616d 6520 6173 2074 6861 740a he same as that. │ │ │ │ +00025f90: 7265 7475 726e 6564 2062 7920 4368 6572 returned by Cher │ │ │ │ +00025fa0: 6e2e 2057 6865 6e20 7573 696e 6720 7468 n. When using th │ │ │ │ +00025fb0: 6520 2a6e 6f74 6520 4353 4d3a 2043 534d e *note CSM: CSM │ │ │ │ +00025fc0: 2c20 2063 6f6d 6d61 6e64 2069 6e20 7468 , command in th │ │ │ │ +00025fd0: 6520 6465 6661 756c 740a 636f 6e66 6967 e default.config │ │ │ │ +00025fe0: 7572 6174 696f 6e73 2028 7468 6174 2069 urations (that i │ │ │ │ +00025ff0: 7320 2a6e 6f74 6520 4d65 7468 6f64 3a20 s *note Method: │ │ │ │ +00026000: 4d65 7468 6f64 2c3d 3e49 6e63 6c75 7369 Method,=>Inclusi │ │ │ │ +00026010: 6f6e 4578 636c 7573 696f 6e2c 202a 6e6f onExclusion, *no │ │ │ │ +00026020: 7465 0a43 6f6d 704d 6574 686f 643a 2043 te.CompMethod: C │ │ │ │ +00026030: 6f6d 704d 6574 686f 642c 3d3e 5072 6f6a ompMethod,=>Proj │ │ │ │ +00026040: 6563 7469 7665 4465 6772 6565 2920 7468 ectiveDegree) th │ │ │ │ +00026050: 6572 6520 6973 2074 6865 2061 6464 6974 ere is the addit │ │ │ │ +00026060: 696f 6e61 6c20 6f70 7469 6f6e 2074 6f0a ional option to. │ │ │ │ +00026070: 7365 7420 4f75 7470 7574 3d3e 4861 7368 set Output=>Hash │ │ │ │ +00026080: 466f 726d 584c 2e20 5468 6973 2072 6574 FormXL. This ret │ │ │ │ +00026090: 7572 6e73 2061 6c6c 2074 6865 2075 7375 urns all the usu │ │ │ │ +000260a0: 616c 2069 6e66 6f72 6d61 7469 6f6e 2074 al information t │ │ │ │ +000260b0: 6861 740a 4f75 7470 7574 3d3e 4861 7368 hat.Output=>Hash │ │ │ │ +000260c0: 466f 726d 2077 6f75 6c64 2066 6f72 2074 Form would for t │ │ │ │ +000260d0: 6869 7320 636f 6e66 6967 7572 6174 696f his configuratio │ │ │ │ +000260e0: 6e20 7769 7468 2074 6865 2061 6464 6974 n with the addit │ │ │ │ +000260f0: 696f 6e20 6f66 2074 6865 0a70 726f 6a65 ion of the.proje │ │ │ │ +00026100: 6374 6976 6520 6465 6772 6565 7320 616e ctive degrees an │ │ │ │ +00026110: 6420 5365 6772 6520 636c 6173 7365 7320 d Segre classes │ │ │ │ +00026120: 6f66 2073 696e 6775 6c61 7269 7479 2073 of singularity s │ │ │ │ +00026130: 7562 7363 6865 6d65 7320 6765 6e65 7261 ubschemes genera │ │ │ │ +00026140: 7465 6420 6279 2074 6865 0a68 7970 6572 ted by the.hyper │ │ │ │ +00026150: 7375 7266 6163 6573 2063 6f6e 7369 6465 surfaces conside │ │ │ │ +00026160: 7265 6420 696e 2074 6865 2069 6e63 6c75 red in the inclu │ │ │ │ +00026170: 7369 6f6e 2f65 7863 6c75 7369 6f6e 2070 sion/exclusion p │ │ │ │ +00026180: 726f 6365 6475 7265 2c20 7468 6174 2069 rocedure, that i │ │ │ │ +00026190: 7320 696e 0a66 696e 6469 6e67 2074 6865 s in.finding the │ │ │ │ +000261a0: 2043 534d 2063 6c61 7373 206f 6620 616c CSM class of al │ │ │ │ +000261b0: 6c20 6879 7065 7273 7572 6661 6365 7320 l hypersurfaces │ │ │ │ +000261c0: 6765 6e65 7261 7465 6420 6279 2074 616b generated by tak │ │ │ │ +000261d0: 696e 6720 6120 7072 6f64 7563 7420 6f66 ing a product of │ │ │ │ +000261e0: 0a73 6f6d 6520 7375 6273 6574 7320 6f66 .some subsets of │ │ │ │ +000261f0: 2067 656e 6572 6174 6f72 7320 6f66 2074 generators of t │ │ │ │ +00026200: 6865 2069 6e70 7574 2069 6465 616c 2e20 he input ideal. │ │ │ │ +00026210: 4e6f 7465 2074 6861 742c 2073 696e 6365 Note that, since │ │ │ │ +00026220: 2074 6865 2043 534d 2063 6c61 7373 0a6f the CSM class.o │ │ │ │ +00026230: 6620 6120 7375 6273 6368 656d 6520 6571 f a subscheme eq │ │ │ │ +00026240: 7561 6c73 2074 6865 2043 534d 2063 6c61 uals the CSM cla │ │ │ │ +00026250: 7373 206f 6620 6974 7320 7265 6475 6365 ss of its reduce │ │ │ │ +00026260: 6420 7363 6865 6d65 2c20 6f72 2065 7175 d scheme, or equ │ │ │ │ +00026270: 6976 616c 656e 746c 7920 666f 720a 7573 ivalently for.us │ │ │ │ +00026280: 2074 6865 2043 534d 2063 6c61 7373 2063 the CSM class c │ │ │ │ +00026290: 6f72 7265 7370 6f6e 6469 6e67 2074 6f20 orresponding to │ │ │ │ +000262a0: 616e 2069 6465 616c 2049 2065 7175 616c an ideal I equal │ │ │ │ +000262b0: 7320 7468 6520 4353 4d20 636c 6173 7320 s the CSM class │ │ │ │ +000262c0: 6f66 2074 6865 0a72 6164 6963 616c 206f of the.radical o │ │ │ │ +000262d0: 6620 492c 2074 6865 6e20 696e 7465 726e f I, then intern │ │ │ │ +000262e0: 616c 6c79 2077 6520 616c 7761 7973 2077 ally we always w │ │ │ │ +000262f0: 6f72 6b20 7769 7468 2072 6164 6963 616c ork with radical │ │ │ │ +00026300: 2069 6465 616c 7320 2866 6f72 0a65 6666 ideals (for.eff │ │ │ │ +00026310: 6963 6965 6e63 7920 7265 6173 6f6e 7329 iciency reasons) │ │ │ │ +00026320: 2e20 4865 6e63 6520 7468 6520 7072 6f6a . 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6f31 |.|o1 │ │ │ │ -000264c0: 3420 3d20 4d75 7461 626c 6548 6173 6854 4 = MutableHashT │ │ │ │ -000264d0: 6162 6c65 7b2e 2e2e 3130 2e2e 2e7d 2020 able{...10...} │ │ │ │ +000264a0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026500: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00026510: 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...} │ │ │ │ 00026520: 2020 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 207c 0a7c 6f31 |.|o1 │ │ │ │ -00026560: 3420 3a20 4d75 7461 626c 6548 6173 6854 4 : MutableHashT │ │ │ │ -00026570: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ +00026540: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000265a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -000265b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00026600: 3520 3a20 7065 656b 2063 736d 584c 6861 5 : peek csmXLha │ │ │ │ -00026610: 7368 2020 2020 2020 2020 2020 2020 2020 sh │ │ │ │ -00026620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026640: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00026590: 2020 2020 7c0a 7c6f 3134 203a 204d 7574 |.|o14 : Mut │ │ │ │ +000265a0: 6162 6c65 4861 7368 5461 626c 6520 2020 ableHashTable │ │ │ │ +000265b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000265e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000265f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00026630: 2d2d 2d2d 2b0a 7c69 3135 203a 2070 6565 ----+.|i15 : pee │ │ │ │ +00026640: 6b20 6373 6d58 4c68 6173 6820 2020 2020 k csmXLhash │ │ │ │ 00026650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026690: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000266a0: 3520 3d20 4d75 7461 626c 6548 6173 6854 5 = MutableHashT │ │ │ │ -000266b0: 6162 6c65 7b47 284a 6163 6f62 6961 6e29 able{G(Jacobian) │ │ │ │ -000266c0: 7b30 7d20 3d3e 2030 2020 2020 2020 2020 {0} => 0 │ │ │ │ -000266d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000266e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000266f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026700: 2020 2020 2053 6567 7265 284a 6163 6f62 Segre(Jacob │ │ │ │ -00026710: 6961 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2020 2020 2020 2020 2020 │ │ │ │ -00026810: 2020 3120 2020 2020 2020 3120 2020 2020 1 1 │ │ │ │ -00026820: 2020 3120 2020 2020 2020 207c 0a7c 2020 1 |.| │ │ │ │ +000267a0: 2020 2020 2020 2020 2020 2036 2020 2020 6 │ │ │ │ +000267b0: 2020 2035 2020 2020 2020 2034 2020 2020 5 4 │ │ │ │ +000267c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000267d0: 2020 2020 2020 2020 2020 2020 2020 5365 Se │ │ │ │ +000267e0: 6772 6528 4a61 636f 6269 616e 297b 302c gre(Jacobian){0, │ │ │ │ +000267f0: 2031 7d20 3d3e 2033 3930 6820 202d 2033 1} => 390h - 3 │ │ │ │ +00026800: 3836 6820 202b 2031 3530 6820 202d 2020 86h + 150h - │ │ │ │ +00026810: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026860: 2036 2020 2020 2020 3520 2020 2020 3320 6 5 3 │ │ │ │ -00026870: 2020 2020 3220 2020 2020 207c 0a7c 2020 2 |.| │ │ │ │ +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 2053 6567 7265 284a 6163 6f62 Segre(Jacob │ │ │ │ -000268a0: 6961 6e29 7b31 7d20 3d3e 202d 2031 3630 ian){1} => - 160 │ │ │ │ -000268b0: 6820 202b 2033 3268 2020 2d20 3468 2020 h + 32h - 4h │ │ │ │ -000268c0: 2b20 3268 2020 2020 2020 207c 0a7c 2020 + 2h |.| │ │ │ │ -000268d0: 2020 2020 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: 2031 2020 2020 2020 3120 2020 2020 3120 1 1 1 │ │ │ │ -00026910: 2020 2020 3120 2020 2020 207c 0a7c 2020 1 |.| │ │ │ │ +00026890: 2020 2020 2020 2020 2020 3620 2020 2020 6 │ │ │ │ +000268a0: 2035 2020 2020 2033 2020 2020 2032 2020 5 3 2 │ │ │ │ +000268b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000268c0: 2020 2020 2020 2020 2020 2020 2020 5365 Se │ │ │ │ +000268d0: 6772 6528 4a61 636f 6269 616e 297b 317d gre(Jacobian){1} │ │ │ │ +000268e0: 203d 3e20 2d20 3136 3068 2020 2b20 3332 => - 160h + 32 │ │ │ │ +000268f0: 6820 202d 2034 6820 202b 2032 6820 2020 h - 4h + 2h │ │ │ │ +00026900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026940: 2020 2020 2020 2020 2020 2020 2036 2020 6 │ │ │ │ -00026950: 2020 2020 3520 2020 2020 2034 2020 2020 5 4 │ │ │ │ -00026960: 2020 3320 2020 2020 2020 207c 0a7c 2020 3 |.| │ │ │ │ +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 2047 284a 6163 6f62 6961 6e29 G(Jacobian) │ │ │ │ -00026990: 7b30 2c20 317d 203d 3e20 3130 6820 202b {0, 1} => 10h + │ │ │ │ -000269a0: 2031 3068 2020 2b20 3130 6820 202b 2031 10h + 10h + 1 │ │ │ │ -000269b0: 3068 2020 2b20 2020 2020 207c 0a7c 2020 0h + |.| │ │ │ │ -000269c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000269d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000269e0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -000269f0: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ -00026a00: 2020 3120 2020 2020 2020 207c 0a7c 2020 1 |.| │ │ │ │ +00026980: 2020 2020 2020 3620 2020 2020 2035 2020 6 5 │ │ │ │ +00026990: 2020 2020 3420 2020 2020 2033 2020 2020 4 3 │ │ │ │ +000269a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000269b0: 2020 2020 2020 2020 2020 2020 2020 4728 G( │ │ │ │ +000269c0: 4a61 636f 6269 616e 297b 302c 2031 7d20 Jacobian){0, 1} │ │ │ │ +000269d0: 3d3e 2031 3068 2020 2b20 3130 6820 202b => 10h + 10h + │ │ │ │ +000269e0: 2031 3068 2020 2b20 3130 6820 202b 2020 10h + 10h + │ │ │ │ +000269f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a30: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -00026a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00026a20: 2020 2020 2020 3120 2020 2020 2031 2020 1 1 │ │ │ │ +00026a30: 2020 2020 3120 2020 2020 2031 2020 2020 1 1 │ │ │ │ +00026a40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00026a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026a70: 2020 2020 2047 284a 6163 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│ │ │ │ +000274b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000274d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00027500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00027550: 3820 3a20 4a3d 6964 6561 6c20 6a61 636f 8 : J=ideal jaco │ │ │ │ -00027560: 6269 616e 2072 6164 6963 616c 204b 3b20 bian radical K; │ │ │ │ -00027570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027590: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000275a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000274e0: 2020 2020 7c0a 7c6f 3137 203d 2074 7275 |.|o17 = tru │ │ │ │ +000274f0: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00027500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027580: 2d2d 2d2d 2b0a 7c69 3138 203a 204a 3d69 ----+.|i18 : J=i │ │ │ │ +00027590: 6465 616c 206a 6163 6f62 6961 6e20 7261 deal jacobian ra │ │ │ │ +000275a0: 6469 6361 6c20 4b3b 2020 2020 2020 2020 dical K; │ │ │ │ 000275b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000275f0: 3820 3a20 4964 6561 6c20 6f66 2052 2020 8 : Ideal of R │ │ │ │ +000275d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000275f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027630: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00027640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00027690: 3920 3a20 7365 674a 3d53 6567 7265 2841 9 : segJ=Segre(A │ │ │ │ -000276a0: 2c4a 2c4f 7574 7075 743d 3e48 6173 6846 ,J,Output=>HashF │ │ │ │ -000276b0: 6f72 6d29 2020 2020 2020 2020 2020 2020 orm) │ │ │ │ -000276c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000276e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027620: 2020 2020 7c0a 7c6f 3138 203a 2049 6465 |.|o18 : Ide │ │ │ │ +00027630: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +00027640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027670: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000276c0: 2d2d 2d2d 2b0a 7c69 3139 203a 2073 6567 ----+.|i19 : seg │ │ │ │ +000276d0: 4a3d 5365 6772 6528 412c 4a2c 4f75 7470 J=Segre(A,J,Outp │ │ │ │ +000276e0: 7574 3d3e 4861 7368 466f 726d 2920 2020 ut=>HashForm) │ │ │ │ 000276f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027720: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00027730: 3920 3d20 4d75 7461 626c 6548 6173 6854 9 = MutableHashT │ │ │ │ -00027740: 6162 6c65 7b2e 2e2e 342e 2e2e 7d20 2020 able{...4...} │ │ │ │ +00027710: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027770: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00027780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027760: 2020 2020 7c0a 7c6f 3139 203d 204d 7574 |.|o19 = Mut │ │ │ │ +00027770: 6162 6c65 4861 7368 5461 626c 657b 2e2e ableHashTable{.. │ │ │ │ +00027780: 2e34 2e2e 2e7d 2020 2020 2020 2020 2020 .4...} │ │ │ │ 00027790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000277d0: 3920 3a20 4d75 7461 626c 6548 6173 6854 9 : MutableHashT │ │ │ │ -000277e0: 6162 6c65 2020 2020 2020 2020 2020 2020 able │ │ │ │ +000277b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000277c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000277d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000277e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027810: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00027870: 3020 3a20 6373 6d58 4c68 6173 6823 2822 0 : csmXLhash#(" │ │ │ │ -00027880: 4728 4a61 636f 6269 616e 2922 7c74 6f53 G(Jacobian)"|toS │ │ │ │ -00027890: 7472 696e 6728 7b30 2c31 7d29 293d 3d73 tring({0,1}))==s │ │ │ │ -000278a0: 6567 4a23 2247 2220 2020 2020 2020 2020 egJ#"G" │ │ │ │ -000278b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -000278c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027800: 2020 2020 7c0a 7c6f 3139 203a 204d 7574 |.|o19 : Mut │ │ │ │ +00027810: 6162 6c65 4861 7368 5461 626c 6520 2020 ableHashTable │ │ │ │ +00027820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027850: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000278a0: 2d2d 2d2d 2b0a 7c69 3230 203a 2063 736d ----+.|i20 : csm │ │ │ │ +000278b0: 584c 6861 7368 2328 2247 284a 6163 6f62 XLhash#("G(Jacob │ │ │ │ +000278c0: 6961 6e29 227c 746f 5374 7269 6e67 287b ian)"|toString({ │ │ │ │ +000278d0: 302c 317d 2929 3d3d 7365 674a 2322 4722 0,1}))==segJ#"G" │ │ │ │ 000278e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027900: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00027910: 3020 3d20 7472 7565 2020 2020 2020 2020 0 = true │ │ │ │ +000278f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027950: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -000279b0: 3120 3a20 6373 6d58 4c68 6173 6823 2822 1 : csmXLhash#(" │ │ │ │ -000279c0: 5365 6772 6528 4a61 636f 6269 616e 2922 Segre(Jacobian)" │ │ │ │ -000279d0: 7c74 6f53 7472 696e 6728 7b30 2c31 7d29 |toString({0,1}) │ │ │ │ -000279e0: 293d 3d73 6567 4a23 2253 6567 7265 2220 )==segJ#"Segre" │ │ │ │ -000279f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00027a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a40: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00027a50: 3120 3d20 7472 7565 2020 2020 2020 2020 1 = true │ │ │ │ +00027940: 2020 2020 7c0a 7c6f 3230 203d 2074 7275 |.|o20 = tru │ │ │ │ +00027950: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00027960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027990: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000279a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000279e0: 2d2d 2d2d 2b0a 7c69 3231 203a 2063 736d ----+.|i21 : csm │ │ │ │ +000279f0: 584c 6861 7368 2328 2253 6567 7265 284a XLhash#("Segre(J │ │ │ │ +00027a00: 6163 6f62 6961 6e29 227c 746f 5374 7269 acobian)"|toStri │ │ │ │ +00027a10: 6e67 287b 302c 317d 2929 3d3d 7365 674a ng({0,1}))==segJ │ │ │ │ +00027a20: 2322 5365 6772 6522 2020 2020 2020 2020 #"Segre" │ │ │ │ +00027a30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4675 -----------+..Fu │ │ │ │ -00027af0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -00027b00: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ -00027b10: 616d 6564 204f 7574 7075 743a 0a3d 3d3d amed Output:.=== │ │ │ │ -00027b20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027b40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00027b50: 2022 4368 6572 6e28 2e2e 2e2c 4f75 7470 "Chern(...,Outp │ │ │ │ -00027b60: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ut=>...)" -- see │ │ │ │ -00027b70: 202a 6e6f 7465 2043 6865 726e 3a20 4368 *note Chern: Ch │ │ │ │ -00027b80: 6572 6e2c 202d 2d20 5468 6520 4368 6572 ern, -- The Cher │ │ │ │ -00027b90: 6e20 636c 6173 730a 2020 2a20 2243 534d n class. * "CSM │ │ │ │ -00027ba0: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ -00027bb0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00027bc0: 4353 4d3a 2043 534d 2c20 2d2d 2054 6865 CSM: CSM, -- The │ │ │ │ -00027bd0: 0a20 2020 2043 6865 726e 2d53 6368 7761 . Chern-Schwa │ │ │ │ -00027be0: 7274 7a2d 4d61 6350 6865 7273 6f6e 2063 rtz-MacPherson c │ │ │ │ -00027bf0: 6c61 7373 0a20 202a 2022 4575 6c65 7228 lass. * "Euler( │ │ │ │ -00027c00: 2e2e 2e2c 4f75 7470 7574 3d3e 2e2e 2e29 ...,Output=>...) │ │ │ │ -00027c10: 2220 2d2d 2073 6565 202a 6e6f 7465 2045 " -- see *note E │ │ │ │ -00027c20: 756c 6572 3a20 4575 6c65 722c 202d 2d20 uler: Euler, -- │ │ │ │ -00027c30: 5468 6520 4575 6c65 720a 2020 2020 4368 The Euler. Ch │ │ │ │ -00027c40: 6172 6163 7465 7269 7374 6963 0a20 202a aracteristic. * │ │ │ │ -00027c50: 2022 5365 6772 6528 2e2e 2e2c 4f75 7470 "Segre(...,Outp │ │ │ │ -00027c60: 7574 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ut=>...)" -- see │ │ │ │ -00027c70: 202a 6e6f 7465 2053 6567 7265 3a20 5365 *note Segre: Se │ │ │ │ -00027c80: 6772 652c 202d 2d20 5468 6520 5365 6772 gre, -- The Segr │ │ │ │ -00027c90: 6520 636c 6173 7320 6f66 2061 0a20 2020 e class of a. │ │ │ │ -00027ca0: 2073 7562 7363 6865 6d65 0a0a 466f 7220 subscheme..For │ │ │ │ -00027cb0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00027cc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00027cd0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00027ce0: 6f74 6520 4f75 7470 7574 3a20 4f75 7470 ote Output: Outp │ │ │ │ -00027cf0: 7574 2c20 6973 2061 202a 6e6f 7465 2073 ut, is a *note s │ │ │ │ -00027d00: 796d 626f 6c3a 2028 4d61 6361 756c 6179 ymbol: (Macaulay │ │ │ │ -00027d10: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ -00027d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00027d70: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00027d80: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00027d90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00027da0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00027db0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00027dc0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00027dd0: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ -00027de0: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ -00027df0: 3234 3639 3a30 2e0a 1f0a 4669 6c65 3a20 2469:0....File: │ │ │ │ -00027e00: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ -00027e10: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ -00027e20: 3a20 7072 6f62 6162 696c 6973 7469 6320 : probabilistic │ │ │ │ -00027e30: 616c 676f 7269 7468 6d2c 204e 6578 743a algorithm, Next: │ │ │ │ -00027e40: 2053 6567 7265 2c20 5072 6576 3a20 4f75 Segre, Prev: Ou │ │ │ │ -00027e50: 7470 7574 2c20 5570 3a20 546f 700a 0a70 tput, Up: Top..p │ │ │ │ -00027e60: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ -00027e70: 6f72 6974 686d 0a2a 2a2a 2a2a 2a2a 2a2a orithm.********* │ │ │ │ -00027e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00027e90: 5468 6520 616c 676f 7269 7468 6d73 2075 The algorithms u │ │ │ │ -00027ea0: 7365 6420 666f 7220 7468 6520 636f 6d70 sed for the comp │ │ │ │ -00027eb0: 7574 6174 696f 6e20 6f66 2063 6861 7261 utation of chara │ │ │ │ -00027ec0: 6374 6572 6973 7469 6320 636c 6173 7365 cteristic classe │ │ │ │ -00027ed0: 7320 6172 650a 7072 6f62 6162 696c 6973 s are.probabilis │ │ │ │ -00027ee0: 7469 632e 2054 6865 6f72 6574 6963 616c tic. Theoretical │ │ │ │ -00027ef0: 6c79 2c20 7468 6579 2063 616c 6375 6c61 ly, they calcula │ │ │ │ -00027f00: 7465 2074 6865 2063 6c61 7373 6573 2063 te the classes c │ │ │ │ -00027f10: 6f72 7265 6374 6c79 2066 6f72 2061 0a67 orrectly for a.g │ │ │ │ -00027f20: 656e 6572 616c 2063 686f 6963 6520 6f66 eneral choice of │ │ │ │ -00027f30: 2063 6572 7461 696e 2070 6f6c 796e 6f6d certain polynom │ │ │ │ -00027f40: 6961 6c73 2e20 5468 6174 2069 732c 2074 ials. That is, t │ │ │ │ -00027f50: 6865 7265 2069 7320 616e 206f 7065 6e20 here is an open │ │ │ │ -00027f60: 6465 6e73 6520 5a61 7269 736b 690a 7365 dense Zariski.se │ │ │ │ -00027f70: 7420 666f 7220 7768 6963 6820 7468 6520 t for which the │ │ │ │ -00027f80: 616c 676f 7269 7468 6d20 7969 656c 6473 algorithm yields │ │ │ │ -00027f90: 2074 6865 2063 6f72 7265 6374 2063 6c61 the correct cla │ │ │ │ -00027fa0: 7373 2c20 692e 652e 2c20 7468 6520 636f ss, i.e., the co │ │ │ │ -00027fb0: 7272 6563 7420 636c 6173 730a 6973 2063 rrect class.is c │ │ │ │ -00027fc0: 616c 6375 6c61 7465 6420 7769 7468 2070 alculated with p │ │ │ │ -00027fd0: 726f 6261 6269 6c69 7479 2031 2e20 486f robability 1. Ho │ │ │ │ -00027fe0: 7765 7665 722c 2073 696e 6365 2074 6865 wever, since the │ │ │ │ -00027ff0: 2069 6d70 6c65 6d65 6e74 6174 696f 6e20 implementation │ │ │ │ -00028000: 776f 726b 7320 6f76 6572 0a61 2064 6973 works over.a dis │ │ │ │ -00028010: 6372 6574 6520 7072 6f62 6162 696c 6974 crete probabilit │ │ │ │ -00028020: 7920 7370 6163 6520 7468 6572 6520 6973 y space there is │ │ │ │ -00028030: 2061 2076 6572 7920 736d 616c 6c2c 2062 a very small, b │ │ │ │ -00028040: 7574 206e 6f6e 2d7a 6572 6f2c 2070 726f ut non-zero, pro │ │ │ │ -00028050: 6261 6269 6c69 7479 0a6f 6620 6e6f 7420 bability.of not │ │ │ │ -00028060: 636f 6d70 7574 696e 6720 7468 6520 636f computing the co │ │ │ │ -00028070: 7272 6563 7420 636c 6173 732e 2053 6b65 rrect class. Ske │ │ │ │ -00028080: 7074 6963 616c 2075 7365 7273 2073 686f ptical users sho │ │ │ │ -00028090: 756c 6420 7265 7065 6174 2063 616c 6375 uld repeat calcu │ │ │ │ -000280a0: 6c61 7469 6f6e 730a 7365 7665 7261 6c20 lations.several │ │ │ │ -000280b0: 7469 6d65 7320 746f 2069 6e63 7265 6173 times to increas │ │ │ │ -000280c0: 6520 7468 6520 7072 6f62 6162 696c 6974 e the probabilit │ │ │ │ -000280d0: 7920 6f66 2063 6f6d 7075 7469 6e67 2074 y of computing t │ │ │ │ -000280e0: 6865 2063 6f72 7265 6374 2063 6c61 7373 he correct class │ │ │ │ -000280f0: 2e0a 0a49 6e20 7468 6520 6361 7365 206f ...In the case o │ │ │ │ -00028100: 6620 7468 6520 7379 6d62 6f6c 6963 2069 f the symbolic i │ │ │ │ -00028110: 6d70 6c65 6d65 6e74 6174 696f 6e20 6f66 mplementation of │ │ │ │ -00028120: 2074 6865 2050 726f 6a65 6374 6976 6544 the ProjectiveD │ │ │ │ -00028130: 6567 7265 6520 6d65 7468 6f64 0a70 7261 egree method.pra │ │ │ │ -00028140: 6374 6963 616c 2065 7870 6572 6965 6e63 ctical experienc │ │ │ │ -00028150: 6520 616e 6420 616c 676f 7269 7468 6d20 e and algorithm │ │ │ │ -00028160: 7465 7374 696e 6720 696e 6469 6361 7465 testing indicate │ │ │ │ -00028170: 2074 6861 7420 6120 6669 6e69 7465 2066 that a finite f │ │ │ │ -00028180: 6965 6c64 2077 6974 680a 6f76 6572 2032 ield with.over 2 │ │ │ │ -00028190: 3530 3030 2065 6c65 6d65 6e74 7320 6973 5000 elements is │ │ │ │ -000281a0: 206d 6f72 6520 7468 616e 2073 7566 6669 more than suffi │ │ │ │ -000281b0: 6369 656e 7420 746f 2065 7870 6563 7420 cient to expect │ │ │ │ -000281c0: 6120 636f 7272 6563 7420 7265 7375 6c74 a correct result │ │ │ │ -000281d0: 2077 6974 680a 6869 6768 2070 726f 6261 with.high proba │ │ │ │ -000281e0: 6269 6c69 7479 2c20 692e 652e 2075 7369 bility, i.e. usi │ │ │ │ -000281f0: 6e67 2074 6865 2066 696e 6974 6520 6669 ng the finite fi │ │ │ │ -00028200: 656c 6420 6b6b 3d5a 5a2f 3235 3037 3320 eld kk=ZZ/25073 │ │ │ │ -00028210: 7468 6520 6578 7065 7269 6d65 6e74 616c the experimental │ │ │ │ -00028220: 0a63 6861 6e63 6520 6f66 2066 6169 6c75 .chance of failu │ │ │ │ -00028230: 7265 2077 6974 6820 7468 6520 5072 6f6a re with the Proj │ │ │ │ -00028240: 6563 7469 7665 4465 6772 6565 2061 6c67 ectiveDegree alg │ │ │ │ -00028250: 6f72 6974 686d 206f 6e20 6120 7661 7269 orithm on a vari │ │ │ │ -00028260: 6574 7920 6f66 2065 7861 6d70 6c65 730a ety of examples. │ │ │ │ -00028270: 7761 7320 6c65 7373 2074 6861 6e20 312f was less than 1/ │ │ │ │ -00028280: 3230 3030 2e20 5573 696e 6720 7468 6520 2000. Using the │ │ │ │ -00028290: 6669 6e69 7465 2066 6965 6c64 206b 6b3d finite field kk= │ │ │ │ -000282a0: 5a5a 2f33 3237 3439 2072 6573 756c 7465 ZZ/32749 resulte │ │ │ │ -000282b0: 6420 696e 206e 6f0a 6661 696c 7572 6573 d in no.failures │ │ │ │ -000282c0: 2069 6e20 6f76 6572 2031 3030 3030 2061 in over 10000 a │ │ │ │ -000282d0: 7474 656d 7074 7320 6f66 2073 6576 6572 ttempts of sever │ │ │ │ -000282e0: 616c 2064 6966 6665 7265 6e74 2065 7861 al different exa │ │ │ │ -000282f0: 6d70 6c65 732e 0a0a 5765 2069 6c6c 7573 mples...We illus │ │ │ │ -00028300: 7472 6174 6520 7468 6520 7072 6f62 6162 trate the probab │ │ │ │ -00028310: 696c 6973 7469 6320 6265 6861 7669 6f75 ilistic behaviou │ │ │ │ -00028320: 7220 7769 7468 2061 6e20 6578 616d 706c r with an exampl │ │ │ │ -00028330: 6520 7768 6572 6520 7468 6520 6368 6f73 e where the chos │ │ │ │ -00028340: 656e 0a72 616e 646f 6d20 7365 6564 206c en.random seed l │ │ │ │ -00028350: 6561 6473 2074 6f20 6120 7772 6f6e 6720 eads to a wrong │ │ │ │ -00028360: 7265 7375 6c74 2069 6e20 7468 6520 6669 result in the fi │ │ │ │ -00028370: 7273 7420 6361 6c63 756c 6174 696f 6e2e rst calculation. │ │ │ │ -00028380: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ -00028390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000283b0: 2d2d 2d2b 0a7c 6931 203a 2073 6574 5261 ---+.|i1 : setRa │ │ │ │ -000283c0: 6e64 6f6d 5365 6564 2031 3231 3b20 2020 ndomSeed 121; │ │ │ │ -000283d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283e0: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574 |.| -- set │ │ │ │ -000283f0: 7469 6e67 2072 616e 646f 6d20 7365 6564 ting random seed │ │ │ │ -00028400: 2074 6f20 3132 3120 2020 2020 2020 2020 to 121 │ │ │ │ -00028410: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00028420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00028450: 3220 3a20 5220 3d20 5151 5b78 2c79 2c7a 2 : R = QQ[x,y,z │ │ │ │ -00028460: 2c77 5d20 2020 2020 2020 2020 2020 2020 ,w] │ │ │ │ -00028470: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028480: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00028490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a80: 2020 2020 7c0a 7c6f 3231 203d 2074 7275 |.|o21 = tru │ │ │ │ +00027a90: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00027aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ad0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027b20: 2d2d 2d2d 2b0a 0a46 756e 6374 696f 6e73 ----+..Functions │ │ │ │ +00027b30: 2077 6974 6820 6f70 7469 6f6e 616c 2061 with optional a │ │ │ │ +00027b40: 7267 756d 656e 7420 6e61 6d65 6420 4f75 rgument named Ou │ │ │ │ +00027b50: 7470 7574 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d tput:.========== │ │ │ │ +00027b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027b70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00027b80: 3d3d 3d3d 0a0a 2020 2a20 2243 6865 726e ====.. * "Chern │ │ │ │ +00027b90: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ +00027ba0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00027bb0: 4368 6572 6e3a 2043 6865 726e 2c20 2d2d Chern: Chern, -- │ │ │ │ +00027bc0: 2054 6865 2043 6865 726e 2063 6c61 7373 The Chern class │ │ │ │ +00027bd0: 0a20 202a 2022 4353 4d28 2e2e 2e2c 4f75 . * "CSM(...,Ou │ │ │ │ +00027be0: 7470 7574 3d3e 2e2e 2e29 2220 2d2d 2073 tput=>...)" -- s │ │ │ │ +00027bf0: 6565 202a 6e6f 7465 2043 534d 3a20 4353 ee *note CSM: CS │ │ │ │ +00027c00: 4d2c 202d 2d20 5468 650a 2020 2020 4368 M, -- The. Ch │ │ │ │ +00027c10: 6572 6e2d 5363 6877 6172 747a 2d4d 6163 ern-Schwartz-Mac │ │ │ │ +00027c20: 5068 6572 736f 6e20 636c 6173 730a 2020 Pherson class. │ │ │ │ +00027c30: 2a20 2245 756c 6572 282e 2e2e 2c4f 7574 * "Euler(...,Out │ │ │ │ +00027c40: 7075 743d 3e2e 2e2e 2922 202d 2d20 7365 put=>...)" -- se │ │ │ │ +00027c50: 6520 2a6e 6f74 6520 4575 6c65 723a 2045 e *note Euler: E │ │ │ │ +00027c60: 756c 6572 2c20 2d2d 2054 6865 2045 756c uler, -- The Eul │ │ │ │ +00027c70: 6572 0a20 2020 2043 6861 7261 6374 6572 er. Character │ │ │ │ +00027c80: 6973 7469 630a 2020 2a20 2253 6567 7265 istic. * "Segre │ │ │ │ +00027c90: 282e 2e2e 2c4f 7574 7075 743d 3e2e 2e2e (...,Output=>... │ │ │ │ +00027ca0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ +00027cb0: 5365 6772 653a 2053 6567 7265 2c20 2d2d Segre: Segre, -- │ │ │ │ +00027cc0: 2054 6865 2053 6567 7265 2063 6c61 7373 The Segre class │ │ │ │ +00027cd0: 206f 6620 610a 2020 2020 7375 6273 6368 of a. subsch │ │ │ │ +00027ce0: 656d 650a 0a46 6f72 2074 6865 2070 726f eme..For the pro │ │ │ │ +00027cf0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00027d00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00027d10: 6f62 6a65 6374 202a 6e6f 7465 204f 7574 object *note Out │ │ │ │ +00027d20: 7075 743a 204f 7574 7075 742c 2069 7320 put: Output, is │ │ │ │ +00027d30: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20 a *note symbol: │ │ │ │ +00027d40: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +00027d50: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ +00027d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00027da0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00027db0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00027dc0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00027dd0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00027de0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00027df0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +00027e00: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00027e10: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ +00027e20: 6173 7365 732e 6d32 3a32 3436 393a 302e asses.m2:2469:0. │ │ │ │ +00027e30: 0a1f 0a46 696c 653a 2043 6861 7261 6374 ...File: Charact │ │ │ │ +00027e40: 6572 6973 7469 6343 6c61 7373 6573 2e69 eristicClasses.i │ │ │ │ +00027e50: 6e66 6f2c 204e 6f64 653a 2070 726f 6261 nfo, Node: proba │ │ │ │ +00027e60: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ +00027e70: 686d 2c20 4e65 7874 3a20 5365 6772 652c hm, Next: Segre, │ │ │ │ +00027e80: 2050 7265 763a 204f 7574 7075 742c 2055 Prev: Output, U │ │ │ │ +00027e90: 703a 2054 6f70 0a0a 7072 6f62 6162 696c p: Top..probabil │ │ │ │ +00027ea0: 6973 7469 6320 616c 676f 7269 7468 6d0a istic algorithm. │ │ │ │ +00027eb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00027ec0: 2a2a 2a2a 2a2a 2a0a 0a54 6865 2061 6c67 *******..The alg │ │ │ │ +00027ed0: 6f72 6974 686d 7320 7573 6564 2066 6f72 orithms used for │ │ │ │ +00027ee0: 2074 6865 2063 6f6d 7075 7461 7469 6f6e the computation │ │ │ │ +00027ef0: 206f 6620 6368 6172 6163 7465 7269 7374 of characterist │ │ │ │ +00027f00: 6963 2063 6c61 7373 6573 2061 7265 0a70 ic classes are.p │ │ │ │ +00027f10: 726f 6261 6269 6c69 7374 6963 2e20 5468 robabilistic. Th │ │ │ │ +00027f20: 656f 7265 7469 6361 6c6c 792c 2074 6865 eoretically, the │ │ │ │ +00027f30: 7920 6361 6c63 756c 6174 6520 7468 6520 y calculate the │ │ │ │ +00027f40: 636c 6173 7365 7320 636f 7272 6563 746c classes correctl │ │ │ │ +00027f50: 7920 666f 7220 610a 6765 6e65 7261 6c20 y for a.general │ │ │ │ +00027f60: 6368 6f69 6365 206f 6620 6365 7274 6169 choice of certai │ │ │ │ +00027f70: 6e20 706f 6c79 6e6f 6d69 616c 732e 2054 n polynomials. T │ │ │ │ +00027f80: 6861 7420 6973 2c20 7468 6572 6520 6973 hat is, there is │ │ │ │ +00027f90: 2061 6e20 6f70 656e 2064 656e 7365 205a an open dense Z │ │ │ │ +00027fa0: 6172 6973 6b69 0a73 6574 2066 6f72 2077 ariski.set for w │ │ │ │ +00027fb0: 6869 6368 2074 6865 2061 6c67 6f72 6974 hich the algorit │ │ │ │ +00027fc0: 686d 2079 6965 6c64 7320 7468 6520 636f hm yields the co │ │ │ │ +00027fd0: 7272 6563 7420 636c 6173 732c 2069 2e65 rrect class, i.e │ │ │ │ +00027fe0: 2e2c 2074 6865 2063 6f72 7265 6374 2063 ., the correct c │ │ │ │ +00027ff0: 6c61 7373 0a69 7320 6361 6c63 756c 6174 lass.is calculat │ │ │ │ +00028000: 6564 2077 6974 6820 7072 6f62 6162 696c ed with probabil │ │ │ │ +00028010: 6974 7920 312e 2048 6f77 6576 6572 2c20 ity 1. However, │ │ │ │ +00028020: 7369 6e63 6520 7468 6520 696d 706c 656d since the implem │ │ │ │ +00028030: 656e 7461 7469 6f6e 2077 6f72 6b73 206f entation works o │ │ │ │ +00028040: 7665 720a 6120 6469 7363 7265 7465 2070 ver.a discrete p │ │ │ │ +00028050: 726f 6261 6269 6c69 7479 2073 7061 6365 robability space │ │ │ │ +00028060: 2074 6865 7265 2069 7320 6120 7665 7279 there is a very │ │ │ │ +00028070: 2073 6d61 6c6c 2c20 6275 7420 6e6f 6e2d small, but non- │ │ │ │ +00028080: 7a65 726f 2c20 7072 6f62 6162 696c 6974 zero, probabilit │ │ │ │ +00028090: 790a 6f66 206e 6f74 2063 6f6d 7075 7469 y.of not computi │ │ │ │ +000280a0: 6e67 2074 6865 2063 6f72 7265 6374 2063 ng the correct c │ │ │ │ +000280b0: 6c61 7373 2e20 536b 6570 7469 6361 6c20 lass. Skeptical │ │ │ │ +000280c0: 7573 6572 7320 7368 6f75 6c64 2072 6570 users should rep │ │ │ │ +000280d0: 6561 7420 6361 6c63 756c 6174 696f 6e73 eat calculations │ │ │ │ +000280e0: 0a73 6576 6572 616c 2074 696d 6573 2074 .several times t │ │ │ │ +000280f0: 6f20 696e 6372 6561 7365 2074 6865 2070 o increase the p │ │ │ │ +00028100: 726f 6261 6269 6c69 7479 206f 6620 636f robability of co │ │ │ │ +00028110: 6d70 7574 696e 6720 7468 6520 636f 7272 mputing the corr │ │ │ │ +00028120: 6563 7420 636c 6173 732e 0a0a 496e 2074 ect class...In t │ │ │ │ +00028130: 6865 2063 6173 6520 6f66 2074 6865 2073 he case of the s │ │ │ │ +00028140: 796d 626f 6c69 6320 696d 706c 656d 656e ymbolic implemen │ │ │ │ +00028150: 7461 7469 6f6e 206f 6620 7468 6520 5072 tation of the Pr │ │ │ │ +00028160: 6f6a 6563 7469 7665 4465 6772 6565 206d ojectiveDegree m │ │ │ │ +00028170: 6574 686f 640a 7072 6163 7469 6361 6c20 ethod.practical │ │ │ │ +00028180: 6578 7065 7269 656e 6365 2061 6e64 2061 experience and a │ │ │ │ +00028190: 6c67 6f72 6974 686d 2074 6573 7469 6e67 lgorithm testing │ │ │ │ +000281a0: 2069 6e64 6963 6174 6520 7468 6174 2061 indicate that a │ │ │ │ +000281b0: 2066 696e 6974 6520 6669 656c 6420 7769 finite field wi │ │ │ │ +000281c0: 7468 0a6f 7665 7220 3235 3030 3020 656c th.over 25000 el │ │ │ │ +000281d0: 656d 656e 7473 2069 7320 6d6f 7265 2074 ements is more t │ │ │ │ +000281e0: 6861 6e20 7375 6666 6963 6965 6e74 2074 han sufficient t │ │ │ │ +000281f0: 6f20 6578 7065 6374 2061 2063 6f72 7265 o expect a corre │ │ │ │ +00028200: 6374 2072 6573 756c 7420 7769 7468 0a68 ct result with.h │ │ │ │ +00028210: 6967 6820 7072 6f62 6162 696c 6974 792c igh probability, │ │ │ │ +00028220: 2069 2e65 2e20 7573 696e 6720 7468 6520 i.e. using the │ │ │ │ +00028230: 6669 6e69 7465 2066 6965 6c64 206b 6b3d finite field kk= │ │ │ │ +00028240: 5a5a 2f32 3530 3733 2074 6865 2065 7870 ZZ/25073 the exp │ │ │ │ +00028250: 6572 696d 656e 7461 6c0a 6368 616e 6365 erimental.chance │ │ │ │ +00028260: 206f 6620 6661 696c 7572 6520 7769 7468 of failure with │ │ │ │ +00028270: 2074 6865 2050 726f 6a65 6374 6976 6544 the ProjectiveD │ │ │ │ +00028280: 6567 7265 6520 616c 676f 7269 7468 6d20 egree algorithm │ │ │ │ +00028290: 6f6e 2061 2076 6172 6965 7479 206f 6620 on a variety of │ │ │ │ +000282a0: 6578 616d 706c 6573 0a77 6173 206c 6573 examples.was les │ │ │ │ +000282b0: 7320 7468 616e 2031 2f32 3030 302e 2055 s than 1/2000. U │ │ │ │ +000282c0: 7369 6e67 2074 6865 2066 696e 6974 6520 sing the finite │ │ │ │ +000282d0: 6669 656c 6420 6b6b 3d5a 5a2f 3332 3734 field kk=ZZ/3274 │ │ │ │ +000282e0: 3920 7265 7375 6c74 6564 2069 6e20 6e6f 9 resulted in no │ │ │ │ +000282f0: 0a66 6169 6c75 7265 7320 696e 206f 7665 .failures in ove │ │ │ │ +00028300: 7220 3130 3030 3020 6174 7465 6d70 7473 r 10000 attempts │ │ │ │ +00028310: 206f 6620 7365 7665 7261 6c20 6469 6666 of several diff │ │ │ │ +00028320: 6572 656e 7420 6578 616d 706c 6573 2e0a erent examples.. │ │ │ │ +00028330: 0a57 6520 696c 6c75 7374 7261 7465 2074 .We illustrate t │ │ │ │ +00028340: 6865 2070 726f 6261 6269 6c69 7374 6963 he probabilistic │ │ │ │ +00028350: 2062 6568 6176 696f 7572 2077 6974 6820 behaviour with │ │ │ │ +00028360: 616e 2065 7861 6d70 6c65 2077 6865 7265 an example where │ │ │ │ +00028370: 2074 6865 2063 686f 7365 6e0a 7261 6e64 the chosen.rand │ │ │ │ +00028380: 6f6d 2073 6565 6420 6c65 6164 7320 746f om seed leads to │ │ │ │ +00028390: 2061 2077 726f 6e67 2072 6573 756c 7420 a wrong result │ │ │ │ +000283a0: 696e 2074 6865 2066 6972 7374 2063 616c in the first cal │ │ │ │ +000283b0: 6375 6c61 7469 6f6e 2e0a 0a2b 2d2d 2d2d culation...+---- │ │ │ │ +000283c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000283d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000283e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000283f0: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ +00028400: 6420 3132 313b 2020 2020 2020 2020 2020 d 121; │ │ │ │ +00028410: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028420: 0a7c 202d 2d20 7365 7474 696e 6720 7261 .| -- setting ra │ │ │ │ +00028430: 6e64 6f6d 2073 6565 6420 746f 2031 3231 ndom seed to 121 │ │ │ │ +00028440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028450: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00028460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028480: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 203d -----+.|i2 : R = │ │ │ │ +00028490: 2051 515b 782c 792c 7a2c 775d 2020 2020 QQ[x,y,z,w] │ │ │ │ 000284a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284b0: 2020 7c0a 7c6f 3220 3d20 5220 2020 2020 |.|o2 = R │ │ │ │ +000284b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000284c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000284d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -000284f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000284e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +000284f0: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 00028500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028510: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00028520: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +00028510: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028520: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00028530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00028550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00028580: 7c69 3320 3a20 4920 3d20 6d69 6e6f 7273 |i3 : I = minors │ │ │ │ -00028590: 2832 2c6d 6174 7269 787b 7b78 2c79 2c7a (2,matrix{{x,y,z │ │ │ │ -000285a0: 7d2c 7b79 2c7a 2c77 7d7d 2920 2020 2020 },{y,z,w}}) │ │ │ │ -000285b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000285c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000285e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000285f0: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ -00028600: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00028610: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -00028620: 6465 616c 2028 2d20 7920 202b 2078 2a7a deal (- y + x*z │ │ │ │ -00028630: 2c20 2d20 792a 7a20 2b20 782a 772c 202d , - y*z + x*w, - │ │ │ │ -00028640: 207a 2020 2b20 792a 7729 7c0a 7c20 2020 z + y*w)|.| │ │ │ │ -00028650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028670: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028680: 6f33 203a 2049 6465 616c 206f 6620 5220 o3 : Ideal of R │ │ │ │ +00028540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028550: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +00028560: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +00028570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000285a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000285b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 -------+.|i3 : I │ │ │ │ +000285c0: 203d 206d 696e 6f72 7328 322c 6d61 7472 = minors(2,matr │ │ │ │ +000285d0: 6978 7b7b 782c 792c 7a7d 2c7b 792c 7a2c ix{{x,y,z},{y,z, │ │ │ │ +000285e0: 777d 7d29 2020 2020 2020 7c0a 7c20 2020 w}}) |.| │ │ │ │ +000285f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028600: 2020 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 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+000286c0: 6561 6c20 6f66 2052 2020 2020 2020 2020 eal of R │ │ │ │ +000286d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000286e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000286f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00028720: 3420 3a20 4368 6572 6e20 2849 2c43 6f6d 4 : Chern (I,Com │ │ │ │ +00028730: 704d 6574 686f 643d 3e50 6e52 6573 6964 pMethod=>PnResid │ │ │ │ +00028740: 7561 6c29 2020 2020 2020 2020 2020 207c ual) | │ │ │ │ +00028750: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028770: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00028780: 3420 3d20 3248 2020 2b20 3348 2020 2020 4 = 2H + 3H │ │ │ │ -00028790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000287b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000287c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028780: 2020 7c0a 7c20 2020 2020 2020 3320 2020 |.| 3 │ │ │ │ +00028790: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000287a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000287b0: 2020 2020 207c 0a7c 6f34 203d 2032 4820 |.|o4 = 2H │ │ │ │ +000287c0: 202b 2033 4820 2020 2020 2020 2020 2020 + 3H │ │ │ │ 000287d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000287e0: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20 |.| ZZ[H] │ │ │ │ +000287e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 000287f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028810: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d |.|o4 : --- │ │ │ │ -00028820: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ +00028810: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00028820: 2020 205a 5a5b 485d 2020 2020 2020 2020 ZZ[H] │ │ │ │ 00028830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028840: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028850: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00028840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028850: 7c6f 3420 3a20 2d2d 2d2d 2d20 2020 2020 |o4 : ----- │ │ │ │ 00028860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028870: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028880: 2020 2020 2048 2020 2020 2020 2020 2020 H │ │ │ │ +00028870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028880: 207c 0a7c 2020 2020 2020 2020 3420 2020 |.| 4 │ │ │ │ 00028890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000288a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000288b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -000288c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000288d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000288e0: 2d2b 0a7c 6935 203a 2043 6865 726e 2028 -+.|i5 : Chern ( │ │ │ │ -000288f0: 492c 436f 6d70 4d65 7468 6f64 3d3e 506e I,CompMethod=>Pn │ │ │ │ -00028900: 5265 7369 6475 616c 2920 2020 2020 2020 Residual) │ │ │ │ -00028910: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00028920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028940: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028950: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ +000288a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288b0: 2020 2020 7c0a 7c20 2020 2020 2020 4820 |.| H │ │ │ │ +000288c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000288e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000288f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028910: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +00028920: 3a20 4368 6572 6e20 2849 2c43 6f6d 704d : Chern (I,CompM │ │ │ │ +00028930: 6574 686f 643d 3e50 6e52 6573 6964 7561 ethod=>PnResidua │ │ │ │ +00028940: 6c29 2020 2020 2020 2020 2020 207c 0a7c l) |.| │ │ │ │ +00028950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028970: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00028980: 3d20 3248 2020 2b20 3348 2020 2020 2020 = 2H + 3H │ │ │ │ -00028990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000289b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028980: 7c0a 7c20 2020 2020 2020 3320 2020 2020 |.| 3 │ │ │ │ +00028990: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000289a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000289b0: 2020 207c 0a7c 6f35 203d 2032 4820 202b |.|o5 = 2H + │ │ │ │ +000289c0: 2033 4820 2020 2020 2020 2020 2020 2020 3H │ │ │ │ 000289d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000289e0: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020 |.| ZZ[H] │ │ │ │ +000289e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 000289f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a10: 2020 207c 0a7c 6f35 203a 202d 2d2d 2d2d |.|o5 : ----- │ │ │ │ -00028a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00028a20: 205a 5a5b 485d 2020 2020 2020 2020 2020 ZZ[H] │ │ │ │ 00028a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00028a50: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00028a50: 3520 3a20 2d2d 2d2d 2d20 2020 2020 2020 5 : ----- │ │ │ │ 00028a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028a80: 2020 2048 2020 2020 2020 2020 2020 2020 H │ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028a80: 0a7c 2020 2020 2020 2020 3420 2020 2020 .| 4 │ │ │ │ 00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028aa0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00028ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00028ae0: 0a7c 6936 203a 2043 6865 726e 2028 492c .|i6 : Chern (I, │ │ │ │ -00028af0: 436f 6d70 4d65 7468 6f64 3d3e 506e 5265 CompMethod=>PnRe │ │ │ │ -00028b00: 7369 6475 616c 2920 2020 2020 2020 2020 sidual) │ │ │ │ -00028b10: 2020 7c0a 7c20 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 2020 2020 2020 2033 |.| 3 │ │ │ │ -00028b50: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +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) │ │ │ │ +00028b40: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00028b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b70: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -00028b80: 3248 2020 2b20 3348 2020 2020 2020 2020 2H + 3H │ │ │ │ +00028b70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028b80: 7c20 2020 2020 2020 3320 2020 2020 3220 | 3 2 │ │ │ │ 00028b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ba0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028be0: 7c20 2020 2020 5a5a 5b48 5d20 2020 2020 | ZZ[H] │ │ │ │ +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 │ │ │ │ +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 │ │ │ │ 00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c10: 207c 0a7c 6f36 203a 202d 2d2d 2d2d 2020 |.|o6 : ----- │ │ │ │ -00028c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c10: 2020 2020 2020 207c 0a7c 2020 2020 205a |.| Z │ │ │ │ +00028c20: 5a5b 485d 2020 2020 2020 2020 2020 2020 Z[H] │ │ │ │ 00028c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c40: 2020 2020 7c0a 7c20 2020 2020 2020 2034 |.| 4 │ │ │ │ -00028c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028c40: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +00028c50: 3a20 2d2d 2d2d 2d20 2020 2020 2020 2020 : ----- │ │ │ │ 00028c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00028c80: 2048 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ +00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00028c80: 2020 2020 2020 2020 3420 2020 2020 2020 4 │ │ │ │ 00028c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ca0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00028cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00028ce0: 6937 203a 2043 6865 726e 2849 2c43 6f6d i7 : Chern(I,Com │ │ │ │ -00028cf0: 704d 6574 686f 643d 3e50 726f 6a65 6374 pMethod=>Project │ │ │ │ -00028d00: 6976 6544 6567 7265 6529 2020 2020 2020 iveDegree) │ │ │ │ -00028d10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00028d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028d40: 2020 207c 0a7c 2020 2020 2020 2033 2020 |.| 3 │ │ │ │ -00028d50: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00028ca0: 2020 2020 2020 2020 2020 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2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028d80: 2020 2020 2020 3320 2020 2020 3220 2020 3 2 │ │ │ │ 00028d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028da0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028db0: 2020 2031 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +00028da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028db0: 0a7c 6f37 203d 2032 6820 202b 2033 6820 .|o7 = 2h + 3h │ │ │ │ 00028dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -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 207c | │ │ │ │ -00028e10: 0a7c 2020 2020 205a 5a5b 6820 5d20 2020 .| ZZ[h ] │ │ │ │ +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 │ │ │ │ +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 │ │ │ │ 00028e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e40: 2020 7c0a 7c20 2020 2020 2020 2020 3120 |.| 1 │ │ │ │ -00028e50: 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 ] │ │ │ │ 00028e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028e70: 2020 2020 207c 0a7c 6f37 203a 202d 2d2d |.|o7 : --- │ │ │ │ -00028e80: 2d2d 2d20 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 │ │ │ │ 00028e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ea0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00028eb0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00028ea0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028eb0: 7c6f 3720 3a20 2d2d 2d2d 2d2d 2020 2020 |o7 : ------ │ │ │ │ 00028ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ed0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00028ee0: 2020 2020 2068 2020 2020 2020 2020 2020 h │ │ │ │ +00028ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028ee0: 207c 0a7c 2020 2020 2020 2020 3420 2020 |.| 4 │ │ │ │ 00028ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028f10: 7c20 2020 2020 2020 2031 2020 2020 2020 | 1 │ │ │ │ +00028f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028f10: 2020 2020 7c0a 7c20 2020 2020 2020 6820 |.| h │ │ │ │ 00028f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028f40: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -00028f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028f70: 2d2d 2d2d 2b0a 2d2d 2d2d 2d2d 2d2d 2d2d ----+.---------- │ │ │ │ +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 |.+--- │ │ │ │ 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 2d2d 2d2d ---------------- │ │ │ │ +00028fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a2d -------------+.- │ │ │ │ 00028fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028fc0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -00028fd0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -00028fe0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00028ff0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00029000: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00029010: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -00029020: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -00029030: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -00029040: 7365 732e 6d32 3a32 3337 383a 302e 0a1f ses.m2:2378:0... │ │ │ │ -00029050: 0a46 696c 653a 2043 6861 7261 6374 6572 .File: Character │ │ │ │ -00029060: 6973 7469 6343 6c61 7373 6573 2e69 6e66 isticClasses.inf │ │ │ │ -00029070: 6f2c 204e 6f64 653a 2053 6567 7265 2c20 o, Node: Segre, │ │ │ │ -00029080: 4e65 7874 3a20 546f 7269 6343 686f 7752 Next: ToricChowR │ │ │ │ -00029090: 696e 672c 2050 7265 763a 2070 726f 6261 ing, Prev: proba │ │ │ │ -000290a0: 6269 6c69 7374 6963 2061 6c67 6f72 6974 bilistic algorit │ │ │ │ -000290b0: 686d 2c20 5570 3a20 546f 700a 0a53 6567 hm, Up: Top..Seg │ │ │ │ -000290c0: 7265 202d 2d20 5468 6520 5365 6772 6520 re -- The Segre │ │ │ │ -000290d0: 636c 6173 7320 6f66 2061 2073 7562 7363 class of a subsc │ │ │ │ -000290e0: 6865 6d65 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a heme.*********** │ │ │ │ -000290f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -00029110: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00029120: 2020 5365 6772 6520 490a 2020 2020 2020 Segre I. │ │ │ │ -00029130: 2020 5365 6772 6528 412c 4929 0a20 2020 Segre(A,I). │ │ │ │ -00029140: 2020 2020 2053 6567 7265 2858 2c4a 290a Segre(X,J). │ │ │ │ -00029150: 2020 2020 2020 2020 5365 6772 6528 4368 Segre(Ch │ │ │ │ -00029160: 2c58 2c4a 290a 2020 2a20 496e 7075 7473 ,X,J). * Inputs │ │ │ │ -00029170: 3a0a 2020 2020 2020 2a20 492c 2061 6e20 :. * I, an │ │ │ │ -00029180: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ -00029190: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ -000291a0: 2c2c 2061 206d 756c 7469 2d68 6f6d 6f67 ,, a multi-homog │ │ │ │ -000291b0: 656e 656f 7573 2069 6465 616c 2069 6e20 eneous ideal in │ │ │ │ -000291c0: 610a 2020 2020 2020 2020 6772 6164 6564 a. graded │ │ │ │ -000291d0: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -000291e0: 206f 7665 7220 6120 6669 656c 6420 6465 over a field de │ │ │ │ -000291f0: 6669 6e69 6e67 2061 2063 6c6f 7365 6420 fining a closed │ │ │ │ -00029200: 7375 6273 6368 656d 6520 5620 6f66 0a20 subscheme V of. │ │ │ │ -00029210: 2020 2020 2020 205c 5050 5e7b 6e5f 317d \PP^{n_1} │ │ │ │ -00029220: 782e 2e2e 785c 5050 5e7b 6e5f 6d7d 0a20 x...x\PP^{n_m}. │ │ │ │ -00029230: 2020 2020 202a 2041 2c20 6120 2a6e 6f74 * A, a *not │ │ │ │ -00029240: 6520 7175 6f74 6965 6e74 2072 696e 673a e quotient ring: │ │ │ │ -00029250: 2028 4d61 6361 756c 6179 3244 6f63 2951 (Macaulay2Doc)Q │ │ │ │ -00029260: 756f 7469 656e 7452 696e 672c 2c0a 2020 uotientRing,,. │ │ │ │ -00029270: 2020 2020 2020 413d 5c5a 5a5b 685f 312c A=\ZZ[h_1, │ │ │ │ -00029280: 2e2e 2e2c 685f 6d5d 2f28 685f 315e 7b6e ...,h_m]/(h_1^{n │ │ │ │ -00029290: 5f31 2b31 7d2c 2e2e 2e2c 685f 6d5e 7b6e _1+1},...,h_m^{n │ │ │ │ -000292a0: 5f6d 2b31 7d29 2071 756f 7469 656e 7420 _m+1}) quotient │ │ │ │ -000292b0: 7269 6e67 0a20 2020 2020 2020 2072 6570 ring. rep │ │ │ │ -000292c0: 7265 7365 6e74 696e 6720 7468 6520 4368 resenting the Ch │ │ │ │ -000292d0: 6f77 2072 696e 6720 6f66 205c 5050 5e7b ow ring of \PP^{ │ │ │ │ -000292e0: 6e5f 317d 782e 2e2e 785c 5050 5e7b 6e5f n_1}x...x\PP^{n_ │ │ │ │ -000292f0: 6d7d 2c20 7468 6973 2072 696e 6720 7368 m}, this ring sh │ │ │ │ -00029300: 6f75 6c64 0a20 2020 2020 2020 2062 6520 ould. be │ │ │ │ -00029310: 6275 696c 7420 7573 696e 6720 7468 6520 built using the │ │ │ │ -00029320: 2a6e 6f74 6520 4368 6f77 5269 6e67 3a20 *note ChowRing: │ │ │ │ -00029330: 4368 6f77 5269 6e67 2c20 636f 6d6d 616e ChowRing, comman │ │ │ │ -00029340: 640a 2020 2020 2020 2a20 4a2c 2061 6e20 d. * J, an │ │ │ │ -00029350: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ -00029360: 6361 756c 6179 3244 6f63 2949 6465 616c caulay2Doc)Ideal │ │ │ │ -00029370: 2c2c 2069 6e20 7468 6520 6772 6164 6564 ,, in the graded │ │ │ │ -00029380: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ -00029390: 0a20 2020 2020 2020 2077 6869 6368 2069 . which i │ │ │ │ -000293a0: 7320 636f 6f72 6469 6e61 7465 2072 696e s coordinate rin │ │ │ │ -000293b0: 6720 6f66 2074 6865 204e 6f72 6d61 6c20 g of the Normal │ │ │ │ -000293c0: 546f 7269 6320 5661 7269 6574 7920 580a Toric Variety X. │ │ │ │ -000293d0: 2020 2020 2020 2a20 582c 2061 202a 6e6f * X, a *no │ │ │ │ -000293e0: 7465 206e 6f72 6d61 6c20 746f 7269 6320 te normal toric │ │ │ │ -000293f0: 7661 7269 6574 793a 0a20 2020 2020 2020 variety:. │ │ │ │ -00029400: 2028 4e6f 726d 616c 546f 7269 6356 6172 (NormalToricVar │ │ │ │ -00029410: 6965 7469 6573 294e 6f72 6d61 6c54 6f72 ieties)NormalTor │ │ │ │ -00029420: 6963 5661 7269 6574 792c 2c20 7768 6963 icVariety,, whic │ │ │ │ -00029430: 6820 6973 2074 6865 2061 6d62 6965 6e74 h is the ambient │ │ │ │ -00029440: 2073 7061 6365 0a20 2020 2020 2020 2077 space. w │ │ │ │ -00029450: 6869 6368 2063 6f6e 7461 696e 7320 5628 hich contains V( │ │ │ │ -00029460: 4a29 0a20 2020 2020 202a 2043 682c 2061 J). * Ch, a │ │ │ │ -00029470: 202a 6e6f 7465 2071 756f 7469 656e 7420 *note quotient │ │ │ │ -00029480: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ -00029490: 446f 6329 5175 6f74 6965 6e74 5269 6e67 Doc)QuotientRing │ │ │ │ -000294a0: 2c2c 2074 6865 2043 686f 7720 7269 6e67 ,, the Chow ring │ │ │ │ -000294b0: 0a20 2020 2020 2020 206f 6620 7468 6520 . of the │ │ │ │ -000294c0: 746f 7269 6320 7661 7269 6574 7920 582c toric variety X, │ │ │ │ -000294d0: 2043 683d 2872 696e 6720 4a29 2f28 5352 Ch=(ring J)/(SR │ │ │ │ -000294e0: 2b4c 5229 2077 6865 7265 2053 5220 6973 +LR) where SR is │ │ │ │ -000294f0: 2074 6865 0a20 2020 2020 2020 2053 7461 the. Sta │ │ │ │ -00029500: 6e6c 6579 2d52 6569 736e 6572 2069 6465 nley-Reisner ide │ │ │ │ -00029510: 616c 206f 6620 7468 6520 6661 6e20 6465 al of the fan de │ │ │ │ -00029520: 6669 6e69 6e67 2058 2061 6e64 204c 5220 fining X and LR │ │ │ │ -00029530: 6973 2074 6865 206c 696e 6561 720a 2020 is the linear. │ │ │ │ -00029540: 2020 2020 2020 7265 6c61 7469 6f6e 7320 relations │ │ │ │ -00029550: 6964 6561 6c2c 2074 6869 7320 7269 6e67 ideal, this ring │ │ │ │ -00029560: 2073 686f 756c 6420 6265 2062 7569 6c74 should be built │ │ │ │ -00029570: 2075 7369 6e67 2074 6865 202a 6e6f 7465 using the *note │ │ │ │ -00029580: 0a20 2020 2020 2020 2054 6f72 6963 4368 . ToricCh │ │ │ │ -00029590: 6f77 5269 6e67 3a20 546f 7269 6343 686f owRing: ToricCho │ │ │ │ -000295a0: 7752 696e 672c 2063 6f6d 6d61 6e64 0a20 wRing, command. │ │ │ │ -000295b0: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ -000295c0: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ -000295d0: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ -000295e0: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -000295f0: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ -00029600: 2020 2020 202a 2043 6f6d 704d 6574 686f * CompMetho │ │ │ │ -00029610: 6420 286d 6973 7369 6e67 2064 6f63 756d d (missing docum │ │ │ │ -00029620: 656e 7461 7469 6f6e 2920 3d3e 202e 2e2e entation) => ... │ │ │ │ -00029630: 2c20 6465 6661 756c 7420 7661 6c75 650a , default value. │ │ │ │ -00029640: 2020 2020 2020 2020 5072 6f6a 6563 7469 Projecti │ │ │ │ -00029650: 7665 4465 6772 6565 2c20 5072 6f6a 6563 veDegree, Projec │ │ │ │ -00029660: 7469 7665 4465 6772 6565 2c20 7468 6973 tiveDegree, this │ │ │ │ -00029670: 2061 6c67 6f72 6974 686d 206d 6179 2062 algorithm may b │ │ │ │ -00029680: 6520 7573 6564 2066 6f72 0a20 2020 2020 e used for. │ │ │ │ -00029690: 2020 2073 7562 7363 6865 6d65 7320 6f66 subschemes of │ │ │ │ -000296a0: 2061 6e79 2061 7070 6c69 6361 626c 6520 any applicable │ │ │ │ -000296b0: 746f 7269 6320 7661 7269 6574 7920 2874 toric variety (t │ │ │ │ -000296c0: 6869 7320 6d61 7920 6265 2063 6865 636b his may be check │ │ │ │ -000296d0: 6564 2075 7369 6e67 0a20 2020 2020 2020 ed using. │ │ │ │ -000296e0: 2074 6865 202a 6e6f 7465 2043 6865 636b the *note Check │ │ │ │ -000296f0: 546f 7269 6356 6172 6965 7479 5661 6c69 ToricVarietyVali │ │ │ │ -00029700: 643a 2043 6865 636b 546f 7269 6356 6172 d: CheckToricVar │ │ │ │ -00029710: 6965 7479 5661 6c69 642c 2063 6f6d 6d61 ietyValid, comma │ │ │ │ -00029720: 6e64 290a 2020 2020 2020 2a20 436f 6d70 nd). * Comp │ │ │ │ -00029730: 4d65 7468 6f64 2028 6d69 7373 696e 6720 Method (missing │ │ │ │ -00029740: 646f 6375 6d65 6e74 6174 696f 6e29 203d documentation) = │ │ │ │ -00029750: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -00029760: 616c 7565 0a20 2020 2020 2020 2050 726f alue. Pro │ │ │ │ -00029770: 6a65 6374 6976 6544 6567 7265 652c 2050 jectiveDegree, P │ │ │ │ -00029780: 6e52 6573 6964 7561 6c2c 2074 6869 7320 nResidual, this │ │ │ │ -00029790: 616c 676f 7269 7468 6d20 6d61 7920 6265 algorithm may be │ │ │ │ -000297a0: 2075 7365 6420 666f 7220 7375 6273 6368 used for subsch │ │ │ │ -000297b0: 656d 6573 0a20 2020 2020 2020 206f 6620 emes. of │ │ │ │ -000297c0: 5c50 505e 6e20 6f6e 6c79 0a20 2020 2020 \PP^n only. │ │ │ │ -000297d0: 202a 204f 7574 7075 7420 3d3e 202e 2e2e * Output => ... │ │ │ │ -000297e0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -000297f0: 4368 6f77 5269 6e67 456c 656d 656e 742c ChowRingElement, │ │ │ │ -00029800: 2043 686f 7752 696e 6745 6c65 6d65 6e74 ChowRingElement │ │ │ │ -00029810: 2c20 7265 7475 726e 730a 2020 2020 2020 , returns. │ │ │ │ -00029820: 2020 6120 5269 6e67 456c 656d 656e 7420 a RingElement │ │ │ │ -00029830: 696e 2074 6865 2043 686f 7720 7269 6e67 in the Chow ring │ │ │ │ -00029840: 206f 6620 7468 6520 6170 7072 6f70 7269 of the appropri │ │ │ │ -00029850: 6174 6520 616d 6269 656e 7420 7370 6163 ate ambient spac │ │ │ │ -00029860: 650a 2020 2020 2020 2a20 4f75 7470 7574 e. * Output │ │ │ │ -00029870: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00029880: 2076 616c 7565 2043 686f 7752 696e 6745 value ChowRingE │ │ │ │ -00029890: 6c65 6d65 6e74 2c20 4861 7368 466f 726d lement, HashForm │ │ │ │ -000298a0: 2c20 4861 7368 466f 726d 0a20 2020 2020 , HashForm. │ │ │ │ -000298b0: 2020 2072 6574 7572 6e73 2061 204d 7574 returns a Mut │ │ │ │ -000298c0: 6162 6c65 4861 7368 5461 626c 6520 636f ableHashTable co │ │ │ │ -000298d0: 6e74 6169 6e69 6e67 2074 6865 2066 6f6c ntaining the fol │ │ │ │ -000298e0: 6c6f 7769 6e67 206b 6579 733a 2022 4722 lowing keys: "G" │ │ │ │ -000298f0: 2028 7468 650a 2020 2020 2020 2020 706f (the. po │ │ │ │ -00029900: 6c79 6e6f 6d69 616c 2077 6974 6820 636f lynomial with co │ │ │ │ -00029910: 6566 6669 6369 656e 7473 206f 6620 7468 efficients of th │ │ │ │ -00029920: 6520 6879 7065 7270 6c61 6e65 2063 6c61 e hyperplane cla │ │ │ │ -00029930: 7373 6573 2072 6570 7265 7365 6e74 696e sses representin │ │ │ │ -00029940: 6720 7468 650a 2020 2020 2020 2020 7072 g the. pr │ │ │ │ -00029950: 6f6a 6563 7469 7665 2064 6567 7265 6573 ojective degrees │ │ │ │ -00029960: 292c 2022 476c 6973 7422 2028 7468 6520 ), "Glist" (the │ │ │ │ -00029970: 6c69 7374 2066 6f72 6d20 6f66 2022 4722 list form of "G" │ │ │ │ -00029980: 2920 2c20 2253 6567 7265 2220 2874 6865 ) , "Segre" (the │ │ │ │ -00029990: 0a20 2020 2020 2020 2074 6f74 616c 2053 . total S │ │ │ │ -000299a0: 6567 7265 2063 6c61 7373 206f 6620 7468 egre class of th │ │ │ │ -000299b0: 6520 696e 7075 7429 2c22 5365 6772 654c e input),"SegreL │ │ │ │ -000299c0: 6973 7422 2028 7468 6520 6c69 7374 2066 ist" (the list f │ │ │ │ -000299d0: 6f72 6d20 6f66 2022 5365 6772 6522 290a orm of "Segre"). │ │ │ │ -000299e0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -000299f0: 2020 202a 2061 202a 6e6f 7465 2072 696e * a *note rin │ │ │ │ -00029a00: 6720 656c 656d 656e 743a 2028 4d61 6361 g element: (Maca │ │ │ │ -00029a10: 756c 6179 3244 6f63 2952 696e 6745 6c65 ulay2Doc)RingEle │ │ │ │ -00029a20: 6d65 6e74 2c2c 2074 6865 2070 7573 6866 ment,, the pushf │ │ │ │ -00029a30: 6f72 7761 7264 206f 660a 2020 2020 2020 orward of. │ │ │ │ -00029a40: 2020 7468 6520 746f 7461 6c20 5365 6772 the total Segr │ │ │ │ -00029a50: 6520 636c 6173 7320 6f66 2074 6865 2073 e class of the s │ │ │ │ -00029a60: 6368 656d 6520 5620 6465 6669 6e65 6420 cheme V defined │ │ │ │ -00029a70: 6279 2074 6865 2069 6e70 7574 2069 6465 by the input ide │ │ │ │ -00029a80: 616c 2074 6f20 7468 650a 2020 2020 2020 al to the. │ │ │ │ -00029a90: 2020 6170 7072 6f70 7269 6174 6520 4368 appropriate Ch │ │ │ │ -00029aa0: 6f77 2072 696e 670a 0a44 6573 6372 6970 ow ring..Descrip │ │ │ │ -00029ab0: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00029ac0: 0a0a 466f 7220 6120 7375 6273 6368 656d ..For a subschem │ │ │ │ -00029ad0: 6520 5620 6f66 2061 6e20 6170 706c 6963 e V of an applic │ │ │ │ -00029ae0: 6162 6c65 2074 6f72 6963 2076 6172 6965 able toric varie │ │ │ │ -00029af0: 7479 2058 2074 6869 7320 636f 6d6d 616e ty X this comman │ │ │ │ -00029b00: 6420 636f 6d70 7574 6573 2074 6865 0a70 d computes the.p │ │ │ │ -00029b10: 7573 682d 666f 7277 6172 6420 6f66 2074 ush-forward of t │ │ │ │ -00029b20: 6865 2074 6f74 616c 2053 6567 7265 2063 he total Segre c │ │ │ │ -00029b30: 6c61 7373 2073 2856 2c58 2920 6f66 2056 lass s(V,X) of V │ │ │ │ -00029b40: 2069 6e20 5820 746f 2074 6865 2043 686f in X to the Cho │ │ │ │ -00029b50: 7720 7269 6e67 206f 6620 582e 0a0a 2b2d w ring of X...+- │ │ │ │ -00029b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029b90: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ -00029ba0: 6420 3732 3b20 2020 2020 2020 2020 2020 d 72; │ │ │ │ -00029bb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029bc0: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ -00029bd0: 6d20 7365 6564 2074 6f20 3732 2020 2020 m seed to 72 │ │ │ │ -00029be0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00029bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029c20: 3220 3a20 5220 3d20 5a5a 2f33 3237 3439 2 : R = ZZ/32749 │ │ │ │ -00029c30: 5b77 2c79 2c7a 5d20 2020 2020 2020 2020 [w,y,z] │ │ │ │ -00029c40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029c70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029c80: 3220 3d20 5220 2020 2020 2020 2020 2020 2 = R │ │ │ │ +00028fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00029000: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00029010: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00029020: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00029030: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00029040: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +00029050: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00029060: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ +00029070: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ +00029080: 3233 3738 3a30 2e0a 1f0a 4669 6c65 3a20 2378:0....File: │ │ │ │ +00029090: 4368 6172 6163 7465 7269 7374 6963 436c CharacteristicCl │ │ │ │ +000290a0: 6173 7365 732e 696e 666f 2c20 4e6f 6465 asses.info, Node │ │ │ │ +000290b0: 3a20 5365 6772 652c 204e 6578 743a 2054 : Segre, Next: T │ │ │ │ +000290c0: 6f72 6963 4368 6f77 5269 6e67 2c20 5072 oricChowRing, Pr │ │ │ │ +000290d0: 6576 3a20 7072 6f62 6162 696c 6973 7469 ev: probabilisti │ │ │ │ +000290e0: 6320 616c 676f 7269 7468 6d2c 2055 703a c algorithm, Up: │ │ │ │ +000290f0: 2054 6f70 0a0a 5365 6772 6520 2d2d 2054 Top..Segre -- T │ │ │ │ +00029100: 6865 2053 6567 7265 2063 6c61 7373 206f he Segre class o │ │ │ │ +00029110: 6620 6120 7375 6273 6368 656d 650a 2a2a f a subscheme.** │ │ │ │ +00029120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00029130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00029140: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00029150: 3a20 0a20 2020 2020 2020 2053 6567 7265 : . Segre │ │ │ │ +00029160: 2049 0a20 2020 2020 2020 2053 6567 7265 I. Segre │ │ │ │ +00029170: 2841 2c49 290a 2020 2020 2020 2020 5365 (A,I). Se │ │ │ │ +00029180: 6772 6528 582c 4a29 0a20 2020 2020 2020 gre(X,J). │ │ │ │ +00029190: 2053 6567 7265 2843 682c 582c 4a29 0a20 Segre(Ch,X,J). │ │ │ │ +000291a0: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000291b0: 202a 2049 2c20 616e 202a 6e6f 7465 2069 * I, an *note i │ │ │ │ +000291c0: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ +000291d0: 446f 6329 4964 6561 6c2c 2c20 6120 6d75 Doc)Ideal,, a mu │ │ │ │ +000291e0: 6c74 692d 686f 6d6f 6765 6e65 6f75 7320 lti-homogeneous │ │ │ │ +000291f0: 6964 6561 6c20 696e 2061 0a20 2020 2020 ideal in a. │ │ │ │ +00029200: 2020 2067 7261 6465 6420 706f 6c79 6e6f graded polyno │ │ │ │ +00029210: 6d69 616c 2072 696e 6720 6f76 6572 2061 mial ring over a │ │ │ │ +00029220: 2066 6965 6c64 2064 6566 696e 696e 6720 field defining │ │ │ │ +00029230: 6120 636c 6f73 6564 2073 7562 7363 6865 a closed subsche │ │ │ │ +00029240: 6d65 2056 206f 660a 2020 2020 2020 2020 me V of. │ │ │ │ +00029250: 5c50 505e 7b6e 5f31 7d78 2e2e 2e78 5c50 \PP^{n_1}x...x\P │ │ │ │ +00029260: 505e 7b6e 5f6d 7d0a 2020 2020 2020 2a20 P^{n_m}. * │ │ │ │ +00029270: 412c 2061 202a 6e6f 7465 2071 756f 7469 A, a *note quoti │ │ │ │ +00029280: 656e 7420 7269 6e67 3a20 284d 6163 6175 ent ring: (Macau │ │ │ │ +00029290: 6c61 7932 446f 6329 5175 6f74 6965 6e74 lay2Doc)Quotient │ │ │ │ +000292a0: 5269 6e67 2c2c 0a20 2020 2020 2020 2041 Ring,,. A │ │ │ │ +000292b0: 3d5c 5a5a 5b68 5f31 2c2e 2e2e 2c68 5f6d =\ZZ[h_1,...,h_m │ │ │ │ +000292c0: 5d2f 2868 5f31 5e7b 6e5f 312b 317d 2c2e ]/(h_1^{n_1+1},. │ │ │ │ +000292d0: 2e2e 2c68 5f6d 5e7b 6e5f 6d2b 317d 2920 ..,h_m^{n_m+1}) │ │ │ │ +000292e0: 7175 6f74 6965 6e74 2072 696e 670a 2020 quotient ring. │ │ │ │ +000292f0: 2020 2020 2020 7265 7072 6573 656e 7469 representi │ │ │ │ +00029300: 6e67 2074 6865 2043 686f 7720 7269 6e67 ng the Chow ring │ │ │ │ +00029310: 206f 6620 5c50 505e 7b6e 5f31 7d78 2e2e of \PP^{n_1}x.. │ │ │ │ +00029320: 2e78 5c50 505e 7b6e 5f6d 7d2c 2074 6869 .x\PP^{n_m}, thi │ │ │ │ +00029330: 7320 7269 6e67 2073 686f 756c 640a 2020 s ring should. │ │ │ │ +00029340: 2020 2020 2020 6265 2062 7569 6c74 2075 be built u │ │ │ │ +00029350: 7369 6e67 2074 6865 202a 6e6f 7465 2043 sing the *note C │ │ │ │ +00029360: 686f 7752 696e 673a 2043 686f 7752 696e howRing: ChowRin │ │ │ │ +00029370: 672c 2063 6f6d 6d61 6e64 0a20 2020 2020 g, command. │ │ │ │ +00029380: 202a 204a 2c20 616e 202a 6e6f 7465 2069 * J, an *note i │ │ │ │ +00029390: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ +000293a0: 446f 6329 4964 6561 6c2c 2c20 696e 2074 Doc)Ideal,, in t │ │ │ │ +000293b0: 6865 2067 7261 6465 6420 706f 6c79 6e6f he graded polyno │ │ │ │ +000293c0: 6d69 616c 2072 696e 670a 2020 2020 2020 mial ring. │ │ │ │ +000293d0: 2020 7768 6963 6820 6973 2063 6f6f 7264 which is coord │ │ │ │ +000293e0: 696e 6174 6520 7269 6e67 206f 6620 7468 inate ring of th │ │ │ │ +000293f0: 6520 4e6f 726d 616c 2054 6f72 6963 2056 e Normal Toric V │ │ │ │ +00029400: 6172 6965 7479 2058 0a20 2020 2020 202a ariety X. * │ │ │ │ +00029410: 2058 2c20 6120 2a6e 6f74 6520 6e6f 726d X, a *note norm │ │ │ │ +00029420: 616c 2074 6f72 6963 2076 6172 6965 7479 al toric variety │ │ │ │ +00029430: 3a0a 2020 2020 2020 2020 284e 6f72 6d61 :. (Norma │ │ │ │ +00029440: 6c54 6f72 6963 5661 7269 6574 6965 7329 lToricVarieties) │ │ │ │ +00029450: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +00029460: 7479 2c2c 2077 6869 6368 2069 7320 7468 ty,, which is th │ │ │ │ +00029470: 6520 616d 6269 656e 7420 7370 6163 650a e ambient space. │ │ │ │ +00029480: 2020 2020 2020 2020 7768 6963 6820 636f which co │ │ │ │ +00029490: 6e74 6169 6e73 2056 284a 290a 2020 2020 ntains V(J). │ │ │ │ +000294a0: 2020 2a20 4368 2c20 6120 2a6e 6f74 6520 * Ch, a *note │ │ │ │ +000294b0: 7175 6f74 6965 6e74 2072 696e 673a 2028 quotient ring: ( │ │ │ │ +000294c0: 4d61 6361 756c 6179 3244 6f63 2951 756f Macaulay2Doc)Quo │ │ │ │ +000294d0: 7469 656e 7452 696e 672c 2c20 7468 6520 tientRing,, the │ │ │ │ +000294e0: 4368 6f77 2072 696e 670a 2020 2020 2020 Chow ring. │ │ │ │ +000294f0: 2020 6f66 2074 6865 2074 6f72 6963 2076 of the toric v │ │ │ │ +00029500: 6172 6965 7479 2058 2c20 4368 3d28 7269 ariety X, Ch=(ri │ │ │ │ +00029510: 6e67 204a 292f 2853 522b 4c52 2920 7768 ng J)/(SR+LR) wh │ │ │ │ +00029520: 6572 6520 5352 2069 7320 7468 650a 2020 ere SR is the. │ │ │ │ +00029530: 2020 2020 2020 5374 616e 6c65 792d 5265 Stanley-Re │ │ │ │ +00029540: 6973 6e65 7220 6964 6561 6c20 6f66 2074 isner ideal of t │ │ │ │ +00029550: 6865 2066 616e 2064 6566 696e 696e 6720 he fan defining │ │ │ │ +00029560: 5820 616e 6420 4c52 2069 7320 7468 6520 X and LR is the │ │ │ │ +00029570: 6c69 6e65 6172 0a20 2020 2020 2020 2072 linear. r │ │ │ │ +00029580: 656c 6174 696f 6e73 2069 6465 616c 2c20 elations ideal, │ │ │ │ +00029590: 7468 6973 2072 696e 6720 7368 6f75 6c64 this ring should │ │ │ │ +000295a0: 2062 6520 6275 696c 7420 7573 696e 6720 be built using │ │ │ │ +000295b0: 7468 6520 2a6e 6f74 650a 2020 2020 2020 the *note. │ │ │ │ +000295c0: 2020 546f 7269 6343 686f 7752 696e 673a ToricChowRing: │ │ │ │ +000295d0: 2054 6f72 6963 4368 6f77 5269 6e67 2c20 ToricChowRing, │ │ │ │ +000295e0: 636f 6d6d 616e 640a 2020 2a20 2a6e 6f74 command. * *not │ │ │ │ +000295f0: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ +00029600: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ +00029610: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ +00029620: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ +00029630: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ +00029640: 436f 6d70 4d65 7468 6f64 2028 6d69 7373 CompMethod (miss │ │ │ │ +00029650: 696e 6720 646f 6375 6d65 6e74 6174 696f ing documentatio │ │ │ │ +00029660: 6e29 203d 3e20 2e2e 2e2c 2064 6566 6175 n) => ..., defau │ │ │ │ +00029670: 6c74 2076 616c 7565 0a20 2020 2020 2020 lt value. │ │ │ │ +00029680: 2050 726f 6a65 6374 6976 6544 6567 7265 ProjectiveDegre │ │ │ │ +00029690: 652c 2050 726f 6a65 6374 6976 6544 6567 e, ProjectiveDeg │ │ │ │ +000296a0: 7265 652c 2074 6869 7320 616c 676f 7269 ree, this algori │ │ │ │ +000296b0: 7468 6d20 6d61 7920 6265 2075 7365 6420 thm may be used │ │ │ │ +000296c0: 666f 720a 2020 2020 2020 2020 7375 6273 for. subs │ │ │ │ +000296d0: 6368 656d 6573 206f 6620 616e 7920 6170 chemes of any ap │ │ │ │ +000296e0: 706c 6963 6162 6c65 2074 6f72 6963 2076 plicable toric v │ │ │ │ +000296f0: 6172 6965 7479 2028 7468 6973 206d 6179 ariety (this may │ │ │ │ +00029700: 2062 6520 6368 6563 6b65 6420 7573 696e be checked usin │ │ │ │ +00029710: 670a 2020 2020 2020 2020 7468 6520 2a6e g. the *n │ │ │ │ +00029720: 6f74 6520 4368 6563 6b54 6f72 6963 5661 ote CheckToricVa │ │ │ │ +00029730: 7269 6574 7956 616c 6964 3a20 4368 6563 rietyValid: Chec │ │ │ │ +00029740: 6b54 6f72 6963 5661 7269 6574 7956 616c kToricVarietyVal │ │ │ │ +00029750: 6964 2c20 636f 6d6d 616e 6429 0a20 2020 id, command). │ │ │ │ +00029760: 2020 202a 2043 6f6d 704d 6574 686f 6420 * CompMethod │ │ │ │ +00029770: 286d 6973 7369 6e67 2064 6f63 756d 656e (missing documen │ │ │ │ +00029780: 7461 7469 6f6e 2920 3d3e 202e 2e2e 2c20 tation) => ..., │ │ │ │ +00029790: 6465 6661 756c 7420 7661 6c75 650a 2020 default value. │ │ │ │ +000297a0: 2020 2020 2020 5072 6f6a 6563 7469 7665 Projective │ │ │ │ +000297b0: 4465 6772 6565 2c20 506e 5265 7369 6475 Degree, PnResidu │ │ │ │ +000297c0: 616c 2c20 7468 6973 2061 6c67 6f72 6974 al, this algorit │ │ │ │ +000297d0: 686d 206d 6179 2062 6520 7573 6564 2066 hm may be used f │ │ │ │ +000297e0: 6f72 2073 7562 7363 6865 6d65 730a 2020 or subschemes. │ │ │ │ +000297f0: 2020 2020 2020 6f66 205c 5050 5e6e 206f of \PP^n o │ │ │ │ +00029800: 6e6c 790a 2020 2020 2020 2a20 4f75 7470 nly. * Outp │ │ │ │ +00029810: 7574 203d 3e20 2e2e 2e2c 2064 6566 6175 ut => ..., defau │ │ │ │ +00029820: 6c74 2076 616c 7565 2043 686f 7752 696e lt value ChowRin │ │ │ │ +00029830: 6745 6c65 6d65 6e74 2c20 4368 6f77 5269 gElement, ChowRi │ │ │ │ +00029840: 6e67 456c 656d 656e 742c 2072 6574 7572 ngElement, retur │ │ │ │ +00029850: 6e73 0a20 2020 2020 2020 2061 2052 696e ns. a Rin │ │ │ │ +00029860: 6745 6c65 6d65 6e74 2069 6e20 7468 6520 gElement in the │ │ │ │ +00029870: 4368 6f77 2072 696e 6720 6f66 2074 6865 Chow ring of the │ │ │ │ +00029880: 2061 7070 726f 7072 6961 7465 2061 6d62 appropriate amb │ │ │ │ +00029890: 6965 6e74 2073 7061 6365 0a20 2020 2020 ient space. │ │ │ │ +000298a0: 202a 204f 7574 7075 7420 3d3e 202e 2e2e * Output => ... │ │ │ │ +000298b0: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +000298c0: 4368 6f77 5269 6e67 456c 656d 656e 742c ChowRingElement, │ │ │ │ +000298d0: 2048 6173 6846 6f72 6d2c 2048 6173 6846 HashForm, HashF │ │ │ │ +000298e0: 6f72 6d0a 2020 2020 2020 2020 7265 7475 orm. retu │ │ │ │ +000298f0: 726e 7320 6120 4d75 7461 626c 6548 6173 rns a MutableHas │ │ │ │ +00029900: 6854 6162 6c65 2063 6f6e 7461 696e 696e hTable containin │ │ │ │ +00029910: 6720 7468 6520 666f 6c6c 6f77 696e 6720 g the following │ │ │ │ +00029920: 6b65 7973 3a20 2247 2220 2874 6865 0a20 keys: "G" (the. │ │ │ │ +00029930: 2020 2020 2020 2070 6f6c 796e 6f6d 6961 polynomia │ │ │ │ +00029940: 6c20 7769 7468 2063 6f65 6666 6963 6965 l with coefficie │ │ │ │ +00029950: 6e74 7320 6f66 2074 6865 2068 7970 6572 nts of the hyper │ │ │ │ +00029960: 706c 616e 6520 636c 6173 7365 7320 7265 plane classes re │ │ │ │ +00029970: 7072 6573 656e 7469 6e67 2074 6865 0a20 presenting the. │ │ │ │ +00029980: 2020 2020 2020 2070 726f 6a65 6374 6976 projectiv │ │ │ │ +00029990: 6520 6465 6772 6565 7329 2c20 2247 6c69 e degrees), "Gli │ │ │ │ +000299a0: 7374 2220 2874 6865 206c 6973 7420 666f st" (the list fo │ │ │ │ +000299b0: 726d 206f 6620 2247 2229 202c 2022 5365 rm of "G") , "Se │ │ │ │ +000299c0: 6772 6522 2028 7468 650a 2020 2020 2020 gre" (the. │ │ │ │ +000299d0: 2020 746f 7461 6c20 5365 6772 6520 636c total Segre cl │ │ │ │ +000299e0: 6173 7320 6f66 2074 6865 2069 6e70 7574 ass of the input │ │ │ │ +000299f0: 292c 2253 6567 7265 4c69 7374 2220 2874 ),"SegreList" (t │ │ │ │ +00029a00: 6865 206c 6973 7420 666f 726d 206f 6620 he list form of │ │ │ │ +00029a10: 2253 6567 7265 2229 0a20 202a 204f 7574 "Segre"). * Out │ │ │ │ +00029a20: 7075 7473 3a0a 2020 2020 2020 2a20 6120 puts:. * a │ │ │ │ +00029a30: 2a6e 6f74 6520 7269 6e67 2065 6c65 6d65 *note ring eleme │ │ │ │ +00029a40: 6e74 3a20 284d 6163 6175 6c61 7932 446f nt: (Macaulay2Do │ │ │ │ +00029a50: 6329 5269 6e67 456c 656d 656e 742c 2c20 c)RingElement,, │ │ │ │ +00029a60: 7468 6520 7075 7368 666f 7277 6172 6420 the pushforward │ │ │ │ +00029a70: 6f66 0a20 2020 2020 2020 2074 6865 2074 of. the t │ │ │ │ +00029a80: 6f74 616c 2053 6567 7265 2063 6c61 7373 otal Segre class │ │ │ │ +00029a90: 206f 6620 7468 6520 7363 6865 6d65 2056 of the scheme V │ │ │ │ +00029aa0: 2064 6566 696e 6564 2062 7920 7468 6520 defined by the │ │ │ │ +00029ab0: 696e 7075 7420 6964 6561 6c20 746f 2074 input ideal to t │ │ │ │ +00029ac0: 6865 0a20 2020 2020 2020 2061 7070 726f he. appro │ │ │ │ +00029ad0: 7072 6961 7465 2043 686f 7720 7269 6e67 priate Chow ring │ │ │ │ +00029ae0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00029af0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a46 6f72 2061 =========..For a │ │ │ │ +00029b00: 2073 7562 7363 6865 6d65 2056 206f 6620 subscheme V of │ │ │ │ +00029b10: 616e 2061 7070 6c69 6361 626c 6520 746f an applicable to │ │ │ │ +00029b20: 7269 6320 7661 7269 6574 7920 5820 7468 ric variety X th │ │ │ │ +00029b30: 6973 2063 6f6d 6d61 6e64 2063 6f6d 7075 is command compu │ │ │ │ +00029b40: 7465 7320 7468 650a 7075 7368 2d66 6f72 tes the.push-for │ │ │ │ +00029b50: 7761 7264 206f 6620 7468 6520 746f 7461 ward of the tota │ │ │ │ +00029b60: 6c20 5365 6772 6520 636c 6173 7320 7328 l Segre class s( │ │ │ │ +00029b70: 562c 5829 206f 6620 5620 696e 2058 2074 V,X) of V in X t │ │ │ │ +00029b80: 6f20 7468 6520 4368 6f77 2072 696e 6720 o the Chow ring │ │ │ │ +00029b90: 6f66 2058 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d of X...+-------- │ │ │ │ +00029ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029bc0: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574 -----+.|i1 : set │ │ │ │ +00029bd0: 5261 6e64 6f6d 5365 6564 2037 323b 2020 RandomSeed 72; │ │ │ │ +00029be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029bf0: 2020 2020 207c 0a7c 202d 2d20 7365 7474 |.| -- sett │ │ │ │ +00029c00: 696e 6720 7261 6e64 6f6d 2073 6565 6420 ing random seed │ │ │ │ +00029c10: 746f 2037 3220 2020 2020 2020 2020 2020 to 72 │ │ │ │ +00029c20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029c50: 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 203d -----+.|i2 : R = │ │ │ │ +00029c60: 205a 5a2f 3332 3734 395b 772c 792c 7a5d ZZ/32749[w,y,z] │ │ │ │ +00029c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029c80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00029c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029cb0: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ 00029cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029ce0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -00029cf0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00029d00: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00029d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029d40: 3320 3a20 5365 6772 6528 6964 6561 6c28 3 : Segre(ideal( │ │ │ │ -00029d50: 772a 7929 2c43 6f6d 704d 6574 686f 643d w*y),CompMethod= │ │ │ │ -00029d60: 3e50 6e52 6573 6964 7561 6c29 7c0a 7c20 >PnResidual)|.| │ │ │ │ -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 7c0a 7c20 |.| │ │ │ │ -00029da0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +00029cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ce0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d10: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ +00029d20: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +00029d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029d40: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029d70: 2d2d 2d2d 2d2b 0a7c 6933 203a 2053 6567 -----+.|i3 : Seg │ │ │ │ +00029d80: 7265 2869 6465 616c 2877 2a79 292c 436f re(ideal(w*y),Co │ │ │ │ +00029d90: 6d70 4d65 7468 6f64 3d3e 506e 5265 7369 mpMethod=>PnResi │ │ │ │ +00029da0: 6475 616c 297c 0a7c 2020 2020 2020 2020 dual)|.| │ │ │ │ 00029db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029dd0: 3320 3d20 2d20 3448 2020 2b20 3248 2020 3 = - 4H + 2H │ │ │ │ -00029de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029e30: 2020 2020 5a5a 5b48 5d20 2020 2020 2020 ZZ[H] │ │ │ │ +00029dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029dd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029de0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00029df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e00: 2020 2020 207c 0a7c 6f33 203d 202d 2034 |.|o3 = - 4 │ │ │ │ +00029e10: 4820 202b 2032 4820 2020 2020 2020 2020 H + 2H │ │ │ │ +00029e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e30: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00029e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029e60: 3320 3a20 2d2d 2d2d 2d20 2020 2020 2020 3 : ----- │ │ │ │ -00029e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029e80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029e90: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ -00029ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029ec0: 2020 2020 2020 4820 2020 2020 2020 2020 H │ │ │ │ -00029ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00029ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00029f20: 3420 3a20 413d 4368 6f77 5269 6e67 2852 4 : A=ChowRing(R │ │ │ │ -00029f30: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -00029f40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -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 7c0a 7c6f |.|o │ │ │ │ -00029f80: 3420 3d20 4120 2020 2020 2020 2020 2020 4 = A │ │ │ │ +00029e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e60: 2020 2020 207c 0a7c 2020 2020 205a 5a5b |.| ZZ[ │ │ │ │ +00029e70: 485d 2020 2020 2020 2020 2020 2020 2020 H] │ │ │ │ +00029e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029e90: 2020 2020 207c 0a7c 6f33 203a 202d 2d2d |.|o3 : --- │ │ │ │ +00029ea0: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ +00029eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00029ed0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00029ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029ef0: 2020 2020 207c 0a7c 2020 2020 2020 2048 |.| H │ │ │ │ +00029f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f20: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00029f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00029f50: 2d2d 2d2d 2d2b 0a7c 6934 203a 2041 3d43 -----+.|i4 : A=C │ │ │ │ +00029f60: 686f 7752 696e 6728 5229 2020 2020 2020 howRing(R) │ │ │ │ +00029f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029f80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00029f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00029fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fb0: 2020 2020 207c 0a7c 6f34 203d 2041 2020 |.|o4 = A │ │ │ │ 00029fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00029fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00029fe0: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +00029fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00029fe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00029ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a000: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002a010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002a040: 3520 3a20 5365 6772 6528 412c 6964 6561 5 : Segre(A,idea │ │ │ │ -0002a050: 6c28 775e 322a 792c 772a 795e 3229 2920 l(w^2*y,w*y^2)) │ │ │ │ -0002a060: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a090: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a0a0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0002a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a010: 2020 2020 207c 0a7c 6f34 203a 2051 756f |.|o4 : Quo │ │ │ │ +0002a020: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +0002a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a070: 2d2d 2d2d 2d2b 0a7c 6935 203a 2053 6567 -----+.|i5 : Seg │ │ │ │ +0002a080: 7265 2841 2c69 6465 616c 2877 5e32 2a79 re(A,ideal(w^2*y │ │ │ │ +0002a090: 2c77 2a79 5e32 2929 2020 2020 2020 2020 ,w*y^2)) │ │ │ │ +0002a0a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002a0d0: 3520 3d20 2d20 3368 2020 2b20 3268 2020 5 = - 3h + 2h │ │ │ │ -0002a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a100: 2020 2020 2020 2020 3120 2020 2020 3120 1 1 │ │ │ │ -0002a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a120: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0002a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002a160: 3520 3a20 4120 2020 2020 2020 2020 2020 5 : A │ │ │ │ +0002a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a0d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a0e0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a100: 2020 2020 207c 0a7c 6f35 203d 202d 2033 |.|o5 = - 3 │ │ │ │ +0002a110: 6820 202b 2032 6820 2020 2020 2020 2020 h + 2h │ │ │ │ +0002a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a130: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002a140: 2031 2020 2020 2031 2020 2020 2020 2020 1 1 │ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a160: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a180: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -0002a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a4e ------------+..N │ │ │ │ -0002a1c0: 6f77 2063 6f6e 7369 6465 7220 616e 2065 ow consider an e │ │ │ │ -0002a1d0: 7861 6d70 6c65 2069 6e20 5c50 505e 3220 xample in \PP^2 │ │ │ │ -0002a1e0: 5c74 696d 6573 205c 5050 5e32 2c20 6966 \times \PP^2, if │ │ │ │ -0002a1f0: 2077 6520 696e 7075 7420 7468 6520 4368 we input the Ch │ │ │ │ -0002a200: 6f77 2072 696e 6720 4120 7468 650a 6f75 ow ring A the.ou │ │ │ │ -0002a210: 7470 7574 2077 696c 6c20 6265 2072 6574 tput will be ret │ │ │ │ -0002a220: 7572 6e65 6420 696e 2074 6865 2073 616d urned in the sam │ │ │ │ -0002a230: 6520 7269 6e67 2e20 546f 2065 6e73 7572 e ring. To ensur │ │ │ │ -0002a240: 6520 7072 6f70 6572 2066 756e 6374 696f e proper functio │ │ │ │ -0002a250: 6e20 6f66 2074 6865 0a6d 6574 686f 6473 n of the.methods │ │ │ │ -0002a260: 2077 6520 6275 696c 6420 7468 6520 4368 we build the Ch │ │ │ │ -0002a270: 6f77 2072 696e 6720 7573 696e 6720 7468 ow ring using th │ │ │ │ -0002a280: 6520 2a6e 6f74 6520 4368 6f77 5269 6e67 e *note ChowRing │ │ │ │ -0002a290: 3a20 4368 6f77 5269 6e67 2c20 636f 6d6d : ChowRing, comm │ │ │ │ -0002a2a0: 616e 642e 2057 650a 6d61 7920 616c 736f and. We.may also │ │ │ │ -0002a2b0: 2072 6574 7572 6e20 6120 4d75 7461 626c return a Mutabl │ │ │ │ -0002a2c0: 6548 6173 6854 6162 6c65 2e0a 0a2b 2d2d eHashTable...+-- │ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ -0002a320: 203a 2052 3d4d 756c 7469 5072 6f6a 436f : R=MultiProjCo │ │ │ │ -0002a330: 6f72 6452 696e 6728 7b32 2c32 7d29 2020 ordRing({2,2}) │ │ │ │ -0002a340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a190: 2020 2020 207c 0a7c 6f35 203a 2041 2020 |.|o5 : A │ │ │ │ +0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a1f0: 2d2d 2d2d 2d2b 0a0a 4e6f 7720 636f 6e73 -----+..Now cons │ │ │ │ +0002a200: 6964 6572 2061 6e20 6578 616d 706c 6520 ider an example │ │ │ │ +0002a210: 696e 205c 5050 5e32 205c 7469 6d65 7320 in \PP^2 \times │ │ │ │ +0002a220: 5c50 505e 322c 2069 6620 7765 2069 6e70 \PP^2, if we inp │ │ │ │ +0002a230: 7574 2074 6865 2043 686f 7720 7269 6e67 ut the Chow ring │ │ │ │ +0002a240: 2041 2074 6865 0a6f 7574 7075 7420 7769 A the.output wi │ │ │ │ +0002a250: 6c6c 2062 6520 7265 7475 726e 6564 2069 ll be returned i │ │ │ │ +0002a260: 6e20 7468 6520 7361 6d65 2072 696e 672e n the same ring. │ │ │ │ +0002a270: 2054 6f20 656e 7375 7265 2070 726f 7065 To ensure prope │ │ │ │ +0002a280: 7220 6675 6e63 7469 6f6e 206f 6620 7468 r function of th │ │ │ │ +0002a290: 650a 6d65 7468 6f64 7320 7765 2062 7569 e.methods we bui │ │ │ │ +0002a2a0: 6c64 2074 6865 2043 686f 7720 7269 6e67 ld the Chow ring │ │ │ │ +0002a2b0: 2075 7369 6e67 2074 6865 202a 6e6f 7465 using the *note │ │ │ │ +0002a2c0: 2043 686f 7752 696e 673a 2043 686f 7752 ChowRing: ChowR │ │ │ │ +0002a2d0: 696e 672c 2063 6f6d 6d61 6e64 2e20 5765 ing, command. We │ │ │ │ +0002a2e0: 0a6d 6179 2061 6c73 6f20 7265 7475 726e .may also return │ │ │ │ +0002a2f0: 2061 204d 7574 6162 6c65 4861 7368 5461 a MutableHashTa │ │ │ │ +0002a300: 626c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ble...+--------- │ │ │ │ +0002a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a350: 2d2d 2d2d 2b0a 7c69 3620 3a20 523d 4d75 ----+.|i6 : R=Mu │ │ │ │ +0002a360: 6c74 6950 726f 6a43 6f6f 7264 5269 6e67 ltiProjCoordRing │ │ │ │ +0002a370: 287b 322c 327d 2920 2020 2020 2020 2020 ({2,2}) │ │ │ │ 0002a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3b0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0002a3c0: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ +0002a3a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a400: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002a3f0: 2020 2020 7c0a 7c6f 3620 3d20 5220 2020 |.|o6 = R │ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a450: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0002a460: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -0002a470: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +0002a440: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a4a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0002a500: 203a 2072 3d67 656e 7320 5220 2020 2020 : r=gens R │ │ │ │ -0002a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a540: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002a490: 2020 2020 7c0a 7c6f 3620 3a20 506f 6c79 |.|o6 : Poly │ │ │ │ +0002a4a0: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +0002a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a4e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a530: 2d2d 2d2d 2b0a 7c69 3720 3a20 723d 6765 ----+.|i7 : r=ge │ │ │ │ +0002a540: 6e73 2052 2020 2020 2020 2020 2020 2020 ns R │ │ │ │ 0002a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a590: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0002a5a0: 203d 207b 7820 2c20 7820 2c20 7820 2c20 = {x , x , x , │ │ │ │ -0002a5b0: 7820 2c20 7820 2c20 7820 7d20 2020 2020 x , x , x } │ │ │ │ +0002a580: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a5e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002a5f0: 2020 2020 2030 2020 2031 2020 2032 2020 0 1 2 │ │ │ │ -0002a600: 2033 2020 2034 2020 2035 2020 2020 2020 3 4 5 │ │ │ │ +0002a5d0: 2020 2020 7c0a 7c6f 3720 3d20 7b78 202c |.|o7 = {x , │ │ │ │ +0002a5e0: 2078 202c 2078 202c 2078 202c 2078 202c x , x , x , x , │ │ │ │ +0002a5f0: 2078 207d 2020 2020 2020 2020 2020 2020 x } │ │ │ │ +0002a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a630: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a620: 2020 2020 7c0a 7c20 2020 2020 2020 3020 |.| 0 │ │ │ │ +0002a630: 2020 3120 2020 3220 2020 3320 2020 3420 1 2 3 4 │ │ │ │ +0002a640: 2020 3520 2020 2020 2020 2020 2020 2020 5 │ │ │ │ 0002a650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a680: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -0002a690: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +0002a670: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a6d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002a6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -0002a730: 203a 2041 3d43 686f 7752 696e 6728 5229 : A=ChowRing(R) │ │ │ │ -0002a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a770: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002a6c0: 2020 2020 7c0a 7c6f 3720 3a20 4c69 7374 |.|o7 : List │ │ │ │ +0002a6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a710: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a760: 2d2d 2d2d 2b0a 7c69 3820 3a20 413d 4368 ----+.|i8 : A=Ch │ │ │ │ +0002a770: 6f77 5269 6e67 2852 2920 2020 2020 2020 owRing(R) │ │ │ │ 0002a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a7c0: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ -0002a7d0: 203d 2041 2020 2020 2020 2020 2020 2020 = A │ │ │ │ +0002a7b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a810: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002a800: 2020 2020 7c0a 7c6f 3820 3d20 4120 2020 |.|o8 = A │ │ │ │ +0002a810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a860: 2020 2020 2020 2020 2020 207c 0a7c 6f38 |.|o8 │ │ │ │ -0002a870: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ +0002a850: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a8b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ -0002a910: 203a 2049 3d69 6465 616c 2872 5f30 5e32 : I=ideal(r_0^2 │ │ │ │ -0002a920: 2a72 5f33 2d72 5f34 2a72 5f31 2a72 5f32 *r_3-r_4*r_1*r_2 │ │ │ │ -0002a930: 2c72 5f32 5e32 2a72 5f35 2920 2020 2020 ,r_2^2*r_5) │ │ │ │ -0002a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a950: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8a0: 2020 2020 7c0a 7c6f 3820 3a20 5175 6f74 |.|o8 : Quot │ │ │ │ +0002a8b0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a8f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002a940: 2d2d 2d2d 2b0a 7c69 3920 3a20 493d 6964 ----+.|i9 : I=id │ │ │ │ +0002a950: 6561 6c28 725f 305e 322a 725f 332d 725f eal(r_0^2*r_3-r_ │ │ │ │ +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) │ │ │ │ 0002a980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a9a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002a9b0: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ -0002a9c0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +0002a990: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a9f0: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -0002aa00: 203d 2069 6465 616c 2028 7820 7820 202d = ideal (x x - │ │ │ │ -0002aa10: 2078 2078 2078 202c 2078 2078 2029 2020 x x x , x x ) │ │ │ │ +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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa40: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002aa50: 2020 2020 2020 2020 2020 2030 2033 2020 0 3 │ │ │ │ -0002aa60: 2020 3120 3220 3420 2020 3220 3520 2020 1 2 4 2 5 │ │ │ │ +0002aa30: 2020 2020 7c0a 7c6f 3920 3d20 6964 6561 |.|o9 = idea │ │ │ │ +0002aa40: 6c20 2878 2078 2020 2d20 7820 7820 7820 l (x x - x x x │ │ │ │ +0002aa50: 2c20 7820 7820 2920 2020 2020 2020 2020 , x x ) │ │ │ │ +0002aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aa90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002aaa0: 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 │ │ │ │ 0002aab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002aac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002aae0: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -0002aaf0: 203a 2049 6465 616c 206f 6620 5220 2020 : Ideal of R │ │ │ │ +0002aad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ 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2020 2020 2020 2020 2020 207c 0a7c 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 |.+--------- │ │ │ │ +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 │ │ │ │ 0002abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 5365 Se │ │ │ │ +0002b680: 6772 654c 6973 7420 3d3e 207b 302c 2030 greList => {0, 0 │ │ │ │ +0002b690: 2c20 3468 2020 2b20 3468 2068 2020 2b20 , 4h + 4h h + │ │ │ │ +0002b6a0: 6820 2c20 2d20 2020 2020 2020 2020 2020 h , - │ │ │ │ +0002b6b0: 2020 2020 7c0a 7c20 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: 2032 2032 2020 2020 2020 3220 2020 2020 2 2 2 │ │ │ │ -0002b700: 2020 2020 2032 2020 2020 2032 2020 2020 2 2 │ │ │ │ -0002b710: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b730: 2020 2020 2053 6567 7265 203d 3e20 3732 Segre => 72 │ │ │ │ -0002b740: 6820 6820 202d 2032 3468 2068 2020 2d20 h h - 24h h - │ │ │ │ -0002b750: 3132 6820 6820 202b 2034 6820 2020 2020 12h h + 4h │ │ │ │ -0002b760: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 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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 |.|--------- │ │ │ │ +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 │ │ │ │ 0002b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b890: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b8b0: 2020 2020 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b8e0: 2020 2020 7c0a 7c20 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 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b940: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002b950: 2032 2020 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ -0002b960: 2020 3220 3220 2020 2020 2020 2020 2020 2 2 │ │ │ │ +0002b930: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b990: 2020 2020 2020 2020 2020 207c 0a7c 3234 |.|24 │ │ │ │ -0002b9a0: 6820 6820 202d 2031 3268 2068 202c 2037 h h - 12h h , 7 │ │ │ │ -0002b9b0: 3268 2068 207d 2020 2020 2020 2020 2020 2h h } │ │ │ │ +0002b980: 2020 2020 7c0a 7c20 2020 3220 2020 2020 |.| 2 │ │ │ │ +0002b990: 2020 2020 2032 2020 2020 2032 2032 2020 2 2 2 │ │ │ │ +0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002b9f0: 2031 2032 2020 2020 2020 3120 3220 2020 1 2 1 2 │ │ │ │ -0002ba00: 2020 3120 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ +0002b9d0: 2020 2020 7c0a 7c32 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7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002ba80: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002baa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bad0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002bae0: 2020 3120 3220 2020 2032 2020 2020 2020 1 2 2 │ │ │ │ +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 │ │ │ │ 0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb20: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002bb30: 2d2d 2d2d 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 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0002bb80: 3420 3a20 7331 3d3d 5365 6748 6173 6823 4 : s1==SegHash# │ │ │ │ -0002bb90: 2253 6567 7265 2220 2020 2020 2020 2020 "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 207c 0a7c 2020 |.| │ │ │ │ +0002bb10: 2020 2020 7c0a 7c20 2020 2031 2032 2020 |.| 1 2 │ │ │ │ +0002bb20: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002bb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bbb0: 2d2d 2d2d 2b0a 7c69 3134 203a 2073 313d ----+.|i14 : s1= │ │ │ │ +0002bbc0: 3d53 6567 4861 7368 2322 5365 6772 6522 =SegHash#"Segre" │ │ │ │ 0002bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc10: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002bc20: 3420 3d20 7472 7565 2020 2020 2020 2020 4 = true │ │ │ │ +0002bc00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 496e -----------+..In │ │ │ │ -0002bcc0: 2074 6865 2063 6173 6520 7768 6572 6520 the case where │ │ │ │ -0002bcd0: 7468 6520 616d 6269 656e 7420 7370 6163 the ambient spac │ │ │ │ -0002bce0: 6520 6973 2061 2074 6f72 6963 2076 6172 e is a toric var │ │ │ │ -0002bcf0: 6965 7479 2077 6869 6368 2069 7320 6e6f iety which is no │ │ │ │ -0002bd00: 7420 6120 7072 6f64 7563 740a 6f66 2070 t a product.of p │ │ │ │ -0002bd10: 726f 6a65 6374 6976 6520 7370 6163 6573 rojective spaces │ │ │ │ -0002bd20: 2077 6520 6d75 7374 206c 6f61 6420 7468 we must load th │ │ │ │ -0002bd30: 6520 4e6f 726d 616c 546f 7269 6356 6172 e NormalToricVar │ │ │ │ -0002bd40: 6965 7469 6573 2070 6163 6b61 6765 2061 ieties package a │ │ │ │ -0002bd50: 6e64 206d 7573 740a 616c 736f 2069 6e70 nd must.also inp │ │ │ │ -0002bd60: 7574 2074 6865 2074 6f72 6963 2076 6172 ut the toric var │ │ │ │ -0002bd70: 6965 7479 2e20 4966 2074 6865 2074 6f72 iety. If the tor │ │ │ │ -0002bd80: 6963 2076 6172 6965 7479 2069 7320 6120 ic variety is a │ │ │ │ -0002bd90: 7072 6f64 7563 7420 6f66 2070 726f 6a65 product of proje │ │ │ │ -0002bda0: 6374 6976 650a 7370 6163 6520 6974 2069 ctive.space it i │ │ │ │ -0002bdb0: 7320 7265 636f 6d6d 656e 6465 6420 746f s recommended to │ │ │ │ -0002bdc0: 2075 7365 2074 6865 2066 6f72 6d20 6162 use the form ab │ │ │ │ -0002bdd0: 6f76 6520 7261 7468 6572 2074 6861 6e20 ove rather than │ │ │ │ -0002bde0: 696e 7075 7474 696e 6720 7468 6520 746f inputting the to │ │ │ │ -0002bdf0: 7269 630a 7661 7269 6574 7920 666f 7220 ric.variety for │ │ │ │ -0002be00: 6566 6669 6369 656e 6379 2072 6561 736f efficiency reaso │ │ │ │ -0002be10: 6e73 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ns...+---------- │ │ │ │ -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 2b0a --------------+. │ │ │ │ -0002be60: 7c69 3135 203a 206e 6565 6473 5061 636b |i15 : needsPack │ │ │ │ -0002be70: 6167 6520 224e 6f72 6d61 6c54 6f72 6963 age "NormalToric │ │ │ │ -0002be80: 5661 7269 6574 6965 7322 2020 2020 2020 Varieties" │ │ │ │ -0002be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bea0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0002beb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc50: 2020 2020 7c0a 7c6f 3134 203d 2074 7275 |.|o14 = tru │ │ │ │ +0002bc60: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bca0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002bcf0: 2d2d 2d2d 2b0a 0a49 6e20 7468 6520 6361 ----+..In the ca │ │ │ │ +0002bd00: 7365 2077 6865 7265 2074 6865 2061 6d62 se where the amb │ │ │ │ +0002bd10: 6965 6e74 2073 7061 6365 2069 7320 6120 ient space is a │ │ │ │ +0002bd20: 746f 7269 6320 7661 7269 6574 7920 7768 toric variety wh │ │ │ │ +0002bd30: 6963 6820 6973 206e 6f74 2061 2070 726f ich is not a pro │ │ │ │ +0002bd40: 6475 6374 0a6f 6620 7072 6f6a 6563 7469 duct.of projecti │ │ │ │ +0002bd50: 7665 2073 7061 6365 7320 7765 206d 7573 ve spaces we mus │ │ │ │ +0002bd60: 7420 6c6f 6164 2074 6865 204e 6f72 6d61 t load the Norma │ │ │ │ +0002bd70: 6c54 6f72 6963 5661 7269 6574 6965 7320 lToricVarieties │ │ │ │ +0002bd80: 7061 636b 6167 6520 616e 6420 6d75 7374 package and must │ │ │ │ +0002bd90: 0a61 6c73 6f20 696e 7075 7420 7468 6520 .also input the │ │ │ │ +0002bda0: 746f 7269 6320 7661 7269 6574 792e 2049 toric variety. I │ │ │ │ +0002bdb0: 6620 7468 6520 746f 7269 6320 7661 7269 f the toric vari │ │ │ │ +0002bdc0: 6574 7920 6973 2061 2070 726f 6475 6374 ety is a product │ │ │ │ +0002bdd0: 206f 6620 7072 6f6a 6563 7469 7665 0a73 of projective.s │ │ │ │ +0002bde0: 7061 6365 2069 7420 6973 2072 6563 6f6d pace it is recom │ │ │ │ +0002bdf0: 6d65 6e64 6564 2074 6f20 7573 6520 7468 mended to use th │ │ │ │ +0002be00: 6520 666f 726d 2061 626f 7665 2072 6174 e form above rat │ │ │ │ +0002be10: 6865 7220 7468 616e 2069 6e70 7574 7469 her than inputti │ │ │ │ +0002be20: 6e67 2074 6865 2074 6f72 6963 0a76 6172 ng the toric.var │ │ │ │ +0002be30: 6965 7479 2066 6f72 2065 6666 6963 6965 iety for efficie │ │ │ │ +0002be40: 6e63 7920 7265 6173 6f6e 732e 0a0a 2b2d ncy reasons...+- │ │ │ │ +0002be50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002be90: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ +0002bea0: 6e65 6564 7350 6163 6b61 6765 2022 4e6f needsPackage "No │ │ │ │ +0002beb0: 726d 616c 546f 7269 6356 6172 6965 7469 rmalToricVarieti │ │ │ │ +0002bec0: 6573 2220 2020 2020 2020 2020 2020 2020 es" │ │ │ │ 0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bef0: 2020 2020 7c0a 7c6f 3135 203d 204e 6f72 |.|o15 = Nor │ │ │ │ -0002bf00: 6d61 6c54 6f72 6963 5661 7269 6574 6965 malToricVarietie │ │ │ │ -0002bf10: 7320 2020 2020 2020 2020 2020 2020 2020 s │ │ │ │ -0002bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002bf40: 0a7c 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 │ │ │ │ 0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf80: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -0002bf90: 203a 2050 6163 6b61 6765 2020 2020 2020 : Package │ │ │ │ +0002bf70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0002bfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c020: 2b0a 7c69 3136 203a 2052 686f 203d 207b +.|i16 : Rho = { │ │ │ │ -0002c030: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c {1,0,0},{0,1,0}, │ │ │ │ -0002c040: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30 {0,0,1},{-1,-1,0 │ │ │ │ -0002c050: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020 },{0,0,-1}} │ │ │ │ -0002c060: 2020 2020 2020 2020 2020 207c 0a7c 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 2020 7c0a 7c6f 3136 203d 207b |.|o16 = { │ │ │ │ -0002c0c0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031 {1, 0, 0}, {0, 1 │ │ │ │ -0002c0d0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c , 0}, {0, 0, 1}, │ │ │ │ -0002c0e0: 207b 2d31 2c20 2d31 2c20 307d 2c20 7b30 {-1, -1, 0}, {0 │ │ │ │ -0002c0f0: 2c20 302c 202d 317d 7d20 2020 2020 2020 , 0, -1}} │ │ │ │ -0002c100: 207c 0a7c 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 7c0a 7c6f |.|o │ │ │ │ -0002c150: 3136 203a 204c 6973 7420 2020 2020 2020 16 : List │ │ │ │ +0002bfc0: 2020 207c 0a7c 6f31 3520 3a20 5061 636b |.|o15 : Pack │ │ │ │ +0002bfd0: 6167 6520 2020 2020 2020 2020 2020 2020 age │ │ │ │ +0002bfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c000: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c010: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002c020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c050: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 ---------+.|i16 │ │ │ │ +0002c060: 3a20 5268 6f20 3d20 7b7b 312c 302c 307d : Rho = {{1,0,0} │ │ │ │ +0002c070: 2c7b 302c 312c 307d 2c7b 302c 302c 317d ,{0,1,0},{0,0,1} │ │ │ │ +0002c080: 2c7b 2d31 2c2d 312c 307d 2c7b 302c 302c ,{-1,-1,0},{0,0, │ │ │ │ +0002c090: 2d31 7d7d 2020 2020 2020 2020 2020 2020 -1}} │ │ │ │ +0002c0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002c0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c0e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002c0f0: 0a7c 6f31 3620 3d20 7b7b 312c 2030 2c20 .|o16 = {{1, 0, │ │ │ │ +0002c100: 307d 2c20 7b30 2c20 312c 2030 7d2c 207b 0}, {0, 1, 0}, { │ │ │ │ +0002c110: 302c 2030 2c20 317d 2c20 7b2d 312c 202d 0, 0, 1}, {-1, - │ │ │ │ +0002c120: 312c 2030 7d2c 207b 302c 2030 2c20 2d31 1, 0}, {0, 0, -1 │ │ │ │ +0002c130: 7d7d 2020 2020 2020 2020 7c0a 7c20 2020 }} |.| │ │ │ │ +0002c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c190: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002c1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c1e0: 2d2d 2b0a 7c69 3137 203a 2053 6967 6d61 --+.|i17 : Sigma │ │ │ │ -0002c1f0: 203d 207b 7b30 2c31 2c32 7d2c 7b31 2c32 = {{0,1,2},{1,2 │ │ │ │ -0002c200: 2c33 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 ,3},{0,2,3},{0,1 │ │ │ │ -0002c210: 2c34 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 ,4},{1,3,4},{0,3 │ │ │ │ -0002c220: 2c34 7d7d 2020 2020 2020 2020 207c 0a7c ,4}} |.| │ │ │ │ -0002c230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c270: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -0002c280: 207b 7b30 2c20 312c 2032 7d2c 207b 312c {{0, 1, 2}, {1, │ │ │ │ -0002c290: 2032 2c20 337d 2c20 7b30 2c20 322c 2033 2, 3}, {0, 2, 3 │ │ │ │ -0002c2a0: 7d2c 207b 302c 2031 2c20 347d 2c20 7b31 }, {0, 1, 4}, {1 │ │ │ │ -0002c2b0: 2c20 332c 2034 7d2c 207b 302c 2033 2c20 , 3, 4}, {0, 3, │ │ │ │ -0002c2c0: 347d 7d7c 0a7c 2020 2020 2020 2020 2020 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 7c0a |. │ │ │ │ -0002c310: 7c6f 3137 203a 204c 6973 7420 2020 2020 |o17 : List │ │ │ │ +0002c180: 2020 2020 207c 0a7c 6f31 3620 3a20 4c69 |.|o16 : Li │ │ │ │ +0002c190: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ +0002c1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c1d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0002c220: 3720 3a20 5369 676d 6120 3d20 7b7b 302c 7 : Sigma = {{0, │ │ │ │ +0002c230: 312c 327d 2c7b 312c 322c 337d 2c7b 302c 1,2},{1,2,3},{0, │ │ │ │ +0002c240: 322c 337d 2c7b 302c 312c 347d 2c7b 312c 2,3},{0,1,4},{1, │ │ │ │ +0002c250: 332c 347d 2c7b 302c 332c 347d 7d20 2020 3,4},{0,3,4}} │ │ │ │ +0002c260: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002c270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c2b0: 207c 0a7c 6f31 3720 3d20 7b7b 302c 2031 |.|o17 = {{0, 1 │ │ │ │ +0002c2c0: 2c20 327d 2c20 7b31 2c20 322c 2033 7d2c , 2}, {1, 2, 3}, │ │ │ │ +0002c2d0: 207b 302c 2032 2c20 337d 2c20 7b30 2c20 {0, 2, 3}, {0, │ │ │ │ +0002c2e0: 312c 2034 7d2c 207b 312c 2033 2c20 347d 1, 4}, {1, 3, 4} │ │ │ │ +0002c2f0: 2c20 7b30 2c20 332c 2034 7d7d 7c0a 7c20 , {0, 3, 4}}|.| │ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c3a0: 2d2d 2d2d 2b0a 7c69 3138 203a 2058 203d ----+.|i18 : X = │ │ │ │ -0002c3b0: 206e 6f72 6d61 6c54 6f72 6963 5661 7269 normalToricVari │ │ │ │ -0002c3c0: 6574 7928 5268 6f2c 5369 676d 612c 436f ety(Rho,Sigma,Co │ │ │ │ -0002c3d0: 6566 6669 6369 656e 7452 696e 6720 3d3e efficientRing => │ │ │ │ -0002c3e0: 5a5a 2f33 3237 3439 2920 2020 2020 207c ZZ/32749) | │ │ │ │ -0002c3f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0002c400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c430: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -0002c440: 203d 2058 2020 2020 2020 2020 2020 2020 = X │ │ │ │ +0002c340: 2020 2020 2020 207c 0a7c 6f31 3720 3a20 |.|o17 : │ │ │ │ +0002c350: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +0002c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c390: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002c3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002c3e0: 6931 3820 3a20 5820 3d20 6e6f 726d 616c i18 : X = normal │ │ │ │ +0002c3f0: 546f 7269 6356 6172 6965 7479 2852 686f ToricVariety(Rho │ │ │ │ +0002c400: 2c53 6967 6d61 2c43 6f65 6666 6963 6965 ,Sigma,Coefficie │ │ │ │ +0002c410: 6e74 5269 6e67 203d 3e5a 5a2f 3332 3734 ntRing =>ZZ/3274 │ │ │ │ +0002c420: 3929 2020 2020 2020 7c0a 7c20 2020 2020 9) |.| │ │ │ │ +0002c430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c480: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002c470: 2020 207c 0a7c 6f31 3820 3d20 5820 2020 |.|o18 = X │ │ │ │ +0002c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c4d0: 7c0a 7c6f 3138 203a 204e 6f72 6d61 6c54 |.|o18 : NormalT │ │ │ │ -0002c4e0: 6f72 6963 5661 7269 6574 7920 2020 2020 oricVariety │ │ │ │ +0002c4b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c4c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c510: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -0002c520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c560: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2043 ------+.|i19 : C │ │ │ │ -0002c570: 6865 636b 546f 7269 6356 6172 6965 7479 heckToricVariety │ │ │ │ -0002c580: 5661 6c69 6428 5829 2020 2020 2020 2020 Valid(X) │ │ │ │ -0002c590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002c5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c500: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +0002c510: 3a20 4e6f 726d 616c 546f 7269 6356 6172 : NormalToricVar │ │ │ │ +0002c520: 6965 7479 2020 2020 2020 2020 2020 2020 iety │ │ │ │ +0002c530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002c5a0: 0a7c 6931 3920 3a20 4368 6563 6b54 6f72 .|i19 : CheckTor │ │ │ │ +0002c5b0: 6963 5661 7269 6574 7956 616c 6964 2858 icVarietyValid(X │ │ │ │ +0002c5c0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 0002c5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0002c600: 3139 203d 2074 7275 6520 2020 2020 2020 19 = true │ │ │ │ +0002c5e0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002c5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c640: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ -0002c650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c690: 2d2d 2b0a 7c69 3230 203a 2052 3d72 696e --+.|i20 : R=rin │ │ │ │ -0002c6a0: 6728 5829 2020 2020 2020 2020 2020 2020 g(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 207c 0a7c |.| │ │ │ │ +0002c630: 2020 2020 207c 0a7c 6f31 3920 3d20 7472 |.|o19 = tr │ │ │ │ +0002c640: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +0002c650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c680: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002c6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002c6d0: 3020 3a20 523d 7269 6e67 2858 2920 2020 0 : R=ring(X) │ │ │ │ 0002c6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c720: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ -0002c730: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0002c710: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002c720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c770: 2020 207c 0a7c 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 │ │ │ │ 0002c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c7b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c7c0: 7c6f 3230 203a 2050 6f6c 796e 6f6d 6961 |o20 : Polynomia │ │ │ │ -0002c7d0: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ +0002c7a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002c7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002c7c0: 2020 2020 2020 2020 2020 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2d2d 2d2d ---------------- │ │ │ │ -0002cd30: 2d2d 2d2d 2d2d 2020 2020 2020 2020 2020 ------ │ │ │ │ -0002cd40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002cd50: 0a7c 2020 2020 2020 2878 2078 202c 2078 .| (x x , x │ │ │ │ -0002cd60: 2078 2078 202c 2078 2020 2d20 7820 2c20 x x , x - x , │ │ │ │ -0002cd70: 7820 202d 2078 202c 2078 2020 2d20 7820 x - x , x - x │ │ │ │ -0002cd80: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0002cd90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002cda0: 2020 2020 2032 2034 2020 2030 2031 2033 2 4 0 1 3 │ │ │ │ -0002cdb0: 2020 2030 2020 2020 3320 2020 3120 2020 0 3 1 │ │ │ │ -0002cdc0: 2033 2020 2032 2020 2020 3420 2020 2020 3 2 4 │ │ │ │ -0002cdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cde0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ -0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce10: 2d2d 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│ │ │ │ +0002ce50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ce60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3320 ---------+.|i23 │ │ │ │ +0002ce70: 3a20 4368 3d54 6f72 6963 4368 6f77 5269 : Ch=ToricChowRi │ │ │ │ +0002ce80: 6e67 2858 2920 2020 2020 2020 2020 2020 ng(X) │ │ │ │ 0002ce90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ceb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cec0: 2020 2020 2020 7c0a 7c6f 3233 203d 2043 |.|o23 = C │ │ │ │ -0002ced0: 6820 2020 2020 2020 2020 2020 2020 2020 h │ │ │ │ +0002ceb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002cec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ced0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002cf10: 207c 0a7c 2020 2020 2020 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Note that │ │ │ │ +0002d390: 7468 6520 6265 7274 696e 6920 6f70 7469 the bertini opti │ │ │ │ +0002d3a0: 6f6e 2069 7320 6f6e 6c79 2061 7661 696c on is only avail │ │ │ │ +0002d3b0: 6162 6c65 2066 6f72 0a73 7562 7363 6865 able for.subsche │ │ │ │ +0002d3c0: 6d65 7320 6f66 205c 5050 5e6e 2e0a 0a4f mes of \PP^n...O │ │ │ │ +0002d3d0: 6273 6572 7665 2074 6861 7420 7468 6520 bserve that the │ │ │ │ +0002d3e0: 616c 676f 7269 7468 6d20 6973 2061 2070 algorithm is a p │ │ │ │ +0002d3f0: 726f 6261 6269 6c69 7374 6963 2061 6c67 robabilistic alg │ │ │ │ +0002d400: 6f72 6974 686d 2061 6e64 206d 6179 2067 orithm and may g │ │ │ │ +0002d410: 6976 6520 6120 7772 6f6e 670a 616e 7377 ive a wrong.answ │ │ │ │ +0002d420: 6572 2077 6974 6820 6120 736d 616c 6c20 er with a small │ │ │ │ +0002d430: 6275 7420 6e6f 6e7a 6572 6f20 7072 6f62 but nonzero prob │ │ │ │ +0002d440: 6162 696c 6974 792e 2052 6561 6420 6d6f ability. 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+0002d520: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +0002d530: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +0002d540: 5365 6772 653a 2053 6567 7265 2c20 6973 Segre: Segre, is │ │ │ │ +0002d550: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +0002d560: 6675 6e63 7469 6f6e 2077 6974 6820 6f70 function with op │ │ │ │ +0002d570: 7469 6f6e 733a 0a28 4d61 6361 756c 6179 tions:.(Macaulay │ │ │ │ +0002d580: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0002d590: 696f 6e57 6974 684f 7074 696f 6e73 2c2e ionWithOptions,. │ │ │ │ +0002d5a0: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0002d5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002d5f0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0002d600: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0002d610: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0002d620: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0002d630: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +0002d640: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0002d650: 7061 636b 6167 6573 2f0a 4368 6172 6163 packages/.Charac │ │ │ │ +0002d660: 7465 7269 7374 6963 436c 6173 7365 732e teristicClasses. │ │ │ │ +0002d670: 6d32 3a31 3736 333a 302e 0a1f 0a46 696c m2:1763:0....Fil │ │ │ │ +0002d680: 653a 2043 6861 7261 6374 6572 6973 7469 e: Characteristi │ │ │ │ +0002d690: 6343 6c61 7373 6573 2e69 6e66 6f2c 204e cClasses.info, N │ │ │ │ +0002d6a0: 6f64 653a 2054 6f72 6963 4368 6f77 5269 ode: ToricChowRi │ │ │ │ +0002d6b0: 6e67 2c20 5072 6576 3a20 5365 6772 652c ng, Prev: Segre, │ │ │ │ +0002d6c0: 2055 703a 2054 6f70 0a0a 546f 7269 6343 Up: Top..ToricC │ │ │ │ +0002d6d0: 686f 7752 696e 6720 2d2d 2043 6f6d 7075 howRing -- Compu │ │ │ │ +0002d6e0: 7465 7320 7468 6520 4368 6f77 2072 696e tes the Chow rin │ │ │ │ +0002d6f0: 6720 6f66 2061 206e 6f72 6d61 6c20 746f g of a normal to │ │ │ │ +0002d700: 7269 6320 7661 7269 6574 790a 2a2a 2a2a ric variety.**** │ │ │ │ +0002d710: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d720: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d730: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0002d740: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +0002d750: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +0002d760: 2020 2054 6f72 6963 4368 6f77 5269 6e67 ToricChowRing │ │ │ │ +0002d770: 2058 0a20 202a 2049 6e70 7574 733a 0a20 X. * Inputs:. │ │ │ │ +0002d780: 2020 2020 202a 2052 2c20 6120 2a6e 6f74 * R, a *not │ │ │ │ +0002d790: 6520 6e6f 726d 616c 2074 6f72 6963 2076 e normal toric v │ │ │ │ +0002d7a0: 6172 6965 7479 3a0a 2020 2020 2020 2020 ariety:. │ │ │ │ +0002d7b0: 284e 6f72 6d61 6c54 6f72 6963 5661 7269 (NormalToricVari │ │ │ │ +0002d7c0: 6574 6965 7329 4e6f 726d 616c 546f 7269 eties)NormalTori │ │ │ │ +0002d7d0: 6356 6172 6965 7479 2c2c 2041 206e 6f72 cVariety,, A nor │ │ │ │ +0002d7e0: 6d61 6c20 746f 7269 6320 7661 7269 6574 mal toric variet │ │ │ │ +0002d7f0: 790a 2020 2a20 4f75 7470 7574 733a 0a20 y. * Outputs:. │ │ │ │ +0002d800: 2020 2020 202a 2061 202a 6e6f 7465 2071 * a *note q │ │ │ │ +0002d810: 756f 7469 656e 7420 7269 6e67 3a20 284d uotient ring: (M │ │ │ │ +0002d820: 6163 6175 6c61 7932 446f 6329 5175 6f74 acaulay2Doc)Quot │ │ │ │ +0002d830: 6965 6e74 5269 6e67 2c2c 200a 0a44 6573 ientRing,, ..Des │ │ │ │ +0002d840: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0002d850: 3d3d 3d3d 0a0a 4c65 7420 5820 6265 2061 ====..Let X be a │ │ │ │ +0002d860: 2074 6f72 6963 2076 6172 6965 7479 2077 toric variety w │ │ │ │ +0002d870: 6974 6820 746f 7461 6c20 636f 6f72 6469 ith total coordi │ │ │ │ +0002d880: 6e61 7465 2072 696e 6720 2843 6f78 2072 nate ring (Cox r │ │ │ │ +0002d890: 696e 6729 2052 2e20 5468 6973 206d 6574 ing) R. This met │ │ │ │ +0002d8a0: 686f 640a 636f 6d70 7574 6573 2074 6865 hod.computes the │ │ │ │ +0002d8b0: 2043 686f 7720 7269 6e67 2020 4368 6f77 Chow ring Chow │ │ │ │ +0002d8c0: 2072 696e 6720 4368 3d52 2f28 5352 2b4c ring Ch=R/(SR+L │ │ │ │ +0002d8d0: 5229 206f 6620 583b 2068 6572 6520 5352 R) of X; here SR │ │ │ │ +0002d8e0: 2069 7320 7468 650a 5374 616e 6c65 792d is the.Stanley- │ │ │ │ +0002d8f0: 5265 6973 6e65 7220 6964 6561 6c20 6f66 Reisner ideal of │ │ │ │ +0002d900: 2074 6865 2063 6f72 7265 7370 6f6e 6469 the correspondi │ │ │ │ +0002d910: 6e67 2066 616e 2061 6e64 204c 5220 6973 ng fan and LR is │ │ │ │ +0002d920: 2074 6865 2069 6465 616c 206f 6620 6c69 the ideal of li │ │ │ │ +0002d930: 6e65 6172 0a72 656c 6174 696f 6e73 2061 near.relations a │ │ │ │ +0002d940: 6d6f 756e 7420 7468 6520 7261 7973 2e20 mount the rays. │ │ │ │ +0002d950: 4974 2069 7320 6e65 6564 6564 2066 6f72 It is needed for │ │ │ │ +0002d960: 2069 6e70 7574 2069 6e74 6f20 7468 6520 input into the │ │ │ │ +0002d970: 6d65 7468 6f64 7320 2a6e 6f74 6520 5365 methods *note Se │ │ │ │ +0002d980: 6772 653a 0a53 6567 7265 2c2c 202a 6e6f gre:.Segre,, *no │ │ │ │ +0002d990: 7465 2043 6865 726e 3a20 4368 6572 6e2c te Chern: Chern, │ │ │ │ +0002d9a0: 2061 6e64 202a 6e6f 7465 2043 534d 3a20 and *note CSM: │ │ │ │ +0002d9b0: 4353 4d2c 2069 6e20 7468 6520 6361 7365 CSM, in the case │ │ │ │ +0002d9c0: 7320 7768 6572 6520 6120 746f 7269 630a s where a toric. │ │ │ │ +0002d9d0: 7661 7269 6574 7920 6973 2061 6c73 6f20 variety is also │ │ │ │ +0002d9e0: 696e 7075 7420 746f 2065 6e73 7572 6520 input to ensure │ │ │ │ +0002d9f0: 7468 6174 2074 6865 7365 206d 6574 686f that these metho │ │ │ │ +0002da00: 6473 2072 6574 7572 6e20 7265 7375 6c74 ds return result │ │ │ │ +0002da10: 7320 696e 2074 6865 2073 616d 650a 7269 s in the same.ri │ │ │ │ +0002da20: 6e67 2e20 5765 2067 6976 6520 616e 2065 ng. We give an e │ │ │ │ +0002da30: 7861 6d70 6c65 206f 6620 7468 6520 7573 xample of the us │ │ │ │ +0002da40: 6520 6f66 2074 6869 7320 6d65 7468 6f64 e of this method │ │ │ │ +0002da50: 2074 6f20 776f 726b 2077 6974 6820 656c to work with el │ │ │ │ +0002da60: 656d 656e 7473 206f 6620 7468 650a 4368 ements of the.Ch │ │ │ │ +0002da70: 6f77 2072 696e 6720 6f66 2061 2074 6f72 ow ring of a tor │ │ │ │ +0002da80: 6963 2076 6172 6965 7479 0a0a 2b2d 2d2d ic variety..+--- │ │ │ │ 0002da90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002daa0: 2d2b 0a7c 6931 203a 206e 6565 6473 5061 -+.|i1 : needsPa │ │ │ │ -0002dab0: 636b 6167 6520 224e 6f72 6d61 6c54 6f72 ckage "NormalTor │ │ │ │ -0002dac0: 6963 5661 7269 6574 6965 7322 2020 2020 icVarieties" │ │ │ │ -0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002daf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002daa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +0002dae0: 3a20 6e65 6564 7350 6163 6b61 6765 2022 : needsPackage " │ │ │ │ +0002daf0: 4e6f 726d 616c 546f 7269 6356 6172 6965 NormalToricVarie │ │ │ │ +0002db00: 7469 6573 2220 2020 2020 2020 2020 2020 ties" │ │ │ │ 0002db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002db30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db40: 207c 0a7c 6f31 203d 204e 6f72 6d61 6c54 |.|o1 = NormalT │ │ │ │ -0002db50: 6f72 6963 5661 7269 6574 6965 7320 2020 oricVarieties │ │ │ │ +0002db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002db70: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0002db80: 3d20 4e6f 726d 616c 546f 7269 6356 6172 = NormalToricVar │ │ │ │ +0002db90: 6965 7469 6573 2020 2020 2020 2020 2020 ieties │ │ │ │ 0002dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dbc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbe0: 207c 0a7c 6f31 203a 2050 6163 6b61 6765 |.|o1 : Package │ │ │ │ +0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc30: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002dc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +0002dc20: 3a20 5061 636b 6167 6520 2020 2020 2020 : Package │ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc60: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002dc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dc80: 2d2b 0a7c 6932 203a 2052 686f 203d 207b -+.|i2 : Rho = { │ │ │ │ -0002dc90: 7b31 2c30 2c30 7d2c 7b30 2c31 2c30 7d2c {1,0,0},{0,1,0}, │ │ │ │ -0002dca0: 7b30 2c30 2c31 7d2c 7b2d 312c 2d31 2c30 {0,0,1},{-1,-1,0 │ │ │ │ -0002dcb0: 7d2c 7b30 2c30 2c2d 317d 7d20 2020 2020 },{0,0,-1}} │ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcd0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002dce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +0002dcc0: 3a20 5268 6f20 3d20 7b7b 312c 302c 307d : Rho = {{1,0,0} │ │ │ │ +0002dcd0: 2c7b 302c 312c 307d 2c7b 302c 302c 317d ,{0,1,0},{0,0,1} │ │ │ │ +0002dce0: 2c7b 2d31 2c2d 312c 307d 2c7b 302c 302c ,{-1,-1,0},{0,0, │ │ │ │ +0002dcf0: 2d31 7d7d 2020 2020 2020 2020 2020 2020 -1}} │ │ │ │ +0002dd00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd20: 207c 0a7c 6f32 203d 207b 7b31 2c20 302c |.|o2 = {{1, 0, │ │ │ │ -0002dd30: 2030 7d2c 207b 302c 2031 2c20 307d 2c20 0}, {0, 1, 0}, │ │ │ │ -0002dd40: 7b30 2c20 302c 2031 7d2c 207b 2d31 2c20 {0, 0, 1}, {-1, │ │ │ │ -0002dd50: 2d31 2c20 307d 2c20 7b30 2c20 302c 202d -1, 0}, {0, 0, - │ │ │ │ -0002dd60: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ -0002dd70: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dd50: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0002dd60: 3d20 7b7b 312c 2030 2c20 307d 2c20 7b30 = {{1, 0, 0}, {0 │ │ │ │ +0002dd70: 2c20 312c 2030 7d2c 207b 302c 2030 2c20 , 1, 0}, {0, 0, │ │ │ │ +0002dd80: 317d 2c20 7b2d 312c 202d 312c 2030 7d2c 1}, {-1, -1, 0}, │ │ │ │ +0002dd90: 207b 302c 2030 2c20 2d31 7d7d 2020 2020 {0, 0, -1}} │ │ │ │ +0002dda0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddc0: 207c 0a7c 6f32 203a 204c 6973 7420 2020 |.|o2 : List │ │ │ │ +0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de10: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0002de00: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +0002de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de40: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002de50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de60: 2d2b 0a7c 6933 203a 2053 6967 6d61 203d -+.|i3 : Sigma = │ │ │ │ -0002de70: 207b 7b30 2c31 2c32 7d2c 7b31 2c32 2c33 {{0,1,2},{1,2,3 │ │ │ │ -0002de80: 7d2c 7b30 2c32 2c33 7d2c 7b30 2c31 2c34 },{0,2,3},{0,1,4 │ │ │ │ -0002de90: 7d2c 7b31 2c33 2c34 7d2c 7b30 2c33 2c34 },{1,3,4},{0,3,4 │ │ │ │ -0002dea0: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ -0002deb0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ +0002dea0: 3a20 5369 676d 6120 3d20 7b7b 302c 312c : Sigma = {{0,1, │ │ │ │ +0002deb0: 327d 2c7b 312c 322c 337d 2c7b 302c 322c 2},{1,2,3},{0,2, │ │ │ │ +0002dec0: 337d 2c7b 302c 312c 347d 2c7b 312c 332c 3},{0,1,4},{1,3, │ │ │ │ +0002ded0: 347d 2c7b 302c 332c 347d 7d20 2020 2020 4},{0,3,4}} │ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df00: 207c 0a7c 6f33 203d 207b 7b30 2c20 312c |.|o3 = {{0, 1, │ │ │ │ -0002df10: 2032 7d2c 207b 312c 2032 2c20 337d 2c20 2}, {1, 2, 3}, │ │ │ │ -0002df20: 7b30 2c20 322c 2033 7d2c 207b 302c 2031 {0, 2, 3}, {0, 1 │ │ │ │ -0002df30: 2c20 347d 2c20 7b31 2c20 332c 2034 7d2c , 4}, {1, 3, 4}, │ │ │ │ -0002df40: 207b 302c 2033 2c20 347d 7d20 2020 2020 {0, 3, 4}} │ │ │ │ -0002df50: 207c 0a7c 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 │ │ │ │ +0002df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002df30: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0002df40: 3d20 7b7b 302c 2031 2c20 327d 2c20 7b31 = {{0, 1, 2}, {1 │ │ │ │ +0002df50: 2c20 322c 2033 7d2c 207b 302c 2032 2c20 , 2, 3}, {0, 2, │ │ │ │ +0002df60: 337d 2c20 7b30 2c20 312c 2034 7d2c 207b 3}, {0, 1, 4}, { │ │ │ │ +0002df70: 312c 2033 2c20 347d 2c20 7b30 2c20 332c 1, 3, 4}, {0, 3, │ │ │ │ +0002df80: 2034 7d7d 2020 2020 2020 7c0a 7c20 2020 4}} |.| │ │ │ │ 0002df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfa0: 207c 0a7c 6f33 203a 204c 6973 7420 2020 |.|o3 : List │ │ │ │ +0002dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dff0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002e000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002dfd0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0002dfe0: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e020: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002e030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e040: 2d2b 0a7c 6934 203a 2058 203d 206e 6f72 -+.|i4 : X = nor │ │ │ │ -0002e050: 6d61 6c54 6f72 6963 5661 7269 6574 7928 malToricVariety( │ │ │ │ -0002e060: 5268 6f2c 5369 676d 612c 436f 6566 6669 Rho,Sigma,Coeffi │ │ │ │ -0002e070: 6369 656e 7452 696e 6720 3d3e 5a5a 2f33 cientRing =>ZZ/3 │ │ │ │ -0002e080: 3237 3439 2920 2020 2020 2020 2020 2020 2749) │ │ │ │ -0002e090: 207c 0a7c 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 │ │ │ │ +0002e040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +0002e080: 3a20 5820 3d20 6e6f 726d 616c 546f 7269 : X = normalTori │ │ │ │ +0002e090: 6356 6172 6965 7479 2852 686f 2c53 6967 cVariety(Rho,Sig │ │ │ │ +0002e0a0: 6d61 2c43 6f65 6666 6963 6965 6e74 5269 ma,CoefficientRi │ │ │ │ +0002e0b0: 6e67 203d 3e5a 5a2f 3332 3734 3929 2020 ng =>ZZ/32749) │ │ │ │ +0002e0c0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0e0: 207c 0a7c 6f34 203d 2058 2020 2020 2020 |.|o4 = X │ │ │ │ +0002e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e130: 207c 0a7c 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 │ │ │ │ 0002e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e160: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e180: 207c 0a7c 6f34 203a 204e 6f72 6d61 6c54 |.|o4 : NormalT │ │ │ │ -0002e190: 6f72 6963 5661 7269 6574 7920 2020 2020 oricVariety │ │ │ │ +0002e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e1d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +0002e1c0: 3a20 4e6f 726d 616c 546f 7269 6356 6172 : NormalToricVar │ │ │ │ +0002e1d0: 6965 7479 2020 2020 2020 2020 2020 2020 iety │ │ │ │ +0002e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e200: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002e210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e220: 2d2b 0a7c 6935 203a 2052 3d72 696e 6720 -+.|i5 : R=ring │ │ │ │ -0002e230: 5820 2020 2020 2020 2020 2020 2020 2020 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ +0002e260: 3a20 523d 7269 6e67 2058 2020 2020 2020 : R=ring X │ │ │ │ +0002e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2c0: 207c 0a7c 6f35 203d 2052 2020 2020 2020 |.|o5 = R │ │ │ │ +0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e310: 207c 0a7c 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 │ │ │ │ 0002e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e340: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e360: 207c 0a7c 6f35 203a 2050 6f6c 796e 6f6d |.|o5 : Polynom │ │ │ │ -0002e370: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +0002e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e390: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ +0002e3a0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e3e0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e400: 2d2b 0a7c 6936 203a 2043 683d 546f 7269 -+.|i6 : Ch=Tori │ │ │ │ -0002e410: 6343 686f 7752 696e 6728 5829 2020 2020 cChowRing(X) │ │ │ │ -0002e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e450: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ +0002e440: 3a20 4368 3d54 6f72 6963 4368 6f77 5269 : Ch=ToricChowRi │ │ │ │ +0002e450: 6e67 2858 2920 2020 2020 2020 2020 2020 ng(X) │ │ │ │ 0002e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e480: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4a0: 207c 0a7c 6f36 203d 2043 6820 2020 2020 |.|o6 = Ch │ │ │ │ +0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4f0: 207c 0a7c 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 │ │ │ │ 0002e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e540: 207c 0a7c 6f36 203a 2051 756f 7469 656e |.|o6 : Quotien │ │ │ │ -0002e550: 7452 696e 6720 2020 2020 2020 2020 2020 tRing │ │ │ │ +0002e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e590: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002e5a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e570: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ +0002e580: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0002e590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002e5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e5e0: 2d2b 0a7c 6937 203a 2064 6573 6372 6962 -+.|i7 : describ │ │ │ │ -0002e5f0: 6520 4368 2020 2020 2020 2020 2020 2020 e 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002e610: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ +0002e620: 3a20 6465 7363 7269 6265 2043 6820 2020 : describe Ch │ │ │ │ +0002e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e660: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e680: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002e690: 2020 2020 2020 2020 205a 5a5b 7820 2e2e ZZ[x .. │ │ │ │ -0002e6a0: 7820 5d20 2020 2020 2020 2020 2020 2020 x ] │ │ │ │ -0002e6b0: 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 |.| │ │ │ │ 0002e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e6d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002e6e0: 2020 2020 2020 2020 2020 2020 2030 2020 0 │ │ │ │ -0002e6f0: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e6d0: 2020 5a5a 5b78 202e 2e78 205d 2020 2020 ZZ[x ..x ] │ │ │ │ +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 |.| │ │ │ │ 0002e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e720: 207c 0a7c 6f37 203d 202d 2d2d 2d2d 2d2d |.|o7 = ------- │ │ │ │ -0002e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e750: 2d2d 2020 2020 2020 2020 2020 2020 2020 -- │ │ │ │ -0002e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e770: 207c 0a7c 2020 2020 2028 7820 7820 2c20 |.| (x x , │ │ │ │ -0002e780: 7820 7820 7820 2c20 7820 202d 2078 202c x x x , x - x , │ │ │ │ -0002e790: 2078 2020 2d20 7820 2c20 7820 202d 2078 x - x , x - x │ │ │ │ -0002e7a0: 2029 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e7c0: 207c 0a7c 2020 2020 2020 2032 2034 2020 |.| 2 4 │ │ │ │ -0002e7d0: 2030 2031 2033 2020 2030 2020 2020 3320 0 1 3 0 3 │ │ │ │ -0002e7e0: 2020 3120 2020 2033 2020 2032 2020 2020 1 3 2 │ │ │ │ -0002e7f0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e810: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002e820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 |.+--- │ │ │ │ 0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e860: 2d2b 0a7c 6938 203a 2072 3d67 656e 7320 -+.|i8 : r=gens │ │ │ │ -0002e870: 5220 2020 2020 2020 2020 2020 2020 2020 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 │ │ │ │ 0002e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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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 2020 2020 2020 │ │ │ │ -0002e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9a0: 207c 0a7c 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 │ │ │ │ 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 │ │ │ │ +0002e9d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e9f0: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ +0002e9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ea00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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│ │ -0002eaa0: 2872 616e 646f 6d28 7b31 2c30 7d2c 5229 (random({1,0},R) │ │ │ │ -0002eab0: 2920 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ +0002ead0: 3a20 493d 6964 6561 6c28 7261 6e64 6f6d : I=ideal(random │ │ │ │ +0002eae0: 287b 312c 307d 2c52 2929 2020 2020 2020 ({1,0},R)) │ │ │ │ 0002eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002eb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eb10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb30: 207c 0a7c 6f39 203d 2069 6465 616c 2831 |.|o9 = ideal(1 │ │ │ │ -0002eb40: 3037 7820 202b 2034 3337 3678 2020 2d20 07x + 4376x - │ │ │ │ -0002eb50: 3633 3136 7820 2920 2020 2020 2020 2020 6316x ) │ │ │ │ -0002eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eb80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002eb90: 2020 2030 2020 2020 2020 2020 3120 2020 0 1 │ │ │ │ -0002eba0: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ -0002ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ebd0: 207c 0a7c 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 │ │ │ │ 0002ebe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ebf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec00: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec20: 207c 0a7c 6f39 203a 2049 6465 616c 206f |.|o9 : Ideal o │ │ │ │ -0002ec30: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ +0002ec20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ec40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ec70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002ec80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ec90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ec50: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +0002ec60: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +0002ec70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ec90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002eca0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002ecb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ecc0: 2d2b 0a7c 6931 3020 3a20 4b3d 6964 6561 -+.|i10 : K=idea │ │ │ │ -0002ecd0: 6c28 7261 6e64 6f6d 287b 312c 317d 2c52 l(random({1,1},R │ │ │ │ -0002ece0: 2929 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 ----------+.|i10 │ │ │ │ +0002ed00: 203a 204b 3d69 6465 616c 2872 616e 646f : K=ideal(rando │ │ │ │ +0002ed10: 6d28 7b31 2c31 7d2c 5229 2920 2020 2020 m({1,1},R)) │ │ │ │ 0002ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ed30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ed60: 207c 0a7c 6f31 3020 3d20 6964 6561 6c28 |.|o10 = ideal( │ │ │ │ -0002ed70: 3331 3837 7820 7820 202d 2036 3035 3378 3187x x - 6053x │ │ │ │ -0002ed80: 2078 2020 2d20 3136 3039 3078 2078 2020 x - 16090x x │ │ │ │ -0002ed90: 2b20 3337 3833 7820 7820 202b 2038 3537 + 3783x x + 857 │ │ │ │ -0002eda0: 3078 2078 2020 2b20 3834 3434 7820 7820 0x x + 8444x x │ │ │ │ -0002edb0: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ -0002edc0: 2020 2020 2030 2032 2020 2020 2020 2020 0 2 │ │ │ │ -0002edd0: 3120 3220 2020 2020 2020 2020 3220 3320 1 2 2 3 │ │ │ │ -0002ede0: 2020 2020 2020 2030 2034 2020 2020 2020 0 4 │ │ │ │ -0002edf0: 2020 3120 3420 2020 2020 2020 2033 2034 1 4 3 4 │ │ │ │ -0002ee00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ed90: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ +0002eda0: 203d 2069 6465 616c 2833 3138 3778 2078 = ideal(3187x x │ │ │ │ +0002edb0: 2020 2d20 3630 3533 7820 7820 202d 2031 - 6053x x - 1 │ │ │ │ +0002edc0: 3630 3930 7820 7820 202b 2033 3738 3378 6090x x + 3783x │ │ │ │ +0002edd0: 2078 2020 2b20 3835 3730 7820 7820 202b x + 8570x x + │ │ │ │ +0002ede0: 2038 3434 3478 2078 2029 7c0a 7c20 2020 8444x x )|.| │ │ │ │ +0002edf0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0002ee00: 3220 2020 2020 2020 2031 2032 2020 2020 2 1 2 │ │ │ │ +0002ee10: 2020 2020 2032 2033 2020 2020 2020 2020 2 3 │ │ │ │ +0002ee20: 3020 3420 2020 2020 2020 2031 2034 2020 0 4 1 4 │ │ │ │ +0002ee30: 2020 2020 2020 3320 3420 7c0a 7c20 2020 3 4 |.| │ │ │ │ 0002ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee50: 207c 0a7c 6f31 3020 3a20 4964 6561 6c20 |.|o10 : Ideal │ │ │ │ -0002ee60: 6f66 2052 2020 2020 2020 2020 2020 2020 of R │ │ │ │ +0002ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ee60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ee90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002eea0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 |.+--- │ │ │ │ 0002eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eef0: 2d2b 0a7c 6931 3120 3a20 633d 4368 6572 -+.|i11 : c=Cher │ │ │ │ -0002ef00: 6e28 4368 2c58 2c49 2920 2020 2020 2020 n(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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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) │ │ │ │ 0002ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ef70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ef90: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ -0002efa0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +0002ef90: 2020 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: 207c 0a7c 6f31 3120 3d20 3478 2078 2020 |.|o11 = 4x x │ │ │ │ -0002eff0: 2b20 3278 2020 2b20 3278 2078 2020 2b20 + 2x + 2x x + │ │ │ │ -0002f000: 7820 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ -0002f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f030: 207c 0a7c 2020 2020 2020 2020 3320 3420 |.| 3 4 │ │ │ │ -0002f040: 2020 2020 3320 2020 2020 3320 3420 2020 3 3 4 │ │ │ │ -0002f050: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0002f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f080: 207c 0a7c 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 │ │ │ │ 0002f090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f0d0: 207c 0a7c 6f31 3120 3a20 4368 2020 2020 |.|o11 : Ch │ │ │ │ +0002f0d0: 2020 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: 207c 0a2b 2d2d 2d2d 2d2d 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 ---------------- │ │ │ │ +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 |.+--- │ │ │ │ 0002f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f170: 2d2b 0a7c 6931 3220 3a20 733d 5365 6772 -+.|i12 : s=Segr │ │ │ │ -0002f180: 6528 4368 2c58 2c4b 2920 2020 2020 2020 e(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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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) │ │ │ │ 0002f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f1f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f210: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ -0002f220: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f210: 2020 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: 207c 0a7c 6f31 3220 3d20 3378 2078 2020 |.|o12 = 3x x │ │ │ │ -0002f270: 2d20 7820 202d 2032 7820 7820 202b 2078 - x - 2x x + x │ │ │ │ -0002f280: 2020 2b20 7820 2020 2020 2020 2020 2020 + x │ │ │ │ -0002f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2b0: 207c 0a7c 2020 2020 2020 2020 3320 3420 |.| 3 4 │ │ │ │ -0002f2c0: 2020 2033 2020 2020 2033 2034 2020 2020 3 3 4 │ │ │ │ -0002f2d0: 3320 2020 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ -0002f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f300: 207c 0a7c 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 │ │ │ │ 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 │ │ │ │ +0002f330: 2020 2020 2020 2020 2020 7c0a 7c20 2020 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2020 2020 2020 2020 │ │ │ │ -0002f530: 207c 0a7c 2020 2020 2020 2020 2033 2034 |.| 3 4 │ │ │ │ -0002f540: 2020 2020 2033 2020 2020 2033 2034 2020 3 3 4 │ │ │ │ -0002f550: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f580: 207c 0a7c 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 │ │ │ │ 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 │ │ │ │ +0002f5b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f5d0: 207c 0a7c 6f31 3320 3a20 4368 2020 2020 |.|o13 : Ch │ │ │ │ +0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f620: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 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|.| │ │ │ │ +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 │ │ │ │ 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 │ │ │ │ +0002f6f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ 0002f700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f710: 207c 0a7c 2020 2020 2020 2020 3220 2020 |.| 2 │ │ │ │ -0002f720: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f730: 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: 207c 0a7c 6f31 3420 3d20 3278 2078 2020 |.|o14 = 2x x │ │ │ │ -0002f770: 2b20 7820 202b 2078 2078 2020 2020 2020 + x + x x │ │ │ │ +0002f740: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f750: 2020 2020 2032 2020 2020 2020 3220 2020 2 2 │ │ │ │ +0002f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7b0: 207c 0a7c 2020 2020 2020 2020 3320 3420 |.| 3 4 │ │ │ │ -0002f7c0: 2020 2033 2020 2020 3320 3420 2020 2020 3 3 4 │ │ │ │ +0002f790: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +0002f7a0: 203d 2032 7820 7820 202b 2078 2020 2b20 = 2x x + x + │ │ │ │ +0002f7b0: 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2020 2020 2020 2020 │ │ │ │ 0002f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ -0002f8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f880: 2020 2020 2020 2020 2020 7c0a 7c6f 3134 |.|o14 │ │ │ │ +0002f890: 203a 2043 6820 2020 2020 2020 2020 2020 : Ch │ │ │ │ +0002f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8d0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 0002f8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f8f0: 2d2b 0a0a 466f 7220 7468 6520 7072 6f67 -+..For the prog │ │ │ │ -0002f900: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -0002f910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -0002f920: 626a 6563 7420 2a6e 6f74 6520 546f 7269 bject *note Tori │ │ │ │ -0002f930: 6343 686f 7752 696e 673a 2054 6f72 6963 cChowRing: Toric │ │ │ │ -0002f940: 4368 6f77 5269 6e67 2c20 6973 2061 202a ChowRing, is a * │ │ │ │ -0002f950: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ -0002f960: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ -0002f970: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0002f980: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ -0002f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a46 6f72 ----------+..For │ │ │ │ +0002f930: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0002f940: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002f950: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0002f960: 6e6f 7465 2054 6f72 6963 4368 6f77 5269 note ToricChowRi │ │ │ │ +0002f970: 6e67 3a20 546f 7269 6343 686f 7752 696e ng: ToricChowRin │ │ │ │ +0002f980: 672c 2069 7320 6120 2a6e 6f74 6520 6d65 g, is a *note me │ │ │ │ +0002f990: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0002f9a0: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0002f9b0: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ 0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0002f9e0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0002f9f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0002fa00: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0002fa10: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0002fa20: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -0002fa30: 6179 322f 7061 636b 6167 6573 2f0a 4368 ay2/packages/.Ch │ │ │ │ -0002fa40: 6172 6163 7465 7269 7374 6963 436c 6173 aracteristicClas │ │ │ │ -0002fa50: 7365 732e 6d32 3a31 3935 313a 302e 0a1f ses.m2:1951:0... │ │ │ │ -0002fa60: 0a54 6167 2054 6162 6c65 3a0a 4e6f 6465 .Tag Table:.Node │ │ │ │ -0002fa70: 3a20 546f 707f 3239 310a 4e6f 6465 3a20 : Top.291.Node: │ │ │ │ -0002fa80: 6265 7274 696e 6943 6865 636b 7f31 3637 bertiniCheck.167 │ │ │ │ -0002fa90: 3035 0a4e 6f64 653a 2043 6865 636b 536d 05.Node: 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│ -0002fb70: 534d 7f37 3831 3133 0a4e 6f64 653a 2045 SM.78113.Node: E │ │ │ │ -0002fb80: 756c 6572 7f31 3031 3830 320a 4e6f 6465 uler.101802.Node │ │ │ │ -0002fb90: 3a20 4575 6c65 7241 6666 696e 657f 3131 : EulerAffine.11 │ │ │ │ -0002fba0: 3936 3739 0a4e 6f64 653a 2049 6e64 734f 9679.Node: IndsO │ │ │ │ -0002fbb0: 6653 6d6f 6f74 687f 3132 3231 3639 0a4e fSmooth.122169.N │ │ │ │ -0002fbc0: 6f64 653a 2049 6e70 7574 4973 536d 6f6f ode: InputIsSmoo │ │ │ │ -0002fbd0: 7468 7f31 3236 3132 340a 4e6f 6465 3a20 th.126124.Node: │ │ │ │ -0002fbe0: 6973 4d75 6c74 6948 6f6d 6f67 656e 656f isMultiHomogeneo │ │ │ │ -0002fbf0: 7573 7f31 3330 3130 370a 4e6f 6465 3a20 us.130107.Node: │ │ │ │ -0002fc00: 4d65 7468 6f64 7f31 3334 3138 350a 4e6f Method.134185.No │ │ │ │ -0002fc10: 6465 3a20 4d75 6c74 6950 726f 6a43 6f6f de: MultiProjCoo │ │ │ │ -0002fc20: 7264 5269 6e67 7f31 3338 3133 340a 4e6f rdRing.138134.No │ │ │ │ -0002fc30: 6465 3a20 4f75 7470 7574 7f31 3434 3830 de: Output.14480 │ │ │ │ -0002fc40: 320a 4e6f 6465 3a20 7072 6f62 6162 696c 2.Node: probabil │ │ │ │ -0002fc50: 6973 7469 6320 616c 676f 7269 7468 6d7f istic algorithm. │ │ │ │ -0002fc60: 3136 3333 3230 0a4e 6f64 653a 2053 6567 163320.Node: Seg │ │ │ │ -0002fc70: 7265 7f31 3638 3031 350a 4e6f 6465 3a20 re.168015.Node: │ │ │ │ -0002fc80: 546f 7269 6343 686f 7752 696e 677f 3138 ToricChowRing.18 │ │ │ │ -0002fc90: 3539 3232 0a1f 0a45 6e64 2054 6167 2054 5922...End Tag T │ │ │ │ -0002fca0: 6162 6c65 0a able. │ │ │ │ +0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002fa00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0002fa10: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0002fa20: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0002fa30: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0002fa40: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0002fa50: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +0002fa60: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0002fa70: 6b61 6765 732f 0a43 6861 7261 6374 6572 kages/.Character │ │ │ │ +0002fa80: 6973 7469 6343 6c61 7373 6573 2e6d 323a isticClasses.m2: │ │ │ │ +0002fa90: 3139 3531 3a30 2e0a 1f0a 5461 6720 5461 1951:0....Tag Ta │ │ │ │ +0002faa0: 626c 653a 0a4e 6f64 653a 2054 6f70 7f32 ble:.Node: Top.2 │ │ │ │ +0002fab0: 3931 0a4e 6f64 653a 2062 6572 7469 6e69 91.Node: bertini │ │ │ │ +0002fac0: 4368 6563 6b7f 3136 3730 350a 4e6f 6465 Check.16705.Node │ │ │ │ +0002fad0: 3a20 4368 6563 6b53 6d6f 6f74 687f 3137 : CheckSmooth.17 │ │ │ │ +0002fae0: 3932 320a 4e6f 6465 3a20 4368 6563 6b54 922.Node: CheckT │ │ │ │ +0002faf0: 6f72 6963 5661 7269 6574 7956 616c 6964 oricVarietyValid │ │ │ │ +0002fb00: 7f32 3237 3138 0a4e 6f64 653a 2043 6865 .22718.Node: Che │ │ │ │ +0002fb10: 726e 7f33 3337 3039 0a4e 6f64 653a 2043 rn.33709.Node: C │ │ │ │ +0002fb20: 686f 7752 696e 677f 3531 3231 370a 4e6f howRing.51217.No │ │ │ │ +0002fb30: 6465 3a20 436c 6173 7349 6e43 686f 7752 de: ClassInChowR │ │ │ │ +0002fb40: 696e 677f 3538 3934 380a 4e6f 6465 3a20 ing.58948.Node: │ │ │ │ +0002fb50: 436c 6173 7349 6e54 6f72 6963 4368 6f77 ClassInToricChow │ │ │ │ +0002fb60: 5269 6e67 7f36 3133 3038 0a4e 6f64 653a Ring.61308.Node: │ │ │ │ +0002fb70: 2043 6f6d 704d 6574 686f 647f 3636 3535 CompMethod.6655 │ │ │ │ +0002fb80: 310a 4e6f 6465 3a20 636f 6e66 6967 7572 1.Node: configur │ │ │ │ +0002fb90: 696e 6720 4265 7274 696e 697f 3736 3437 ing Bertini.7647 │ │ │ │ +0002fba0: 380a 4e6f 6465 3a20 4353 4d7f 3738 3131 8.Node: CSM.7811 │ │ │ │ +0002fbb0: 330a 4e6f 6465 3a20 4575 6c65 727f 3130 3.Node: Euler.10 │ │ │ │ +0002fbc0: 3138 3032 0a4e 6f64 653a 2045 756c 6572 1802.Node: Euler │ │ │ │ +0002fbd0: 4166 6669 6e65 7f31 3139 3637 390a 4e6f Affine.119679.No │ │ │ │ +0002fbe0: 6465 3a20 496e 6473 4f66 536d 6f6f 7468 de: IndsOfSmooth │ │ │ │ +0002fbf0: 7f31 3232 3139 310a 4e6f 6465 3a20 496e .122191.Node: In │ │ │ │ +0002fc00: 7075 7449 7353 6d6f 6f74 687f 3132 3631 putIsSmooth.1261 │ │ │ │ +0002fc10: 3436 0a4e 6f64 653a 2069 734d 756c 7469 46.Node: isMulti │ │ │ │ +0002fc20: 486f 6d6f 6765 6e65 6f75 737f 3133 3031 Homogeneous.1301 │ │ │ │ +0002fc30: 3634 0a4e 6f64 653a 204d 6574 686f 647f 64.Node: Method. │ │ │ │ +0002fc40: 3133 3432 3432 0a4e 6f64 653a 204d 756c 134242.Node: Mul │ │ │ │ +0002fc50: 7469 5072 6f6a 436f 6f72 6452 696e 677f tiProjCoordRing. │ │ │ │ +0002fc60: 3133 3831 3931 0a4e 6f64 653a 204f 7574 138191.Node: Out │ │ │ │ +0002fc70: 7075 747f 3134 3438 3539 0a4e 6f64 653a put.144859.Node: │ │ │ │ +0002fc80: 2070 726f 6261 6269 6c69 7374 6963 2061 probabilistic a │ │ │ │ +0002fc90: 6c67 6f72 6974 686d 7f31 3633 3337 370a lgorithm.163377. │ │ │ │ +0002fca0: 4e6f 6465 3a20 5365 6772 657f 3136 3830 Node: Segre.1680 │ │ │ │ +0002fcb0: 3732 0a4e 6f64 653a 2054 6f72 6963 4368 72.Node: ToricCh │ │ │ │ +0002fcc0: 6f77 5269 6e67 7f31 3835 3937 390a 1f0a owRing.185979... │ │ │ │ +0002fcd0: 456e 6420 5461 6720 5461 626c 650a End Tag Table. │ │ ├── ./usr/share/info/CohomCalg.info.gz │ │ │ ├── CohomCalg.info │ │ │ │ @@ -1042,16 +1042,16 @@ │ │ │ │ 00004110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00004120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00004130: 7c69 3230 203a 2065 6c61 7073 6564 5469 |i20 : elapsedTi │ │ │ │ 00004140: 6d65 2068 7665 6373 203d 2063 6f68 6f6d me hvecs = cohom │ │ │ │ 00004150: 4361 6c67 2858 2c20 4432 2920 2020 2020 Calg(X, D2) │ │ │ │ 00004160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00004180: 7c20 2d2d 2033 2e30 3931 3431 7320 656c | -- 3.09141s el │ │ │ │ -00004190: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00004180: 7c20 2d2d 2032 2e37 3633 7320 656c 6170 | -- 2.763s elap │ │ │ │ +00004190: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 000041a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000041d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000041e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000041f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1677,16 +1677,16 @@ │ │ │ │ 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 3338 3039 7320 656c 6170 | -- .3809s elap │ │ │ │ -00006940: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ +00006930: 7c20 2d2d 202e 3437 3537 3536 7320 656c | -- .475756s 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 │ │ │ │ 000069b0: 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 312e 3531 3735 7320 656c | -- 11.5175s el │ │ │ │ +00006b60: 7c20 2d2d 2037 2e39 3938 3238 7320 656c | -- 7.99828s el │ │ │ │ 00006b70: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00006b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006ba0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00006bb0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00006bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00006bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1797,15 +1797,15 @@ │ │ │ │ 00007040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00007050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00007060: 7c69 3237 203a 2065 6c61 7073 6564 5469 |i27 : elapsedTi │ │ │ │ 00007070: 6d65 2063 6f68 6f6d 7665 6331 203d 2063 me cohomvec1 = c │ │ │ │ 00007080: 6f68 6f6d 4361 6c67 2858 5f33 202b 2058 ohomCalg(X_3 + X │ │ │ │ 00007090: 5f37 202d 2058 5f38 2920 2020 2020 2020 _7 - X_8) │ │ │ │ 000070a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000070b0: 7c20 2d2d 202e 3339 3739 3238 7320 656c | -- .397928s el │ │ │ │ +000070b0: 7c20 2d2d 202e 3433 3438 3837 7320 656c | -- .434887s el │ │ │ │ 000070c0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 000070d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000070e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000070f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00007100: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00007110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -1832,20 +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 3535 3339 3738 7320 656c | -- .553978s el │ │ │ │ +000072e0: 7c20 2d2d 202e 3337 3431 3935 7320 656c | -- .374195s el │ │ │ │ 000072f0: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00007300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00007320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00007330: 7c20 2d2d 202e 3535 3430 3039 7320 656c | -- .554009s el │ │ │ │ +00007330: 7c20 2d2d 202e 3337 3432 3137 7320 656c | -- .374217s 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 │ │ │ │ @@ -4344,16 +4344,16 @@ │ │ │ │ 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 3833 3932 7320 2863 7075 293b 2035 .68392s (cpu); 5 │ │ │ │ -00010ff0: 2e30 3138 3533 7320 2874 6872 6561 6429 .01853s (thread) │ │ │ │ +00010fe0: 2e35 3837 3135 7320 2863 7075 293b 2035 .58715s (cpu); 5 │ │ │ │ +00010ff0: 2e30 3635 3233 7320 2874 6872 6561 6429 .06523s (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: 2e32 3332 3835 3873 2028 6370 7529 3b20 .232858s (cpu); │ │ │ │ -000131b0: 302e 3135 3631 3436 7320 2874 6872 6561 0.156146s (threa │ │ │ │ -000131c0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +000131a0: 2e31 3835 3332 3473 2028 6370 7529 3b20 .185324s (cpu); │ │ │ │ +000131b0: 302e 3131 3537 3573 2028 7468 7265 6164 0.11575s (thread │ │ │ │ +000131c0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 000131d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000131e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000131f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013210: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00013220: 2020 3120 2020 2020 2020 3620 2020 2020 1 6 │ │ │ │ 00013230: 2020 3138 2020 2020 2020 2033 3820 2020 18 38 │ │ │ │ @@ -4925,16 +4925,16 @@ │ │ │ │ 000133c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000133d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000133e0: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 4747 -----+.|i21 : GG │ │ │ │ 000133f0: 203d 2074 696d 6520 4569 7365 6e62 7564 = time Eisenbud │ │ │ │ 00013400: 5368 616d 6173 6828 6666 2c46 2c34 2920 Shamash(ff,F,4) │ │ │ │ 00013410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013420: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00013430: 6564 2030 2e39 3830 3632 3273 2028 6370 ed 0.980622s (cp │ │ │ │ -00013440: 7529 3b20 302e 3735 3432 3438 7320 2874 u); 0.754248s (t │ │ │ │ +00013430: 6564 2030 2e39 3930 3731 3473 2028 6370 ed 0.990714s (cp │ │ │ │ +00013440: 7529 3b20 302e 3737 3531 3436 7320 2874 u); 0.775146s (t │ │ │ │ 00013450: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ 00013460: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00013470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000134a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000134b0: 2020 2020 2f20 525c 3120 2020 2020 2f20 / R\1 / │ │ │ │ @@ -4977,24029 +4977,24027 @@ │ │ │ │ 00013700: 5468 6520 6675 6e63 7469 6f6e 2061 6c73 The function als │ │ │ │ 00013710: 6f20 6465 616c 7320 636f 7272 6563 746c o deals correctl │ │ │ │ 00013720: 7920 7769 7468 2063 6f6d 706c 6578 6573 y with complexes │ │ │ │ 00013730: 2046 2077 6865 7265 206d 696e 2046 2069 F where min F i │ │ │ │ 00013740: 7320 6e6f 7420 303a 0a0a 2b2d 2d2d 2d2d s not 0:..+----- │ │ │ │ 00013750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013780: 2b0a 7c69 3232 203a 2047 4720 3d20 7469 +.|i22 : GG = ti │ │ │ │ -00013790: 6d65 2045 6973 656e 6275 6453 6861 6d61 me EisenbudShama │ │ │ │ -000137a0: 7368 2852 312c 465b 325d 2c34 2920 2020 sh(R1,F[2],4) │ │ │ │ -000137b0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ -000137c0: 7365 6420 302e 3939 3436 3636 7320 2863 sed 0.994666s (c │ │ │ │ -000137d0: 7075 293b 2030 2e37 3334 3738 3373 2028 pu); 0.734783s ( │ │ │ │ -000137e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -000137f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00013770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00013780: 0a7c 6932 3220 3a20 4747 203d 2074 696d .|i22 : GG = tim │ │ │ │ +00013790: 6520 4569 7365 6e62 7564 5368 616d 6173 e EisenbudShamas │ │ │ │ +000137a0: 6828 5231 2c46 5b32 5d2c 3429 2020 2020 h(R1,F[2],4) │ │ │ │ +000137b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ +000137c0: 6420 312e 3033 3033 3573 2028 6370 7529 d 1.03035s (cpu) │ │ │ │ +000137d0: 3b20 302e 3833 3430 3339 7320 2874 6872 ; 0.834039s (thr │ │ │ │ +000137e0: 6561 6429 3b20 3073 2028 6763 297c 0a7c ead); 0s (gc)|.| │ │ │ │ +000137f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013820: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00013830: 2020 2031 2020 2020 2020 2036 2020 2020 1 6 │ │ │ │ -00013840: 2020 2031 3820 2020 2020 2020 3338 2020 18 38 │ │ │ │ -00013850: 2020 2020 2036 3620 2020 2020 2020 2020 66 │ │ │ │ -00013860: 7c0a 7c6f 3232 203d 2052 3120 203c 2d2d |.|o22 = R1 <-- │ │ │ │ -00013870: 2052 3120 203c 2d2d 2052 3120 2020 3c2d R1 <-- R1 <- │ │ │ │ -00013880: 2d20 5231 2020 203c 2d2d 2052 3120 2020 - R1 <-- R1 │ │ │ │ -00013890: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00013820: 2020 2020 7c0a 7c20 2020 2020 2020 2031 |.| 1 │ │ │ │ +00013830: 2020 2020 2020 2036 2020 2020 2020 2031 6 1 │ │ │ │ +00013840: 3820 2020 2020 2020 3338 2020 2020 2020 8 38 │ │ │ │ +00013850: 2036 3620 2020 2020 2020 207c 0a7c 6f32 66 |.|o2 │ │ │ │ +00013860: 3220 3d20 5231 2020 3c2d 2d20 5231 2020 2 = R1 <-- R1 │ │ │ │ +00013870: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120 <-- R1 <-- R1 │ │ │ │ +00013880: 2020 3c2d 2d20 5231 2020 2020 2020 2020 <-- R1 │ │ │ │ +00013890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000138a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000138b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000138d0: 7c0a 7c20 2020 2020 202d 3220 2020 2020 |.| -2 │ │ │ │ -000138e0: 202d 3120 2020 2020 2030 2020 2020 2020 -1 0 │ │ │ │ -000138f0: 2020 3120 2020 2020 2020 2032 2020 2020 1 2 │ │ │ │ -00013900: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000138c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000138d0: 2020 2d32 2020 2020 2020 2d31 2020 2020 -2 -1 │ │ │ │ +000138e0: 2020 3020 2020 2020 2020 2031 2020 2020 0 1 │ │ │ │ +000138f0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00013900: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00013910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00013920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013940: 7c0a 7c6f 3232 203a 2043 6f6d 706c 6578 |.|o22 : Complex │ │ │ │ +00013930: 2020 2020 2020 207c 0a7c 6f32 3220 3a20 |.|o22 : │ │ │ │ +00013940: 436f 6d70 6c65 7820 2020 2020 2020 2020 Complex │ │ │ │ 00013950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00013970: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00013960: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00013970: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00013980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000139b0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d +..See also.==== │ │ │ │ -000139c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -000139d0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -000139e0: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -000139f0: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ -00013a00: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ -00013a10: 2020 686f 6d6f 746f 7069 6573 0a20 202a homotopies. * │ │ │ │ -00013a20: 202a 6e6f 7465 2053 6861 6d61 7368 3a20 *note Shamash: │ │ │ │ -00013a30: 5368 616d 6173 682c 202d 2d20 436f 6d70 Shamash, -- Comp │ │ │ │ -00013a40: 7574 6573 2074 6865 2053 6861 6d61 7368 utes the Shamash │ │ │ │ -00013a50: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f Complex. * *no │ │ │ │ -00013a60: 7465 2065 7870 6f3a 2065 7870 6f2c 202d te expo: expo, - │ │ │ │ -00013a70: 2d20 7265 7475 726e 7320 6120 7365 7420 - returns a set │ │ │ │ -00013a80: 636f 7272 6573 706f 6e64 696e 6720 746f corresponding to │ │ │ │ -00013a90: 2074 6865 2062 6173 6973 206f 6620 6120 the basis of a │ │ │ │ -00013aa0: 6469 7669 6465 640a 2020 2020 706f 7765 divided. powe │ │ │ │ -00013ab0: 720a 0a57 6179 7320 746f 2075 7365 2045 r..Ways to use E │ │ │ │ -00013ac0: 6973 656e 6275 6453 6861 6d61 7368 3a0a isenbudShamash:. │ │ │ │ +000139a0: 2d2d 2d2d 2d2b 0a0a 5365 6520 616c 736f -----+..See also │ │ │ │ +000139b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +000139c0: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +000139d0: 6965 733a 206d 616b 6548 6f6d 6f74 6f70 ies: makeHomotop │ │ │ │ +000139e0: 6965 732c 202d 2d20 7265 7475 726e 7320 ies, -- returns │ │ │ │ +000139f0: 6120 7379 7374 656d 206f 6620 6869 6768 a system of high │ │ │ │ +00013a00: 6572 0a20 2020 2068 6f6d 6f74 6f70 6965 er. homotopie │ │ │ │ +00013a10: 730a 2020 2a20 2a6e 6f74 6520 5368 616d s. * *note Sham │ │ │ │ +00013a20: 6173 683a 2053 6861 6d61 7368 2c20 2d2d ash: Shamash, -- │ │ │ │ +00013a30: 2043 6f6d 7075 7465 7320 7468 6520 5368 Computes the Sh │ │ │ │ +00013a40: 616d 6173 6820 436f 6d70 6c65 780a 2020 amash Complex. │ │ │ │ +00013a50: 2a20 2a6e 6f74 6520 6578 706f 3a20 6578 * *note expo: ex │ │ │ │ +00013a60: 706f 2c20 2d2d 2072 6574 7572 6e73 2061 po, -- returns a │ │ │ │ +00013a70: 2073 6574 2063 6f72 7265 7370 6f6e 6469 set correspondi │ │ │ │ +00013a80: 6e67 2074 6f20 7468 6520 6261 7369 7320 ng to the basis │ │ │ │ +00013a90: 6f66 2061 2064 6976 6964 6564 0a20 2020 of a divided. │ │ │ │ +00013aa0: 2070 6f77 6572 0a0a 5761 7973 2074 6f20 power..Ways to │ │ │ │ +00013ab0: 7573 6520 4569 7365 6e62 7564 5368 616d use EisenbudSham │ │ │ │ +00013ac0: 6173 683a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ash:.=========== │ │ │ │ 00013ad0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00013ae0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -00013af0: 2a20 2245 6973 656e 6275 6453 6861 6d61 * "EisenbudShama │ │ │ │ -00013b00: 7368 284d 6174 7269 782c 436f 6d70 6c65 sh(Matrix,Comple │ │ │ │ -00013b10: 782c 5a5a 2922 0a20 202a 2022 4569 7365 x,ZZ)". * "Eise │ │ │ │ -00013b20: 6e62 7564 5368 616d 6173 6828 5269 6e67 nbudShamash(Ring │ │ │ │ -00013b30: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46 ,Complex,ZZ)"..F │ │ │ │ -00013b40: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00013b50: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00013b60: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00013b70: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -00013b80: 6861 6d61 7368 3a20 4569 7365 6e62 7564 hamash: Eisenbud │ │ │ │ -00013b90: 5368 616d 6173 682c 2069 7320 6120 2a6e Shamash, is a *n │ │ │ │ -00013ba0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00013bb0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00013bc0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00013bd0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00013ae0: 3d0a 0a20 202a 2022 4569 7365 6e62 7564 =.. * "Eisenbud │ │ │ │ +00013af0: 5368 616d 6173 6828 4d61 7472 6978 2c43 Shamash(Matrix,C │ │ │ │ +00013b00: 6f6d 706c 6578 2c5a 5a29 220a 2020 2a20 omplex,ZZ)". * │ │ │ │ +00013b10: 2245 6973 656e 6275 6453 6861 6d61 7368 "EisenbudShamash │ │ │ │ +00013b20: 2852 696e 672c 436f 6d70 6c65 782c 5a5a (Ring,Complex,ZZ │ │ │ │ +00013b30: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00013b40: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00013b50: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00013b60: 626a 6563 7420 2a6e 6f74 6520 4569 7365 bject *note Eise │ │ │ │ +00013b70: 6e62 7564 5368 616d 6173 683a 2045 6973 nbudShamash: Eis │ │ │ │ +00013b80: 656e 6275 6453 6861 6d61 7368 2c20 6973 enbudShamash, is │ │ │ │ +00013b90: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00013ba0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00013bb0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00013bc0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00013bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00013c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00013c20: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00013c30: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00013c40: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00013c50: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00013c60: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00013c70: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00013c80: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ -00013c90: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -00013ca0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ -00013cb0: 3438 3432 3a30 2e0a 1f0a 4669 6c65 3a20 4842:0....File: │ │ │ │ -00013cc0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00013cd0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00013ce0: 696e 666f 2c20 4e6f 6465 3a20 4569 7365 info, Node: Eise │ │ │ │ -00013cf0: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ -00013d00: 2c20 4e65 7874 3a20 6576 656e 4578 744d , Next: evenExtM │ │ │ │ -00013d10: 6f64 756c 652c 2050 7265 763a 2045 6973 odule, Prev: Eis │ │ │ │ -00013d20: 656e 6275 6453 6861 6d61 7368 2c20 5570 enbudShamash, Up │ │ │ │ -00013d30: 3a20 546f 700a 0a45 6973 656e 6275 6453 : Top..EisenbudS │ │ │ │ -00013d40: 6861 6d61 7368 546f 7461 6c20 2d2d 2050 hamashTotal -- P │ │ │ │ -00013d50: 7265 6375 7273 6f72 2063 6f6d 706c 6578 recursor complex │ │ │ │ -00013d60: 206f 6620 746f 7461 6c20 4578 740a 2a2a of total Ext.** │ │ │ │ +00013c10: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00013c20: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00013c30: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00013c40: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00013c50: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00013c60: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +00013c70: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00013c80: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00013c90: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00013ca0: 732e 6d32 3a34 3834 323a 302e 0a1f 0a46 s.m2:4842:0....F │ │ │ │ +00013cb0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00013cc0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00013cd0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00013ce0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ +00013cf0: 546f 7461 6c2c 204e 6578 743a 2065 7665 Total, Next: eve │ │ │ │ +00013d00: 6e45 7874 4d6f 6475 6c65 2c20 5072 6576 nExtModule, Prev │ │ │ │ +00013d10: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00013d20: 682c 2055 703a 2054 6f70 0a0a 4569 7365 h, Up: Top..Eise │ │ │ │ +00013d30: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00013d40: 202d 2d20 5072 6563 7572 736f 7220 636f -- Precursor co │ │ │ │ +00013d50: 6d70 6c65 7820 6f66 2074 6f74 616c 2045 mplex of total E │ │ │ │ +00013d60: 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a xt.************* │ │ │ │ 00013d70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00013d80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013d90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00013da0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00013db0: 200a 2020 2020 2020 2020 2864 302c 6431 . (d0,d1 │ │ │ │ -00013dc0: 2920 3d20 2045 6973 656e 6275 6453 6861 ) = EisenbudSha │ │ │ │ -00013dd0: 6d61 7368 546f 7461 6c20 4d0a 2020 2a20 mashTotal M. * │ │ │ │ -00013de0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -00013df0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -00013e00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00013e10: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -00013e20: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00013e30: 6563 7469 6f6e 0a20 202a 202a 6e6f 7465 ection. * *note │ │ │ │ -00013e40: 204f 7074 696f 6e61 6c20 696e 7075 7473 Optional inputs │ │ │ │ -00013e50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00013e60: 7573 696e 6720 6675 6e63 7469 6f6e 7320 using functions │ │ │ │ -00013e70: 7769 7468 206f 7074 696f 6e61 6c20 696e with optional in │ │ │ │ -00013e80: 7075 7473 2c3a 0a20 2020 2020 202a 2043 puts,:. * C │ │ │ │ -00013e90: 6865 636b 203d 3e20 2e2e 2e2c 2064 6566 heck => ..., def │ │ │ │ -00013ea0: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ -00013eb0: 0a20 2020 2020 202a 2047 7261 6469 6e67 . * Grading │ │ │ │ -00013ec0: 203d 3e20 2e2e 2e2c 2064 6566 6175 6c74 => ..., default │ │ │ │ -00013ed0: 2076 616c 7565 2032 0a20 2020 2020 202a value 2. * │ │ │ │ -00013ee0: 2056 6172 6961 626c 6573 203d 3e20 2e2e Variables => .. │ │ │ │ -00013ef0: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00013f00: 2073 0a20 202a 204f 7574 7075 7473 3a0a s. * Outputs:. │ │ │ │ -00013f10: 2020 2020 2020 2a20 6430 2c20 6120 2a6e * d0, a *n │ │ │ │ -00013f20: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00013f30: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00013f40: 2c2c 206d 6170 206f 6620 6672 6565 206d ,, map of free m │ │ │ │ -00013f50: 6f64 756c 6573 206f 7665 7220 616e 0a20 odules over an. │ │ │ │ -00013f60: 2020 2020 2020 2065 6e6c 6172 6765 6420 enlarged │ │ │ │ -00013f70: 7269 6e67 0a20 2020 2020 202a 2064 312c ring. * d1, │ │ │ │ -00013f80: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ -00013f90: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -00013fa0: 6174 7269 782c 2c20 6d61 7020 6f66 2066 atrix,, map of f │ │ │ │ -00013fb0: 7265 6520 6d6f 6475 6c65 7320 6f76 6572 ree modules over │ │ │ │ -00013fc0: 2061 6e0a 2020 2020 2020 2020 656e 6c61 an. enla │ │ │ │ -00013fd0: 7267 6564 2072 696e 670a 0a44 6573 6372 rged ring..Descr │ │ │ │ -00013fe0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00013ff0: 3d3d 0a0a 4173 7375 6d65 2074 6861 7420 ==..Assume that │ │ │ │ -00014000: 4d20 6973 2064 6566 696e 6564 206f 7665 M is defined ove │ │ │ │ -00014010: 7220 6120 7269 6e67 206f 6620 7468 6520 r a ring of the │ │ │ │ -00014020: 666f 726d 2052 6261 7220 3d20 522f 2866 form Rbar = R/(f │ │ │ │ -00014030: 5f30 2e2e 665f 7b63 2d31 7d29 2c20 610a _0..f_{c-1}), a. │ │ │ │ -00014040: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ -00014050: 6374 696f 6e2c 2061 6e64 2074 6861 7420 ction, and that │ │ │ │ -00014060: 4d20 6861 7320 6120 6669 6e69 7465 2066 M has a finite f │ │ │ │ -00014070: 7265 6520 7265 736f 6c75 7469 6f6e 2047 ree resolution G │ │ │ │ -00014080: 206f 7665 7220 522e 2049 6e0a 7468 6973 over R. In.this │ │ │ │ -00014090: 2063 6173 6520 4d20 6861 7320 6120 6672 case M has a fr │ │ │ │ -000140a0: 6565 2072 6573 6f6c 7574 696f 6e20 4620 ee resolution F │ │ │ │ -000140b0: 6f76 6572 2052 6261 7220 7768 6f73 6520 over Rbar whose │ │ │ │ -000140c0: 6475 616c 2c20 465e 2a20 6973 2061 2066 dual, F^* is a f │ │ │ │ -000140d0: 696e 6974 656c 790a 6765 6e65 7261 7465 initely.generate │ │ │ │ -000140e0: 642c 205a 2d67 7261 6465 6420 6672 6565 d, Z-graded free │ │ │ │ -000140f0: 206d 6f64 756c 6520 6f76 6572 2061 2072 module over a r │ │ │ │ -00014100: 696e 6720 5362 6172 5c63 6f6e 6720 6b6b ing Sbar\cong kk │ │ │ │ -00014110: 5b73 5f30 2e2e 735f 7b63 2d31 7d2c 6765 [s_0..s_{c-1},ge │ │ │ │ -00014120: 6e73 0a52 6261 725d 2c20 7768 6572 6520 ns.Rbar], where │ │ │ │ -00014130: 7468 6520 6465 6772 6565 7320 6f66 2074 the degrees of t │ │ │ │ -00014140: 6865 2073 5f69 2061 7265 207b 2d32 2c20 he s_i are {-2, │ │ │ │ -00014150: 2d64 6567 7265 6520 665f 697d 2e20 5468 -degree f_i}. Th │ │ │ │ -00014160: 6973 2072 6573 6f6c 7574 696f 6e20 6973 is resolution is │ │ │ │ -00014170: 0a69 7320 636f 6e73 7472 7563 7465 6420 .is constructed │ │ │ │ -00014180: 6672 6f6d 2074 6865 2064 7561 6c20 6f66 from the dual of │ │ │ │ -00014190: 2047 2c20 746f 6765 7468 6572 2077 6974 G, together wit │ │ │ │ -000141a0: 6820 7468 6520 6475 616c 7320 6f66 2074 h the duals of t │ │ │ │ -000141b0: 6865 2068 6967 6865 720a 686f 6d6f 746f he higher.homoto │ │ │ │ -000141c0: 7069 6573 206f 6e20 4720 6465 6669 6e65 pies on G define │ │ │ │ -000141d0: 6420 6279 2045 6973 656e 6275 642e 0a0a d by Eisenbud... │ │ │ │ -000141e0: 5468 6520 6675 6e63 7469 6f6e 2072 6574 The function ret │ │ │ │ -000141f0: 7572 6e73 2074 6865 2064 6966 6665 7265 urns the differe │ │ │ │ -00014200: 6e74 6961 6c73 2064 303a 465e 2a5f 7b65 ntials d0:F^*_{e │ │ │ │ -00014210: 7665 6e7d 205c 746f 2046 5e2a 5f7b 6f64 ven} \to F^*_{od │ │ │ │ -00014220: 647d 2061 6e64 0a64 313a 465e 2a5f 7b6f d} and.d1:F^*_{o │ │ │ │ -00014230: 6464 7d5c 746f 2046 5e2a 5f7b 6576 656e dd}\to F^*_{even │ │ │ │ -00014240: 7d2e 0a0a 5468 6520 6d61 7073 2064 302c }...The maps d0, │ │ │ │ -00014250: 6431 2066 6f72 6d20 6120 6d61 7472 6978 d1 form a matrix │ │ │ │ -00014260: 2066 6163 746f 7269 7a61 7469 6f6e 206f factorization o │ │ │ │ -00014270: 6620 7375 6d28 632c 2069 2d3e 735f 692a f sum(c, i->s_i* │ │ │ │ -00014280: 665f 6929 2e20 5468 6520 6861 7665 2074 f_i). The have t │ │ │ │ -00014290: 6865 0a70 726f 7065 7274 7920 7468 6174 he.property that │ │ │ │ -000142a0: 2066 6f72 2061 6e79 2052 6261 7220 6d6f for any Rbar mo │ │ │ │ -000142b0: 6475 6c65 204e 2c0a 0a48 485f 3120 636f dule N,..HH_1 co │ │ │ │ -000142c0: 6d70 6c65 7820 5c7b 6430 2a2a 4e2c 2064 mplex \{d0**N, d │ │ │ │ -000142d0: 312a 2a4e 5c7d 203d 2045 7874 5e7b 6576 1**N\} = Ext^{ev │ │ │ │ -000142e0: 656e 7d5f 7b52 6261 727d 284d 2c4e 290a en}_{Rbar}(M,N). │ │ │ │ -000142f0: 0a53 5e7b 7b31 2c30 7d7d 2a2a 4848 5f31 .S^{{1,0}}**HH_1 │ │ │ │ -00014300: 2063 6f6d 706c 6578 205c 7b53 5e7b 7b2d complex \{S^{{- │ │ │ │ -00014310: 322c 307d 7d2a 2a64 312a 2a4e 2c20 6430 2,0}}**d1**N, d0 │ │ │ │ -00014320: 2a2a 4e5c 7d20 3d20 4578 745e 7b6f 6464 **N\} = Ext^{odd │ │ │ │ -00014330: 7d5f 7b52 6261 727d 284d 2c4e 290a 0a54 }_{Rbar}(M,N)..T │ │ │ │ -00014340: 6869 7320 6973 2065 6e63 6f64 6564 2069 his is encoded i │ │ │ │ -00014350: 6e20 7468 6520 7363 7269 7074 206e 6577 n the script new │ │ │ │ -00014360: 4578 740a 0a4f 7074 696f 6e20 6465 6661 Ext..Option defa │ │ │ │ -00014370: 756c 7473 3a20 4368 6563 6b3d 3e66 616c ults: Check=>fal │ │ │ │ -00014380: 7365 2056 6172 6961 626c 6573 3d3e 6765 se Variables=>ge │ │ │ │ -00014390: 7453 796d 626f 6c20 2273 222c 2047 7261 tSymbol "s", Gra │ │ │ │ -000143a0: 6469 6e67 203d 3e32 7d0a 0a49 6620 4772 ding =>2}..If Gr │ │ │ │ -000143b0: 6164 696e 6720 3d3e 312c 2074 6865 6e20 ading =>1, then │ │ │ │ -000143c0: 6120 7369 6e67 6c79 2067 7261 6465 6420 a singly graded │ │ │ │ -000143d0: 7265 7375 6c74 2069 7320 7265 7475 726e result is return │ │ │ │ -000143e0: 6564 2028 6a75 7374 2066 6f72 6765 7474 ed (just forgett │ │ │ │ -000143f0: 696e 6720 7468 650a 686f 6d6f 6c6f 6769 ing the.homologi │ │ │ │ -00014400: 6361 6c20 6772 6164 696e 672e 290a 0a0a cal grading.)... │ │ │ │ -00014410: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00013d90: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00013da0: 7361 6765 3a20 0a20 2020 2020 2020 2028 sage: . ( │ │ │ │ +00013db0: 6430 2c64 3129 203d 2020 4569 7365 6e62 d0,d1) = Eisenb │ │ │ │ +00013dc0: 7564 5368 616d 6173 6854 6f74 616c 204d udShamashTotal M │ │ │ │ +00013dd0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +00013de0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +00013df0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +00013e00: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +00013e10: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +00013e20: 6e74 6572 7365 6374 696f 6e0a 2020 2a20 ntersection. * │ │ │ │ +00013e30: 2a6e 6f74 6520 4f70 7469 6f6e 616c 2069 *note Optional i │ │ │ │ +00013e40: 6e70 7574 733a 2028 4d61 6361 756c 6179 nputs: (Macaulay │ │ │ │ +00013e50: 3244 6f63 2975 7369 6e67 2066 756e 6374 2Doc)using funct │ │ │ │ +00013e60: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +00013e70: 616c 2069 6e70 7574 732c 3a0a 2020 2020 al inputs,:. │ │ │ │ +00013e80: 2020 2a20 4368 6563 6b20 3d3e 202e 2e2e * Check => ... │ │ │ │ +00013e90: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +00013ea0: 6661 6c73 650a 2020 2020 2020 2a20 4772 false. * Gr │ │ │ │ +00013eb0: 6164 696e 6720 3d3e 202e 2e2e 2c20 6465 ading => ..., de │ │ │ │ +00013ec0: 6661 756c 7420 7661 6c75 6520 320a 2020 fault value 2. │ │ │ │ +00013ed0: 2020 2020 2a20 5661 7269 6162 6c65 7320 * Variables │ │ │ │ +00013ee0: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00013ef0: 7661 6c75 6520 730a 2020 2a20 4f75 7470 value s. * Outp │ │ │ │ +00013f00: 7574 733a 0a20 2020 2020 202a 2064 302c uts:. * d0, │ │ │ │ +00013f10: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00013f20: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00013f30: 6174 7269 782c 2c20 6d61 7020 6f66 2066 atrix,, map of f │ │ │ │ +00013f40: 7265 6520 6d6f 6475 6c65 7320 6f76 6572 ree modules over │ │ │ │ +00013f50: 2061 6e0a 2020 2020 2020 2020 656e 6c61 an. enla │ │ │ │ +00013f60: 7267 6564 2072 696e 670a 2020 2020 2020 rged ring. │ │ │ │ +00013f70: 2a20 6431 2c20 6120 2a6e 6f74 6520 6d61 * d1, a *note ma │ │ │ │ +00013f80: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ +00013f90: 446f 6329 4d61 7472 6978 2c2c 206d 6170 Doc)Matrix,, map │ │ │ │ +00013fa0: 206f 6620 6672 6565 206d 6f64 756c 6573 of free modules │ │ │ │ +00013fb0: 206f 7665 7220 616e 0a20 2020 2020 2020 over an. │ │ │ │ +00013fc0: 2065 6e6c 6172 6765 6420 7269 6e67 0a0a enlarged ring.. │ │ │ │ +00013fd0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00013fe0: 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d 6520 =======..Assume │ │ │ │ +00013ff0: 7468 6174 204d 2069 7320 6465 6669 6e65 that M is define │ │ │ │ +00014000: 6420 6f76 6572 2061 2072 696e 6720 6f66 d over a ring of │ │ │ │ +00014010: 2074 6865 2066 6f72 6d20 5262 6172 203d the form Rbar = │ │ │ │ +00014020: 2052 2f28 665f 302e 2e66 5f7b 632d 317d R/(f_0..f_{c-1} │ │ │ │ +00014030: 292c 2061 0a63 6f6d 706c 6574 6520 696e ), a.complete in │ │ │ │ +00014040: 7465 7273 6563 7469 6f6e 2c20 616e 6420 tersection, and │ │ │ │ +00014050: 7468 6174 204d 2068 6173 2061 2066 696e that M has a fin │ │ │ │ +00014060: 6974 6520 6672 6565 2072 6573 6f6c 7574 ite free resolut │ │ │ │ +00014070: 696f 6e20 4720 6f76 6572 2052 2e20 496e ion G over R. In │ │ │ │ +00014080: 0a74 6869 7320 6361 7365 204d 2068 6173 .this case M has │ │ │ │ +00014090: 2061 2066 7265 6520 7265 736f 6c75 7469 a free resoluti │ │ │ │ +000140a0: 6f6e 2046 206f 7665 7220 5262 6172 2077 on F over Rbar w │ │ │ │ +000140b0: 686f 7365 2064 7561 6c2c 2046 5e2a 2069 hose dual, F^* i │ │ │ │ +000140c0: 7320 6120 6669 6e69 7465 6c79 0a67 656e s a finitely.gen │ │ │ │ +000140d0: 6572 6174 6564 2c20 5a2d 6772 6164 6564 erated, Z-graded │ │ │ │ +000140e0: 2066 7265 6520 6d6f 6475 6c65 206f 7665 free module ove │ │ │ │ +000140f0: 7220 6120 7269 6e67 2053 6261 725c 636f r a ring Sbar\co │ │ │ │ +00014100: 6e67 206b 6b5b 735f 302e 2e73 5f7b 632d ng kk[s_0..s_{c- │ │ │ │ +00014110: 317d 2c67 656e 730a 5262 6172 5d2c 2077 1},gens.Rbar], w │ │ │ │ +00014120: 6865 7265 2074 6865 2064 6567 7265 6573 here the degrees │ │ │ │ +00014130: 206f 6620 7468 6520 735f 6920 6172 6520 of the s_i are │ │ │ │ +00014140: 7b2d 322c 202d 6465 6772 6565 2066 5f69 {-2, -degree f_i │ │ │ │ +00014150: 7d2e 2054 6869 7320 7265 736f 6c75 7469 }. This resoluti │ │ │ │ +00014160: 6f6e 2069 730a 6973 2063 6f6e 7374 7275 on is.is constru │ │ │ │ +00014170: 6374 6564 2066 726f 6d20 7468 6520 6475 cted from the du │ │ │ │ +00014180: 616c 206f 6620 472c 2074 6f67 6574 6865 al of G, togethe │ │ │ │ +00014190: 7220 7769 7468 2074 6865 2064 7561 6c73 r with the duals │ │ │ │ +000141a0: 206f 6620 7468 6520 6869 6768 6572 0a68 of the higher.h │ │ │ │ +000141b0: 6f6d 6f74 6f70 6965 7320 6f6e 2047 2064 omotopies on G d │ │ │ │ +000141c0: 6566 696e 6564 2062 7920 4569 7365 6e62 efined by Eisenb │ │ │ │ +000141d0: 7564 2e0a 0a54 6865 2066 756e 6374 696f ud...The functio │ │ │ │ +000141e0: 6e20 7265 7475 726e 7320 7468 6520 6469 n returns the di │ │ │ │ +000141f0: 6666 6572 656e 7469 616c 7320 6430 3a46 fferentials d0:F │ │ │ │ +00014200: 5e2a 5f7b 6576 656e 7d20 5c74 6f20 465e ^*_{even} \to F^ │ │ │ │ +00014210: 2a5f 7b6f 6464 7d20 616e 640a 6431 3a46 *_{odd} and.d1:F │ │ │ │ +00014220: 5e2a 5f7b 6f64 647d 5c74 6f20 465e 2a5f ^*_{odd}\to F^*_ │ │ │ │ +00014230: 7b65 7665 6e7d 2e0a 0a54 6865 206d 6170 {even}...The map │ │ │ │ +00014240: 7320 6430 2c64 3120 666f 726d 2061 206d s d0,d1 form a m │ │ │ │ +00014250: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ +00014260: 696f 6e20 6f66 2073 756d 2863 2c20 692d ion of sum(c, i- │ │ │ │ +00014270: 3e73 5f69 2a66 5f69 292e 2054 6865 2068 >s_i*f_i). The h │ │ │ │ +00014280: 6176 6520 7468 650a 7072 6f70 6572 7479 ave the.property │ │ │ │ +00014290: 2074 6861 7420 666f 7220 616e 7920 5262 that for any Rb │ │ │ │ +000142a0: 6172 206d 6f64 756c 6520 4e2c 0a0a 4848 ar module N,..HH │ │ │ │ +000142b0: 5f31 2063 6f6d 706c 6578 205c 7b64 302a _1 complex \{d0* │ │ │ │ +000142c0: 2a4e 2c20 6431 2a2a 4e5c 7d20 3d20 4578 *N, d1**N\} = Ex │ │ │ │ +000142d0: 745e 7b65 7665 6e7d 5f7b 5262 6172 7d28 t^{even}_{Rbar}( │ │ │ │ +000142e0: 4d2c 4e29 0a0a 535e 7b7b 312c 307d 7d2a M,N)..S^{{1,0}}* │ │ │ │ +000142f0: 2a48 485f 3120 636f 6d70 6c65 7820 5c7b *HH_1 complex \{ │ │ │ │ +00014300: 535e 7b7b 2d32 2c30 7d7d 2a2a 6431 2a2a S^{{-2,0}}**d1** │ │ │ │ +00014310: 4e2c 2064 302a 2a4e 5c7d 203d 2045 7874 N, d0**N\} = Ext │ │ │ │ +00014320: 5e7b 6f64 647d 5f7b 5262 6172 7d28 4d2c ^{odd}_{Rbar}(M, │ │ │ │ +00014330: 4e29 0a0a 5468 6973 2069 7320 656e 636f N)..This is enco │ │ │ │ +00014340: 6465 6420 696e 2074 6865 2073 6372 6970 ded in the scrip │ │ │ │ +00014350: 7420 6e65 7745 7874 0a0a 4f70 7469 6f6e t newExt..Option │ │ │ │ +00014360: 2064 6566 6175 6c74 733a 2043 6865 636b defaults: Check │ │ │ │ +00014370: 3d3e 6661 6c73 6520 5661 7269 6162 6c65 =>false Variable │ │ │ │ +00014380: 733d 3e67 6574 5379 6d62 6f6c 2022 7322 s=>getSymbol "s" │ │ │ │ +00014390: 2c20 4772 6164 696e 6720 3d3e 327d 0a0a , Grading =>2}.. │ │ │ │ +000143a0: 4966 2047 7261 6469 6e67 203d 3e31 2c20 If Grading =>1, │ │ │ │ +000143b0: 7468 656e 2061 2073 696e 676c 7920 6772 then a singly gr │ │ │ │ +000143c0: 6164 6564 2072 6573 756c 7420 6973 2072 aded result is r │ │ │ │ +000143d0: 6574 7572 6e65 6420 286a 7573 7420 666f eturned (just fo │ │ │ │ +000143e0: 7267 6574 7469 6e67 2074 6865 0a68 6f6d rgetting the.hom │ │ │ │ +000143f0: 6f6c 6f67 6963 616c 2067 7261 6469 6e67 ological grading │ │ │ │ +00014400: 2e29 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d .)....+--------- │ │ │ │ +00014410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014460: 0a7c 6931 203a 206e 203d 2033 2020 2020 .|i1 : n = 3 │ │ │ │ +00014450: 2d2d 2d2d 2b0a 7c69 3120 3a20 6e20 3d20 ----+.|i1 : n = │ │ │ │ +00014460: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00014470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000144b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000144a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000144b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000144f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014500: 0a7c 6f31 203d 2033 2020 2020 2020 2020 .|o1 = 3 │ │ │ │ +000144f0: 2020 2020 7c0a 7c6f 3120 3d20 3320 2020 |.|o1 = 3 │ │ │ │ +00014500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014540: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014550: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014540: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000145a0: 0a7c 6932 203a 2063 203d 2032 2020 2020 .|i2 : c = 2 │ │ │ │ +00014590: 2d2d 2d2d 2b0a 7c69 3220 3a20 6320 3d20 ----+.|i2 : c = │ │ │ │ +000145a0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 000145b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000145c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000145d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000145e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000145f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000145e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000145f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014640: 0a7c 6f32 203d 2032 2020 2020 2020 2020 .|o2 = 2 │ │ │ │ +00014630: 2020 2020 7c0a 7c6f 3220 3d20 3220 2020 |.|o2 = 2 │ │ │ │ +00014640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014690: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000146c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000146d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000146e0: 0a7c 6933 203a 206b 6b20 3d20 5a5a 2f31 .|i3 : kk = ZZ/1 │ │ │ │ -000146f0: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ +000146d0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6b6b 203d ----+.|i3 : kk = │ │ │ │ +000146e0: 205a 5a2f 3130 3120 2020 2020 2020 2020 ZZ/101 │ │ │ │ +000146f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014730: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014780: 0a7c 6f33 203d 206b 6b20 2020 2020 2020 .|o3 = kk │ │ │ │ +00014770: 2020 2020 7c0a 7c6f 3320 3d20 6b6b 2020 |.|o3 = kk │ │ │ │ +00014780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000147c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000147d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000147c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000147d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000147f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014820: 0a7c 6f33 203a 2051 756f 7469 656e 7452 .|o3 : QuotientR │ │ │ │ -00014830: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00014810: 2020 2020 7c0a 7c6f 3320 3a20 5175 6f74 |.|o3 : Quot │ │ │ │ +00014820: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00014830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014860: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000148a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000148b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000148c0: 0a7c 6934 203a 2052 203d 206b 6b5b 785f .|i4 : R = kk[x_ │ │ │ │ -000148d0: 302e 2e78 5f28 6e2d 3129 5d20 2020 2020 0..x_(n-1)] │ │ │ │ +000148b0: 2d2d 2d2d 2b0a 7c69 3420 3a20 5220 3d20 ----+.|i4 : R = │ │ │ │ +000148c0: 6b6b 5b78 5f30 2e2e 785f 286e 2d31 295d kk[x_0..x_(n-1)] │ │ │ │ +000148d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000148f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014900: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014910: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014900: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014950: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014960: 0a7c 6f34 203d 2052 2020 2020 2020 2020 .|o4 = R │ │ │ │ +00014950: 2020 2020 7c0a 7c6f 3420 3d20 5220 2020 |.|o4 = R │ │ │ │ +00014960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000149b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000149a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014a00: 0a7c 6f34 203a 2050 6f6c 796e 6f6d 6961 .|o4 : Polynomia │ │ │ │ -00014a10: 6c52 696e 6720 2020 2020 2020 2020 2020 lRing │ │ │ │ +000149f0: 2020 2020 7c0a 7c6f 3420 3a20 506f 6c79 |.|o4 : Poly │ │ │ │ +00014a00: 6e6f 6d69 616c 5269 6e67 2020 2020 2020 nomialRing │ │ │ │ +00014a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014a40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014a50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014a40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014aa0: 0a7c 6935 203a 2049 203d 2069 6465 616c .|i5 : I = ideal │ │ │ │ -00014ab0: 2878 5f30 5e32 2c20 785f 325e 3329 2020 (x_0^2, x_2^3) │ │ │ │ +00014a90: 2d2d 2d2d 2b0a 7c69 3520 3a20 4920 3d20 ----+.|i5 : I = │ │ │ │ +00014aa0: 6964 6561 6c28 785f 305e 322c 2078 5f32 ideal(x_0^2, x_2 │ │ │ │ +00014ab0: 5e33 2920 2020 2020 2020 2020 2020 2020 ^3) │ │ │ │ 00014ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ae0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014af0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014ae0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2032 .| 2 │ │ │ │ -00014b50: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00014b30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014b40: 2020 2020 3220 2020 3320 2020 2020 2020 2 3 │ │ │ │ +00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014b90: 0a7c 6f35 203d 2069 6465 616c 2028 7820 .|o5 = ideal (x │ │ │ │ -00014ba0: 2c20 7820 2920 2020 2020 2020 2020 2020 , x ) │ │ │ │ +00014b80: 2020 2020 7c0a 7c6f 3520 3d20 6964 6561 |.|o5 = idea │ │ │ │ +00014b90: 6c20 2878 202c 2078 2029 2020 2020 2020 l (x , x ) │ │ │ │ +00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014be0: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -00014bf0: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00014bd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014be0: 2020 2020 3020 2020 3220 2020 2020 2020 0 2 │ │ │ │ +00014bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014c30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014c20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014c80: 0a7c 6f35 203a 2049 6465 616c 206f 6620 .|o5 : Ideal of │ │ │ │ -00014c90: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00014c70: 2020 2020 7c0a 7c6f 3520 3a20 4964 6561 |.|o5 : Idea │ │ │ │ +00014c80: 6c20 6f66 2052 2020 2020 2020 2020 2020 l of R │ │ │ │ +00014c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 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2020 2020 2020 2020 2020 207c | │ │ │ │ -00014d70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014d60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014db0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014dc0: 0a7c 6f36 203d 207c 2078 5f30 5e32 2078 .|o6 = | x_0^2 x │ │ │ │ -00014dd0: 5f32 5e33 207c 2020 2020 2020 2020 2020 _2^3 | │ │ │ │ +00014db0: 2020 2020 7c0a 7c6f 3620 3d20 7c20 785f |.|o6 = | x_ │ │ │ │ +00014dc0: 305e 3220 785f 325e 3320 7c20 2020 2020 0^2 x_2^3 | │ │ │ │ +00014dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014e10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014e00: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014e50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014e60: 0a7c 2020 2020 2020 2020 2020 2020 2031 .| 1 │ │ │ │ -00014e70: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +00014e50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014e60: 2020 2020 3120 2020 2020 2032 2020 2020 1 2 │ │ │ │ +00014e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ea0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014eb0: 0a7c 6f36 203a 204d 6174 7269 7820 5220 .|o6 : Matrix R │ │ │ │ -00014ec0: 203c 2d2d 2052 2020 2020 2020 2020 2020 <-- R │ │ │ │ +00014ea0: 2020 2020 7c0a 7c6f 3620 3a20 4d61 7472 |.|o6 : Matr │ │ │ │ +00014eb0: 6978 2052 2020 3c2d 2d20 5220 2020 2020 ix R <-- R │ │ │ │ +00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ef0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014f00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014ef0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00014f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00014f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00014f50: 0a7c 6937 203a 2052 6261 7220 3d20 522f .|i7 : Rbar = R/ │ │ │ │ -00014f60: 4920 2020 2020 2020 2020 2020 2020 2020 I │ │ │ │ +00014f40: 2d2d 2d2d 2b0a 7c69 3720 3a20 5262 6172 ----+.|i7 : Rbar │ │ │ │ +00014f50: 203d 2052 2f49 2020 2020 2020 2020 2020 = R/I │ │ │ │ +00014f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014fa0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014f90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00014fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014fe0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014ff0: 0a7c 6f37 203d 2052 6261 7220 2020 2020 .|o7 = Rbar │ │ │ │ +00014fe0: 2020 2020 7c0a 7c6f 3720 3d20 5262 6172 |.|o7 = Rbar │ │ │ │ +00014ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015030: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015040: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015030: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015090: 0a7c 6f37 203a 2051 756f 7469 656e 7452 .|o7 : QuotientR │ │ │ │ -000150a0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +00015080: 2020 2020 7c0a 7c6f 3720 3a20 5175 6f74 |.|o7 : Quot │ │ │ │ +00015090: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000150c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000150d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000150e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000150d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000150e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000150f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015130: 0a7c 6938 203a 2062 6172 203d 206d 6170 .|i8 : bar = map │ │ │ │ -00015140: 2852 6261 722c 2052 2920 2020 2020 2020 (Rbar, R) │ │ │ │ +00015120: 2d2d 2d2d 2b0a 7c69 3820 3a20 6261 7220 ----+.|i8 : bar │ │ │ │ +00015130: 3d20 6d61 7028 5262 6172 2c20 5229 2020 = map(Rbar, R) │ │ │ │ +00015140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015170: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015180: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015170: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000151b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000151c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000151d0: 0a7c 6f38 203d 206d 6170 2028 5262 6172 .|o8 = map (Rbar │ │ │ │ -000151e0: 2c20 522c 207b 7820 2c20 7820 2c20 7820 , R, {x , x , x │ │ │ │ -000151f0: 7d29 2020 2020 2020 2020 2020 2020 2020 }) │ │ │ │ +000151c0: 2020 2020 7c0a 7c6f 3820 3d20 6d61 7020 |.|o8 = map │ │ │ │ +000151d0: 2852 6261 722c 2052 2c20 7b78 202c 2078 (Rbar, R, {x , x │ │ │ │ +000151e0: 202c 2078 207d 2920 2020 2020 2020 2020 , x }) │ │ │ │ +000151f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015210: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00015230: 2020 2020 2020 2030 2020 2031 2020 2032 0 1 2 │ │ │ │ +00015210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015220: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015230: 3120 2020 3220 2020 2020 2020 2020 2020 1 2 │ │ │ │ 00015240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015260: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015270: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015260: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000152b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000152c0: 0a7c 6f38 203a 2052 696e 674d 6170 2052 .|o8 : RingMap R │ │ │ │ -000152d0: 6261 7220 3c2d 2d20 5220 2020 2020 2020 bar <-- R │ │ │ │ +000152b0: 2020 2020 7c0a 7c6f 3820 3a20 5269 6e67 |.|o8 : Ring │ │ │ │ +000152c0: 4d61 7020 5262 6172 203c 2d2d 2052 2020 Map Rbar <-- R │ │ │ │ +000152d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000152f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015300: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015300: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015360: 0a7c 6939 203a 204d 6261 7220 3d20 7072 .|i9 : Mbar = pr │ │ │ │ -00015370: 756e 6520 636f 6b65 7220 7261 6e64 6f6d une coker random │ │ │ │ -00015380: 2852 6261 725e 312c 2052 6261 725e 7b2d (Rbar^1, Rbar^{- │ │ │ │ -00015390: 327d 2920 2020 2020 2020 2020 2020 2020 2}) │ │ │ │ -000153a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000153b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015350: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d62 6172 ----+.|i9 : Mbar │ │ │ │ +00015360: 203d 2070 7275 6e65 2063 6f6b 6572 2072 = prune coker r │ │ │ │ +00015370: 616e 646f 6d28 5262 6172 5e31 2c20 5262 andom(Rbar^1, Rb │ │ │ │ +00015380: 6172 5e7b 2d32 7d29 2020 2020 2020 2020 ar^{-2}) │ │ │ │ +00015390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000153a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000153e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015400: 0a7c 6f39 203d 2063 6f6b 6572 6e65 6c20 .|o9 = cokernel │ │ │ │ -00015410: 7c20 785f 3078 5f31 2b32 3478 5f31 5e32 | x_0x_1+24x_1^2 │ │ │ │ -00015420: 2b34 3978 5f30 785f 322b 3378 5f31 785f +49x_0x_2+3x_1x_ │ │ │ │ -00015430: 322b 3578 5f32 5e32 207c 2020 2020 2020 2+5x_2^2 | │ │ │ │ -00015440: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015450: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000153f0: 2020 2020 7c0a 7c6f 3920 3d20 636f 6b65 |.|o9 = coke │ │ │ │ +00015400: 726e 656c 207c 2078 5f30 785f 312b 3234 rnel | x_0x_1+24 │ │ │ │ +00015410: 785f 315e 322b 3439 785f 3078 5f32 2b33 x_1^2+49x_0x_2+3 │ │ │ │ +00015420: 785f 3178 5f32 2b35 785f 325e 3220 7c20 x_1x_2+5x_2^2 | │ │ │ │ +00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015440: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00015450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015490: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000154b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154c0: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ +00015490: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000154a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000154b0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +000154c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000154d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000154e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000154f0: 0a7c 6f39 203a 2052 6261 722d 6d6f 6475 .|o9 : Rbar-modu │ │ │ │ -00015500: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ -00015510: 5262 6172 2020 2020 2020 2020 2020 2020 Rbar │ │ │ │ +000154e0: 2020 2020 7c0a 7c6f 3920 3a20 5262 6172 |.|o9 : Rbar │ │ │ │ +000154f0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ +00015500: 7420 6f66 2052 6261 7220 2020 2020 2020 t of Rbar │ │ │ │ +00015510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015530: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015540: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00015530: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00015540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00015580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00015590: 0a7c 6931 3020 3a20 2864 302c 6431 2920 .|i10 : (d0,d1) │ │ │ │ -000155a0: 3d20 4569 7365 6e62 7564 5368 616d 6173 = EisenbudShamas │ │ │ │ -000155b0: 6854 6f74 616c 284d 6261 722c 4772 6164 hTotal(Mbar,Grad │ │ │ │ -000155c0: 696e 6720 3d3e 3129 2020 2020 2020 2020 ing =>1) │ │ │ │ -000155d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000155e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00015580: 2d2d 2d2d 2b0a 7c69 3130 203a 2028 6430 ----+.|i10 : (d0 │ │ │ │ +00015590: 2c64 3129 203d 2045 6973 656e 6275 6453 ,d1) = EisenbudS │ │ │ │ +000155a0: 6861 6d61 7368 546f 7461 6c28 4d62 6172 hamashTotal(Mbar │ │ │ │ +000155b0: 2c47 7261 6469 6e67 203d 3e31 2920 2020 ,Grading =>1) │ │ │ │ +000155c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000155d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000155e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000155f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015620: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015630: 0a7c 6f31 3020 3d20 287b 2d32 7d20 7c20 .|o10 = ({-2} | │ │ │ │ -00015640: 785f 305e 3220 2020 2020 2020 2020 2020 x_0^2 │ │ │ │ -00015650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015660: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ -00015670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015680: 0a7c 2020 2020 2020 207b 2d32 7d20 7c20 .| {-2} | │ │ │ │ -00015690: 785f 3078 5f31 2b32 3478 5f31 5e32 2b34 x_0x_1+24x_1^2+4 │ │ │ │ -000156a0: 3978 5f30 785f 322b 3378 5f31 785f 322b 9x_0x_2+3x_1x_2+ │ │ │ │ -000156b0: 3578 5f32 5e32 2033 3073 5f30 2020 2020 5x_2^2 30s_0 │ │ │ │ -000156c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000156d0: 0a7c 2020 2020 2020 207b 2d33 7d20 7c20 .| {-3} | │ │ │ │ -000156e0: 785f 325e 3320 2020 2020 2020 2020 2020 x_2^3 │ │ │ │ -000156f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015700: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ -00015710: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015720: 0a7c 2020 2020 2020 207b 2d37 7d20 7c20 .| {-7} | │ │ │ │ -00015730: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00015740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015750: 2020 2020 2020 2078 5f32 5e33 2020 2020 x_2^3 │ │ │ │ -00015760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00015770: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015620: 2020 2020 7c0a 7c6f 3130 203d 2028 7b2d |.|o10 = ({- │ │ │ │ +00015630: 327d 207c 2078 5f30 5e32 2020 2020 2020 2} | x_0^2 │ │ │ │ +00015640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015650: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015670: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +00015680: 327d 207c 2078 5f30 785f 312b 3234 785f 2} | x_0x_1+24x_ │ │ │ │ +00015690: 315e 322b 3439 785f 3078 5f32 2b33 785f 1^2+49x_0x_2+3x_ │ │ │ │ +000156a0: 3178 5f32 2b35 785f 325e 3220 3330 735f 1x_2+5x_2^2 30s_ │ │ │ │ +000156b0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +000156c0: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +000156d0: 337d 207c 2078 5f32 5e33 2020 2020 2020 3} | x_2^3 │ │ │ │ +000156e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000156f0: 2020 2020 2020 2020 2020 2020 3020 2020 0 │ │ │ │ +00015700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015710: 2020 2020 7c0a 7c20 2020 2020 2020 7b2d |.| {- │ │ │ │ +00015720: 377d 207c 2030 2020 2020 2020 2020 2020 7} | 0 │ │ │ │ +00015730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015740: 2020 2020 2020 2020 2020 2020 785f 325e x_2^ │ │ │ │ +00015750: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00015760: 2020 2020 7c0a 7c20 2020 2020 202d 2d2d |.| --- │ │ │ │ +00015770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000157a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000157b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -000157c0: 0a7c 2020 2020 2020 2d73 5f31 2020 2020 .| -s_1 │ │ │ │ +000157b0: 2d2d 2d2d 7c0a 7c20 2020 2020 202d 735f ----|.| -s_ │ │ │ │ +000157c0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 000157d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000157f0: 2020 3020 2020 2020 2020 207c 2c20 7b30 0 |, {0 │ │ │ │ -00015800: 7d20 207c 2020 2020 2020 2020 2020 207c } | | │ │ │ │ -00015810: 0a7c 2020 2020 2020 3020 2020 2020 2020 .| 0 │ │ │ │ +000157e0: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +000157f0: 7c2c 207b 307d 2020 7c20 2020 2020 2020 |, {0} | │ │ │ │ +00015800: 2020 2020 7c0a 7c20 2020 2020 2030 2020 |.| 0 │ │ │ │ +00015810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015840: 2020 2d73 5f31 2020 2020 207c 2020 7b2d -s_1 | {- │ │ │ │ -00015850: 347d 207c 2020 2020 2020 2020 2020 207c 4} | | │ │ │ │ -00015860: 0a7c 2020 2020 2020 735f 3020 2020 2020 .| s_0 │ │ │ │ +00015830: 2020 2020 2020 202d 735f 3120 2020 2020 -s_1 │ │ │ │ +00015840: 7c20 207b 2d34 7d20 7c20 2020 2020 2020 | {-4} | │ │ │ │ +00015850: 2020 2020 7c0a 7c20 2020 2020 2073 5f30 |.| s_0 │ │ │ │ +00015860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015890: 2020 3020 2020 2020 2020 207c 2020 7b2d 0 | {- │ │ │ │ -000158a0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -000158b0: 0a7c 2020 2020 2020 3337 785f 3078 5f31 .| 37x_0x_1 │ │ │ │ -000158c0: 2d32 3178 5f31 5e32 2d35 785f 3078 5f32 -21x_1^2-5x_0x_2 │ │ │ │ -000158d0: 2b31 3078 5f31 785f 322d 3137 785f 325e +10x_1x_2-17x_2^ │ │ │ │ -000158e0: 3220 2d33 3778 5f30 5e32 207c 2020 7b2d 2 -37x_0^2 | {- │ │ │ │ -000158f0: 357d 207c 2020 2020 2020 2020 2020 207c 5} | | │ │ │ │ -00015900: 0a7c 2020 2020 2020 2d2d 2d2d 2d2d 2d2d .| -------- │ │ │ │ +00015880: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +00015890: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | {-5} | │ │ │ │ +000158a0: 2020 2020 7c0a 7c20 2020 2020 2033 3778 |.| 37x │ │ │ │ +000158b0: 5f30 785f 312d 3231 785f 315e 322d 3578 _0x_1-21x_1^2-5x │ │ │ │ +000158c0: 5f30 785f 322b 3130 785f 3178 5f32 2d31 _0x_2+10x_1x_2-1 │ │ │ │ +000158d0: 3778 5f32 5e32 202d 3337 785f 305e 3220 7x_2^2 -37x_0^2 │ │ │ │ +000158e0: 7c20 207b 2d35 7d20 7c20 2020 2020 2020 | 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---------------- │ │ │ │ -00016d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00016da0: 0a7c 6931 3620 3a20 5362 6172 203d 2053 .|i16 : Sbar = S │ │ │ │ -00016db0: 2f49 5320 2020 2020 2020 2020 2020 2020 /IS │ │ │ │ +00016d90: 2d2d 2d2d 2b0a 7c69 3136 203a 2053 6261 ----+.|i16 : Sba │ │ │ │ +00016da0: 7220 3d20 532f 4953 2020 2020 2020 2020 r = S/IS │ │ │ │ +00016db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016de0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016df0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016de0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016e40: 0a7c 6f31 3620 3d20 5362 6172 2020 2020 .|o16 = Sbar │ │ │ │ +00016e30: 2020 2020 7c0a 7c6f 3136 203d 2053 6261 |.|o16 = Sba │ │ │ │ +00016e40: 7220 2020 2020 2020 2020 2020 2020 2020 r │ │ │ │ 00016e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016e80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016e90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016e80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016ed0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016ee0: 0a7c 6f31 3620 3a20 5175 6f74 6965 6e74 .|o16 : Quotient │ │ │ │ -00016ef0: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +00016ed0: 2020 2020 7c0a 7c6f 3136 203a 2051 756f |.|o16 : Quo │ │ │ │ +00016ee0: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +00016ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016f20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016f30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00016f20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00016f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00016f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00016f80: 0a7c 6931 3720 3a20 534d 6261 7220 3d20 .|i17 : SMbar = │ │ │ │ -00016f90: 5362 6172 2a2a 4d62 6172 2020 2020 2020 Sbar**Mbar │ │ │ │ +00016f70: 2d2d 2d2d 2b0a 7c69 3137 203a 2053 4d62 ----+.|i17 : SMb │ │ │ │ +00016f80: 6172 203d 2053 6261 722a 2a4d 6261 7220 ar = Sbar**Mbar │ │ │ │ +00016f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00016fc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00016fd0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00016fc0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00016fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017020: 0a7c 6f31 3720 3d20 636f 6b65 726e 656c .|o17 = cokernel │ │ │ │ -00017030: 207c 2078 5f30 785f 312b 3234 785f 315e | x_0x_1+24x_1^ │ │ │ │ -00017040: 322b 3439 785f 3078 5f32 2b33 785f 3178 2+49x_0x_2+3x_1x │ │ │ │ -00017050: 5f32 2b35 785f 325e 3220 7c20 2020 2020 _2+5x_2^2 | │ │ │ │ -00017060: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017070: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00017010: 2020 2020 7c0a 7c6f 3137 203d 2063 6f6b |.|o17 = cok │ │ │ │ +00017020: 6572 6e65 6c20 7c20 785f 3078 5f31 2b32 ernel | x_0x_1+2 │ │ │ │ +00017030: 3478 5f31 5e32 2b34 3978 5f30 785f 322b 4x_1^2+49x_0x_2+ │ │ │ │ +00017040: 3378 5f31 785f 322b 3578 5f32 5e32 207c 3x_1x_2+5x_2^2 | │ │ │ │ +00017050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017060: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00017070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000170a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000170b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000170c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000170d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000170e0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +000170b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000170c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000170d0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +000170e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000170f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017100: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017110: 0a7c 6f31 3720 3a20 5362 6172 2d6d 6f64 .|o17 : Sbar-mod │ │ │ │ -00017120: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ -00017130: 2053 6261 7220 2020 2020 2020 2020 2020 Sbar │ │ │ │ +00017100: 2020 2020 7c0a 7c6f 3137 203a 2053 6261 |.|o17 : Sba │ │ │ │ +00017110: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ +00017120: 6e74 206f 6620 5362 6172 2020 2020 2020 nt of Sbar │ │ │ │ +00017130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017150: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00017160: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00017150: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00017160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000171a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000171b0: 0a0a 486f 6d28 6430 2c53 6261 7229 2061 ..Hom(d0,Sbar) a │ │ │ │ -000171c0: 6e64 2048 6f6d 2864 312c 5362 6172 2920 nd Hom(d1,Sbar) │ │ │ │ -000171d0: 746f 6765 7468 6572 2066 6f72 6d20 7468 together form th │ │ │ │ -000171e0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ -000171f0: 4d62 6172 3b20 7468 7573 2074 6865 0a68 Mbar; thus the.h │ │ │ │ -00017200: 6f6d 6f6c 6f67 7920 6f66 206f 6e65 2063 omology of one c │ │ │ │ -00017210: 6f6d 706f 7369 7469 6f6e 2069 7320 302c omposition is 0, │ │ │ │ -00017220: 2077 6869 6c65 2074 6865 206f 7468 6572 while the other │ │ │ │ -00017230: 2069 7320 4d62 6172 0a0a 2b2d 2d2d 2d2d is Mbar..+----- │ │ │ │ +000171a0: 2d2d 2d2d 2b0a 0a48 6f6d 2864 302c 5362 ----+..Hom(d0,Sb │ │ │ │ +000171b0: 6172 2920 616e 6420 486f 6d28 6431 2c53 ar) and Hom(d1,S │ │ │ │ +000171c0: 6261 7229 2074 6f67 6574 6865 7220 666f bar) together fo │ │ │ │ +000171d0: 726d 2074 6865 2072 6573 6f6c 7574 696f rm the resolutio │ │ │ │ +000171e0: 6e20 6f66 204d 6261 723b 2074 6875 7320 n of Mbar; thus │ │ │ │ +000171f0: 7468 650a 686f 6d6f 6c6f 6779 206f 6620 the.homology of │ │ │ │ +00017200: 6f6e 6520 636f 6d70 6f73 6974 696f 6e20 one composition │ │ │ │ +00017210: 6973 2030 2c20 7768 696c 6520 7468 6520 is 0, while the │ │ │ │ +00017220: 6f74 6865 7220 6973 204d 6261 720a 0a2b other is Mbar..+ │ │ │ │ +00017230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017280: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ -00017290: 2070 7275 6e65 2048 485f 3120 636f 6d70 prune HH_1 comp │ │ │ │ -000172a0: 6c65 787b 6475 616c 2028 5362 6172 2a2a lex{dual (Sbar** │ │ │ │ -000172b0: 6430 292c 2064 7561 6c28 5362 6172 2a2a d0), dual(Sbar** │ │ │ │ -000172c0: 6431 297d 203d 3d20 3020 2020 2020 2020 d1)} == 0 │ │ │ │ -000172d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00017280: 6931 3820 3a20 7072 756e 6520 4848 5f31 i18 : prune HH_1 │ │ │ │ +00017290: 2063 6f6d 706c 6578 7b64 7561 6c20 2853 complex{dual (S │ │ │ │ +000172a0: 6261 722a 2a64 3029 2c20 6475 616c 2853 bar**d0), dual(S │ │ │ │ +000172b0: 6261 722a 2a64 3129 7d20 3d3d 2030 2020 bar**d1)} == 0 │ │ │ │ +000172c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000172d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000172f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017320: 2020 2020 2020 2020 7c0a 7c6f 3138 203d |.|o18 = │ │ │ │ -00017330: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017310: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017320: 6f31 3820 3d20 7472 7565 2020 2020 2020 o18 = true │ │ │ │ +00017330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017370: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017360: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000173a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000173c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a --------+.|i19 : │ │ │ │ -000173d0: 204d 6261 7227 203d 2053 6261 725e 312f Mbar' = Sbar^1/ │ │ │ │ -000173e0: 2853 6261 725f 302c 2053 6261 725f 3129 (Sbar_0, Sbar_1) │ │ │ │ -000173f0: 2a2a 534d 6261 7220 2020 2020 2020 2020 **SMbar │ │ │ │ -00017400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017410: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000173b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000173c0: 6931 3920 3a20 4d62 6172 2720 3d20 5362 i19 : Mbar' = Sb │ │ │ │ +000173d0: 6172 5e31 2f28 5362 6172 5f30 2c20 5362 ar^1/(Sbar_0, Sb │ │ │ │ +000173e0: 6172 5f31 292a 2a53 4d62 6172 2020 2020 ar_1)**SMbar │ │ │ │ +000173f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00017400: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017460: 2020 2020 2020 2020 7c0a 7c6f 3139 203d |.|o19 = │ │ │ │ -00017470: 2063 6f6b 6572 6e65 6c20 7c20 785f 3078 cokernel | x_0x │ │ │ │ -00017480: 5f31 2b32 3478 5f31 5e32 2b34 3978 5f30 _1+24x_1^2+49x_0 │ │ │ │ -00017490: 785f 322b 3378 5f31 785f 322b 3578 5f32 x_2+3x_1x_2+5x_2 │ │ │ │ -000174a0: 5e32 2073 5f30 2073 5f31 207c 2020 2020 ^2 s_0 s_1 | │ │ │ │ -000174b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017460: 6f31 3920 3d20 636f 6b65 726e 656c 207c o19 = cokernel | │ │ │ │ +00017470: 2078 5f30 785f 312b 3234 785f 315e 322b x_0x_1+24x_1^2+ │ │ │ │ +00017480: 3439 785f 3078 5f32 2b33 785f 3178 5f32 49x_0x_2+3x_1x_2 │ │ │ │ +00017490: 2b35 785f 325e 3220 735f 3020 735f 3120 +5x_2^2 s_0 s_1 │ │ │ │ +000174a0: 7c20 2020 2020 2020 2020 2020 207c 0a7c | |.| │ │ │ │ +000174b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000174e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000174f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017500: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000174f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017520: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +00017520: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00017530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017550: 2020 2020 2020 2020 7c0a 7c6f 3139 203a |.|o19 : │ │ │ │ -00017560: 2053 6261 722d 6d6f 6475 6c65 2c20 7175 Sbar-module, qu │ │ │ │ -00017570: 6f74 6965 6e74 206f 6620 5362 6172 2020 otient of Sbar │ │ │ │ +00017540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017550: 6f31 3920 3a20 5362 6172 2d6d 6f64 756c o19 : Sbar-modul │ │ │ │ +00017560: 652c 2071 756f 7469 656e 7420 6f66 2053 e, quotient of S │ │ │ │ +00017570: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 00017580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000175a0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017590: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000175a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000175d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000175f0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ -00017600: 2069 6465 616c 2070 7265 7365 6e74 6174 ideal presentat │ │ │ │ -00017610: 696f 6e20 7072 756e 6520 4848 5f31 2063 ion prune HH_1 c │ │ │ │ -00017620: 6f6d 706c 6578 7b64 7561 6c20 2853 6261 omplex{dual (Sba │ │ │ │ -00017630: 722a 2a64 3129 2c20 6475 616c 2853 6261 r**d1), dual(Sba │ │ │ │ -00017640: 722a 2a64 3029 7d20 7c0a 7c20 2020 2020 r**d0)} |.| │ │ │ │ +000175e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000175f0: 6932 3020 3a20 6964 6561 6c20 7072 6573 i20 : ideal pres │ │ │ │ +00017600: 656e 7461 7469 6f6e 2070 7275 6e65 2048 entation prune H │ │ │ │ +00017610: 485f 3120 636f 6d70 6c65 787b 6475 616c H_1 complex{dual │ │ │ │ +00017620: 2028 5362 6172 2a2a 6431 292c 2064 7561 (Sbar**d1), dua │ │ │ │ +00017630: 6c28 5362 6172 2a2a 6430 297d 207c 0a7c l(Sbar**d0)} |.| │ │ │ │ +00017640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017690: 2020 2020 2020 2020 7c0a 7c6f 3230 203d |.|o20 = │ │ │ │ -000176a0: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ +00017680: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017690: 6f32 3020 3d20 7472 7565 2020 2020 2020 o20 = true │ │ │ │ +000176a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000176c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000176e0: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2d2d |.|----- │ │ │ │ +000176d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000176e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000176f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017730: 2d2d 2d2d 2d2d 2d2d 7c0a 7c3d 3d20 6964 --------|.|== id │ │ │ │ -00017740: 6561 6c20 7072 6573 656e 7461 7469 6f6e eal presentation │ │ │ │ -00017750: 204d 6261 7227 2020 2020 2020 2020 2020 Mbar' │ │ │ │ +00017720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00017730: 3d3d 2069 6465 616c 2070 7265 7365 6e74 == ideal present │ │ │ │ +00017740: 6174 696f 6e20 4d62 6172 2720 2020 2020 ation Mbar' │ │ │ │ +00017750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00017770: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00017780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000177b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000177d0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -000177e0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -000177f0: 2a20 2a6e 6f74 6520 4578 743a 2028 4d61 * *note Ext: (Ma │ │ │ │ -00017800: 6361 756c 6179 3244 6f63 2945 7874 2c20 caulay2Doc)Ext, │ │ │ │ -00017810: 2d2d 2063 6f6d 7075 7465 2061 6e20 4578 -- compute an Ex │ │ │ │ -00017820: 7420 6d6f 6475 6c65 0a20 202a 202a 6e6f t module. * *no │ │ │ │ -00017830: 7465 206e 6577 4578 743a 206e 6577 4578 te newExt: newEx │ │ │ │ -00017840: 742c 202d 2d20 476c 6f62 616c 2045 7874 t, -- Global Ext │ │ │ │ -00017850: 2066 6f72 206d 6f64 756c 6573 206f 7665 for modules ove │ │ │ │ -00017860: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00017870: 2049 6e74 6572 7365 6374 696f 6e0a 2020 Intersection. │ │ │ │ -00017880: 2a20 2a6e 6f74 6520 6d61 6b65 486f 6d6f * *note makeHomo │ │ │ │ -00017890: 746f 7069 6573 3a20 6d61 6b65 486f 6d6f topies: makeHomo │ │ │ │ -000178a0: 746f 7069 6573 2c20 2d2d 2072 6574 7572 topies, -- retur │ │ │ │ -000178b0: 6e73 2061 2073 7973 7465 6d20 6f66 2068 ns a system of h │ │ │ │ -000178c0: 6967 6865 720a 2020 2020 686f 6d6f 746f igher. homoto │ │ │ │ -000178d0: 7069 6573 0a0a 5761 7973 2074 6f20 7573 pies..Ways to us │ │ │ │ -000178e0: 6520 4569 7365 6e62 7564 5368 616d 6173 e EisenbudShamas │ │ │ │ -000178f0: 6854 6f74 616c 3a0a 3d3d 3d3d 3d3d 3d3d hTotal:.======== │ │ │ │ -00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00017910: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00017920: 4569 7365 6e62 7564 5368 616d 6173 6854 EisenbudShamashT │ │ │ │ -00017930: 6f74 616c 284d 6f64 756c 6529 220a 0a46 otal(Module)"..F │ │ │ │ -00017940: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00017950: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00017960: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00017970: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -00017980: 6861 6d61 7368 546f 7461 6c3a 2045 6973 hamashTotal: Eis │ │ │ │ -00017990: 656e 6275 6453 6861 6d61 7368 546f 7461 enbudShamashTota │ │ │ │ -000179a0: 6c2c 2069 7320 6120 2a6e 6f74 6520 6d65 l, is a *note me │ │ │ │ -000179b0: 7468 6f64 0a66 756e 6374 696f 6e20 7769 thod.function wi │ │ │ │ -000179c0: 7468 206f 7074 696f 6e73 3a20 284d 6163 th options: (Mac │ │ │ │ -000179d0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -000179e0: 4675 6e63 7469 6f6e 5769 7468 4f70 7469 FunctionWithOpti │ │ │ │ -000179f0: 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ons,...--------- │ │ │ │ +000177c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +000177d0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +000177e0: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +000177f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00017800: 4578 742c 202d 2d20 636f 6d70 7574 6520 Ext, -- compute │ │ │ │ +00017810: 616e 2045 7874 206d 6f64 756c 650a 2020 an Ext module. │ │ │ │ +00017820: 2a20 2a6e 6f74 6520 6e65 7745 7874 3a20 * *note newExt: │ │ │ │ +00017830: 6e65 7745 7874 2c20 2d2d 2047 6c6f 6261 newExt, -- Globa │ │ │ │ +00017840: 6c20 4578 7420 666f 7220 6d6f 6475 6c65 l Ext for module │ │ │ │ +00017850: 7320 6f76 6572 2061 2063 6f6d 706c 6574 s over a complet │ │ │ │ +00017860: 650a 2020 2020 496e 7465 7273 6563 7469 e. Intersecti │ │ │ │ +00017870: 6f6e 0a20 202a 202a 6e6f 7465 206d 616b on. * *note mak │ │ │ │ +00017880: 6548 6f6d 6f74 6f70 6965 733a 206d 616b eHomotopies: mak │ │ │ │ +00017890: 6548 6f6d 6f74 6f70 6965 732c 202d 2d20 eHomotopies, -- │ │ │ │ +000178a0: 7265 7475 726e 7320 6120 7379 7374 656d returns a system │ │ │ │ +000178b0: 206f 6620 6869 6768 6572 0a20 2020 2068 of higher. h │ │ │ │ +000178c0: 6f6d 6f74 6f70 6965 730a 0a57 6179 7320 omotopies..Ways │ │ │ │ +000178d0: 746f 2075 7365 2045 6973 656e 6275 6453 to use EisenbudS │ │ │ │ +000178e0: 6861 6d61 7368 546f 7461 6c3a 0a3d 3d3d hamashTotal:.=== │ │ │ │ +000178f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00017900: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +00017910: 2020 2a20 2245 6973 656e 6275 6453 6861 * "EisenbudSha │ │ │ │ +00017920: 6d61 7368 546f 7461 6c28 4d6f 6475 6c65 mashTotal(Module │ │ │ │ +00017930: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00017940: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00017950: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00017960: 626a 6563 7420 2a6e 6f74 6520 4569 7365 bject *note Eise │ │ │ │ +00017970: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00017980: 3a20 4569 7365 6e62 7564 5368 616d 6173 : EisenbudShamas │ │ │ │ +00017990: 6854 6f74 616c 2c20 6973 2061 202a 6e6f hTotal, is a *no │ │ │ │ +000179a0: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ +000179b0: 6f6e 2077 6974 6820 6f70 7469 6f6e 733a on with options: │ │ │ │ +000179c0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +000179d0: 6574 686f 6446 756e 6374 696f 6e57 6974 ethodFunctionWit │ │ │ │ +000179e0: 684f 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d hOptions,...---- │ │ │ │ +000179f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017a40: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00017a50: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00017a60: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00017a70: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00017a80: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00017a90: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00017aa0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00017ab0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00017ac0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00017ad0: 323a 3531 3639 3a30 2e0a 1f0a 4669 6c65 2:5169:0....File │ │ │ │ -00017ae0: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -00017af0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00017b00: 732e 696e 666f 2c20 4e6f 6465 3a20 6576 s.info, Node: ev │ │ │ │ -00017b10: 656e 4578 744d 6f64 756c 652c 204e 6578 enExtModule, Nex │ │ │ │ -00017b20: 743a 2065 7870 6f2c 2050 7265 763a 2045 t: expo, Prev: E │ │ │ │ -00017b30: 6973 656e 6275 6453 6861 6d61 7368 546f isenbudShamashTo │ │ │ │ -00017b40: 7461 6c2c 2055 703a 2054 6f70 0a0a 6576 tal, Up: Top..ev │ │ │ │ -00017b50: 656e 4578 744d 6f64 756c 6520 2d2d 2065 enExtModule -- e │ │ │ │ -00017b60: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ -00017b70: 2a28 4d2c 6b29 206f 7665 7220 6120 636f *(M,k) over a co │ │ │ │ -00017b80: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ -00017b90: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -00017ba0: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -00017bb0: 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ing.************ │ │ │ │ +00017a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00017a40: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00017a50: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00017a60: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00017a70: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00017a80: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +00017a90: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00017aa0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00017ab0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00017ac0: 6f6e 732e 6d32 3a35 3136 393a 302e 0a1f ons.m2:5169:0... │ │ │ │ +00017ad0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00017ae0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00017af0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00017b00: 653a 2065 7665 6e45 7874 4d6f 6475 6c65 e: evenExtModule │ │ │ │ +00017b10: 2c20 4e65 7874 3a20 6578 706f 2c20 5072 , Next: expo, Pr │ │ │ │ +00017b20: 6576 3a20 4569 7365 6e62 7564 5368 616d ev: EisenbudSham │ │ │ │ +00017b30: 6173 6854 6f74 616c 2c20 5570 3a20 546f ashTotal, Up: To │ │ │ │ +00017b40: 700a 0a65 7665 6e45 7874 4d6f 6475 6c65 p..evenExtModule │ │ │ │ +00017b50: 202d 2d20 6576 656e 2070 6172 7420 6f66 -- even part of │ │ │ │ +00017b60: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ +00017b70: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ +00017b80: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +00017b90: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +00017ba0: 746f 7220 7269 6e67 0a2a 2a2a 2a2a 2a2a tor ring.******* │ │ │ │ +00017bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017be0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00017bf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00017c10: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00017c20: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -00017c30: 203d 2065 7665 6e45 7874 4d6f 6475 6c65 = evenExtModule │ │ │ │ -00017c40: 204d 0a20 202a 2049 6e70 7574 733a 0a20 M. * Inputs:. │ │ │ │ -00017c50: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ -00017c60: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -00017c70: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -00017c80: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ -00017c90: 2069 6e74 6572 7365 6374 696f 6e0a 2020 intersection. │ │ │ │ -00017ca0: 2020 2020 2020 7269 6e67 0a20 202a 202a ring. * * │ │ │ │ -00017cb0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ -00017cc0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ -00017cd0: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ -00017ce0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ -00017cf0: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ -00017d00: 202a 204f 7574 5269 6e67 203d 3e20 2e2e * OutRing => .. │ │ │ │ -00017d10: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ -00017d20: 2030 0a20 202a 204f 7574 7075 7473 3a0a 0. * Outputs:. │ │ │ │ -00017d30: 2020 2020 2020 2a20 452c 2061 202a 6e6f * E, a *no │ │ │ │ -00017d40: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -00017d50: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -00017d60: 2c20 6f76 6572 2061 2070 6f6c 796e 6f6d , over a polynom │ │ │ │ -00017d70: 6961 6c20 7269 6e67 2077 6974 680a 2020 ial ring with. │ │ │ │ -00017d80: 2020 2020 2020 6765 6e73 2069 6e20 6465 gens in de │ │ │ │ -00017d90: 6772 6565 2031 0a0a 4465 7363 7269 7074 gree 1..Descript │ │ │ │ -00017da0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00017db0: 0a45 7874 7261 6374 7320 7468 6520 6576 .Extracts the ev │ │ │ │ -00017dc0: 656e 2064 6567 7265 6520 7061 7274 2066 en degree part f │ │ │ │ -00017dd0: 726f 6d20 4578 744d 6f64 756c 6520 4d20 rom ExtModule M │ │ │ │ -00017de0: 4966 2074 6865 206f 7074 696f 6e61 6c20 If the optional │ │ │ │ -00017df0: 6172 6775 6d65 6e74 204f 7574 5269 6e67 argument OutRing │ │ │ │ -00017e00: 0a3d 3e20 5420 6973 2067 6976 656e 2c20 .=> T is given, │ │ │ │ -00017e10: 616e 6420 636c 6173 7320 5420 3d3d 3d20 and class T === │ │ │ │ -00017e20: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ -00017e30: 7468 656e 2074 6865 206f 7574 7075 7420 then the output │ │ │ │ -00017e40: 7769 6c6c 2062 6520 6120 6d6f 6475 6c65 will be a module │ │ │ │ -00017e50: 0a6f 7665 7220 542e 0a0a 2b2d 2d2d 2d2d .over T...+----- │ │ │ │ +00017c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00017c10: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00017c20: 2020 2020 4520 3d20 6576 656e 4578 744d E = evenExtM │ │ │ │ +00017c30: 6f64 756c 6520 4d0a 2020 2a20 496e 7075 odule M. * Inpu │ │ │ │ +00017c40: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061 ts:. * M, a │ │ │ │ +00017c50: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +00017c60: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +00017c70: 756c 652c 2c20 6f76 6572 2061 2063 6f6d ule,, over a com │ │ │ │ +00017c80: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +00017c90: 6f6e 0a20 2020 2020 2020 2072 696e 670a on. ring. │ │ │ │ +00017ca0: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ +00017cb0: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ +00017cc0: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ +00017cd0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ +00017ce0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ +00017cf0: 2020 2020 2020 2a20 4f75 7452 696e 6720 * OutRing │ │ │ │ +00017d00: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ +00017d10: 7661 6c75 6520 300a 2020 2a20 4f75 7470 value 0. * Outp │ │ │ │ +00017d20: 7574 733a 0a20 2020 2020 202a 2045 2c20 uts:. * E, │ │ │ │ +00017d30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +00017d40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +00017d50: 6475 6c65 2c2c 206f 7665 7220 6120 706f dule,, over a po │ │ │ │ +00017d60: 6c79 6e6f 6d69 616c 2072 696e 6720 7769 lynomial ring wi │ │ │ │ +00017d70: 7468 0a20 2020 2020 2020 2067 656e 7320 th. gens │ │ │ │ +00017d80: 696e 2064 6567 7265 6520 310a 0a44 6573 in degree 1..Des │ │ │ │ +00017d90: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00017da0: 3d3d 3d3d 0a0a 4578 7472 6163 7473 2074 ====..Extracts t │ │ │ │ +00017db0: 6865 2065 7665 6e20 6465 6772 6565 2070 he even degree p │ │ │ │ +00017dc0: 6172 7420 6672 6f6d 2045 7874 4d6f 6475 art from ExtModu │ │ │ │ +00017dd0: 6c65 204d 2049 6620 7468 6520 6f70 7469 le M If the opti │ │ │ │ +00017de0: 6f6e 616c 2061 7267 756d 656e 7420 4f75 onal argument Ou │ │ │ │ +00017df0: 7452 696e 670a 3d3e 2054 2069 7320 6769 tRing.=> T is gi │ │ │ │ +00017e00: 7665 6e2c 2061 6e64 2063 6c61 7373 2054 ven, and class T │ │ │ │ +00017e10: 203d 3d3d 2050 6f6c 796e 6f6d 6961 6c52 === PolynomialR │ │ │ │ +00017e20: 696e 672c 2074 6865 6e20 7468 6520 6f75 ing, then the ou │ │ │ │ +00017e30: 7470 7574 2077 696c 6c20 6265 2061 206d tput will be a m │ │ │ │ +00017e40: 6f64 756c 650a 6f76 6572 2054 2e0a 0a2b odule.over T...+ │ │ │ │ +00017e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017e90: 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b 3d20 ----+.|i1 : kk= │ │ │ │ -00017ea0: 5a5a 2f31 3031 2020 2020 2020 2020 2020 ZZ/101 │ │ │ │ +00017e80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ +00017e90: 206b 6b3d 205a 5a2f 3130 3120 2020 2020 kk= ZZ/101 │ │ │ │ +00017ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017ed0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00017ec0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00017ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00017f10: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +00017f00: 207c 0a7c 6f31 203d 206b 6b20 2020 2020 |.|o1 = kk │ │ │ │ +00017f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00017f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017f80: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74 |.|o1 : Quot │ │ │ │ -00017f90: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00017f70: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ +00017f80: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00017f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00017fc0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00017fb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00017fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00017ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00018000: 3220 3a20 5320 3d20 6b6b 5b78 2c79 2c7a 2 : S = kk[x,y,z │ │ │ │ -00018010: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -00018020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018030: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00017ff0: 2d2b 0a7c 6932 203a 2053 203d 206b 6b5b -+.|i2 : S = kk[ │ │ │ │ +00018000: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ +00018010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018070: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020 |.|o2 = S │ │ │ │ +00018060: 2020 2020 2020 2020 207c 0a7c 6f32 203d |.|o2 = │ │ │ │ +00018070: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00018080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000180a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000180b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000180c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000180d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000180e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000180f0: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -00018100: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00018110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018120: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000180e0: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +000180f0: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +00018100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018110: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00018120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018160: 2d2d 2d2d 2b0a 7c69 3320 3a20 4932 203d ----+.|i3 : I2 = │ │ │ │ -00018170: 2069 6465 616c 2278 332c 797a 2220 2020 ideal"x3,yz" │ │ │ │ +00018150: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ +00018160: 2049 3220 3d20 6964 6561 6c22 7833 2c79 I2 = ideal"x3,y │ │ │ │ +00018170: 7a22 2020 2020 2020 2020 2020 2020 2020 z" │ │ │ │ 00018180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000181a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018190: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000181a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000181c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000181d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000181e0: 2020 2020 2020 2020 2020 2020 3320 2020 3 │ │ │ │ +000181d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000181e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000181f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018210: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00018220: 6964 6561 6c20 2878 202c 2079 2a7a 2920 ideal (x , y*z) │ │ │ │ +00018200: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018210: 6f33 203d 2069 6465 616c 2028 7820 2c20 o3 = ideal (x , │ │ │ │ +00018220: 792a 7a29 2020 2020 2020 2020 2020 2020 y*z) │ │ │ │ 00018230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018250: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018240: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018290: 7c0a 7c6f 3320 3a20 4964 6561 6c20 6f66 |.|o3 : Ideal of │ │ │ │ -000182a0: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00018280: 2020 2020 207c 0a7c 6f33 203a 2049 6465 |.|o3 : Ide │ │ │ │ +00018290: 616c 206f 6620 5320 2020 2020 2020 2020 al of S │ │ │ │ +000182a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000182b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000182c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000182c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000182d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000182e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018300: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00018310: 5232 203d 2053 2f49 3220 2020 2020 2020 R2 = S/I2 │ │ │ │ +000182f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00018300: 6934 203a 2052 3220 3d20 532f 4932 2020 i4 : R2 = S/I2 │ │ │ │ +00018310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018340: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018330: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018380: 7c0a 7c6f 3420 3d20 5232 2020 2020 2020 |.|o4 = R2 │ │ │ │ +00018370: 2020 2020 207c 0a7c 6f34 203d 2052 3220 |.|o4 = R2 │ │ │ │ +00018380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000183a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000183b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000183c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000183d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000183f0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ -00018400: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +000183e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000183f0: 6f34 203a 2051 756f 7469 656e 7452 696e o4 : QuotientRin │ │ │ │ +00018400: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 00018410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018430: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00018420: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00018430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018470: 2b0a 7c69 3520 3a20 4d32 203d 2052 325e +.|i5 : M2 = R2^ │ │ │ │ -00018480: 312f 6964 6561 6c22 7832 2c79 2c7a 2220 1/ideal"x2,y,z" │ │ │ │ +00018460: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3220 -----+.|i5 : M2 │ │ │ │ +00018470: 3d20 5232 5e31 2f69 6465 616c 2278 322c = R2^1/ideal"x2, │ │ │ │ +00018480: 792c 7a22 2020 2020 2020 2020 2020 2020 y,z" │ │ │ │ 00018490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000184a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000184b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000184c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000184e0: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -000184f0: 636f 6b65 726e 656c 207c 2078 3220 7920 cokernel | x2 y │ │ │ │ -00018500: 7a20 7c20 2020 2020 2020 2020 2020 2020 z | │ │ │ │ -00018510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018520: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000184d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000184e0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ +000184f0: 7832 2079 207a 207c 2020 2020 2020 2020 x2 y z | │ │ │ │ +00018500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018510: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018560: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00018570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018580: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00018590: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000185a0: 3520 3a20 5232 2d6d 6f64 756c 652c 2071 5 : R2-module, q │ │ │ │ -000185b0: 756f 7469 656e 7420 6f66 2052 3220 2020 uotient of R2 │ │ │ │ -000185c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000185d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00018550: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018570: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00018580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018590: 207c 0a7c 6f35 203a 2052 322d 6d6f 6475 |.|o5 : R2-modu │ │ │ │ +000185a0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ +000185b0: 5232 2020 2020 2020 2020 2020 2020 2020 R2 │ │ │ │ +000185c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000185d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000185f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018610: 2d2d 2d2d 2b0a 7c69 3620 3a20 6265 7474 ----+.|i6 : bett │ │ │ │ -00018620: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ -00018630: 2028 4d32 2c20 4c65 6e67 7468 4c69 6d69 (M2, LengthLimi │ │ │ │ -00018640: 7420 3d3e 3130 2920 2020 2020 2020 2020 t =>10) │ │ │ │ -00018650: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018600: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +00018610: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ +00018620: 7574 696f 6e20 284d 322c 204c 656e 6774 ution (M2, Lengt │ │ │ │ +00018630: 684c 696d 6974 203d 3e31 3029 2020 2020 hLimit =>10) │ │ │ │ +00018640: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018680: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018690: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ -000186a0: 2033 2034 2020 3520 2036 2020 3720 2038 3 4 5 6 7 8 │ │ │ │ -000186b0: 2020 3920 3130 2020 2020 2020 2020 2020 9 10 │ │ │ │ -000186c0: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -000186d0: 746f 7461 6c3a 2031 2033 2035 2037 2039 total: 1 3 5 7 9 │ │ │ │ -000186e0: 2031 3120 3133 2031 3520 3137 2031 3920 11 13 15 17 19 │ │ │ │ -000186f0: 3231 2020 2020 2020 2020 2020 2020 2020 21 │ │ │ │ -00018700: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018710: 303a 2031 2032 2032 2032 2032 2020 3220 0: 1 2 2 2 2 2 │ │ │ │ -00018720: 2032 2020 3220 2032 2020 3220 2032 2020 2 2 2 2 2 │ │ │ │ -00018730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018740: 7c0a 7c20 2020 2020 2020 2020 313a 202e |.| 1: . │ │ │ │ -00018750: 2031 2033 2034 2034 2020 3420 2034 2020 1 3 4 4 4 4 │ │ │ │ -00018760: 3420 2034 2020 3420 2034 2020 2020 2020 4 4 4 4 │ │ │ │ -00018770: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018780: 2020 2020 2020 2020 323a 202e 202e 202e 2: . . . │ │ │ │ -00018790: 2031 2033 2020 3420 2034 2020 3420 2034 1 3 4 4 4 4 │ │ │ │ -000187a0: 2020 3420 2034 2020 2020 2020 2020 2020 4 4 │ │ │ │ -000187b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000187c0: 2020 2020 333a 202e 202e 202e 202e 202e 3: . . . . . │ │ │ │ -000187d0: 2020 3120 2033 2020 3420 2034 2020 3420 1 3 4 4 4 │ │ │ │ -000187e0: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -000187f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018800: 343a 202e 202e 202e 202e 202e 2020 2e20 4: . . . . . . │ │ │ │ -00018810: 202e 2020 3120 2033 2020 3420 2034 2020 . 1 3 4 4 │ │ │ │ -00018820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018830: 7c0a 7c20 2020 2020 2020 2020 353a 202e |.| 5: . │ │ │ │ -00018840: 202e 202e 202e 202e 2020 2e20 202e 2020 . . . . . . │ │ │ │ -00018850: 2e20 202e 2020 3120 2033 2020 2020 2020 . . 1 3 │ │ │ │ -00018860: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018680: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018690: 3020 3120 3220 3320 3420 2035 2020 3620 0 1 2 3 4 5 6 │ │ │ │ +000186a0: 2037 2020 3820 2039 2031 3020 2020 2020 7 8 9 10 │ │ │ │ +000186b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000186c0: 6f36 203d 2074 6f74 616c 3a20 3120 3320 o6 = total: 1 3 │ │ │ │ +000186d0: 3520 3720 3920 3131 2031 3320 3135 2031 5 7 9 11 13 15 1 │ │ │ │ +000186e0: 3720 3139 2032 3120 2020 2020 2020 2020 7 19 21 │ │ │ │ +000186f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018700: 2020 2020 2030 3a20 3120 3220 3220 3220 0: 1 2 2 2 │ │ │ │ +00018710: 3220 2032 2020 3220 2032 2020 3220 2032 2 2 2 2 2 2 │ │ │ │ +00018720: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00018730: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018740: 2031 3a20 2e20 3120 3320 3420 3420 2034 1: . 1 3 4 4 4 │ │ │ │ +00018750: 2020 3420 2034 2020 3420 2034 2020 3420 4 4 4 4 4 │ │ │ │ +00018760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018770: 207c 0a7c 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ +00018780: 2e20 2e20 2e20 3120 3320 2034 2020 3420 . . . 1 3 4 4 │ │ │ │ +00018790: 2034 2020 3420 2034 2020 3420 2020 2020 4 4 4 4 │ │ │ │ +000187a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000187b0: 2020 2020 2020 2020 2033 3a20 2e20 2e20 3: . . │ │ │ │ +000187c0: 2e20 2e20 2e20 2031 2020 3320 2034 2020 . . . 1 3 4 │ │ │ │ +000187d0: 3420 2034 2020 3420 2020 2020 2020 2020 4 4 4 │ │ │ │ +000187e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000187f0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20 4: . . . . │ │ │ │ +00018800: 2e20 202e 2020 2e20 2031 2020 3320 2034 . . . 1 3 4 │ │ │ │ +00018810: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00018820: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018830: 2035 3a20 2e20 2e20 2e20 2e20 2e20 202e 5: . . . . . . │ │ │ │ +00018840: 2020 2e20 202e 2020 2e20 2031 2020 3320 . . . 1 3 │ │ │ │ +00018850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188a0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ -000188b0: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +00018890: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000188a0: 6f36 203a 2042 6574 7469 5461 6c6c 7920 o6 : BettiTally │ │ │ │ +000188b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000188c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000188e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000188d0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000188e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000188f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018920: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d +.|i7 : E = ExtM │ │ │ │ -00018930: 6f64 756c 6520 4d32 2020 2020 2020 2020 odule M2 │ │ │ │ +00018910: 2d2d 2d2d 2d2b 0a7c 6937 203a 2045 203d -----+.|i7 : E = │ │ │ │ +00018920: 2045 7874 4d6f 6475 6c65 204d 3220 2020 ExtModule M2 │ │ │ │ +00018930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018950: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018950: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018990: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000189a0: 2020 2020 2020 2020 2020 2020 3820 2020 8 │ │ │ │ +00018980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000189a0: 2038 2020 2020 2020 2020 2020 2020 2020 8 │ │ │ │ 000189b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000189d0: 2020 2020 7c0a 7c6f 3720 3d20 286b 6b5b |.|o7 = (kk[ │ │ │ │ -000189e0: 5820 2e2e 5820 5d29 2020 2020 2020 2020 X ..X ]) │ │ │ │ +000189c0: 2020 2020 2020 2020 207c 0a7c 6f37 203d |.|o7 = │ │ │ │ +000189d0: 2028 6b6b 5b58 202e 2e58 205d 2920 2020 (kk[X ..X ]) │ │ │ │ +000189e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000189f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a10: 7c0a 7c20 2020 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ -00018a20: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00018a00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018a10: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +00018a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00018a40: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018a80: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20 |.|o7 : │ │ │ │ -00018a90: 6b6b 5b58 202e 2e58 205d 2d6d 6f64 756c kk[X ..X ]-modul │ │ │ │ -00018aa0: 652c 2066 7265 652c 2064 6567 7265 6573 e, free, degrees │ │ │ │ -00018ab0: 207b 302e 2e31 2c20 323a 312c 2033 3a32 {0..1, 2:1, 3:2 │ │ │ │ -00018ac0: 2c20 337d 7c0a 7c20 2020 2020 2020 2020 , 3}|.| │ │ │ │ -00018ad0: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00018a70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018a80: 6f37 203a 206b 6b5b 5820 2e2e 5820 5d2d o7 : kk[X ..X ]- │ │ │ │ +00018a90: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465 module, free, de │ │ │ │ +00018aa0: 6772 6565 7320 7b30 2e2e 312c 2032 3a31 grees {0..1, 2:1 │ │ │ │ +00018ab0: 2c20 333a 322c 2033 7d7c 0a7c 2020 2020 , 3:2, 3}|.| │ │ │ │ +00018ac0: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00018ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018b00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00018af0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00018b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00018b40: 3820 3a20 6170 706c 7928 746f 4c69 7374 8 : apply(toList │ │ │ │ -00018b50: 2830 2e2e 3130 292c 2069 2d3e 6869 6c62 (0..10), i->hilb │ │ │ │ -00018b60: 6572 7446 756e 6374 696f 6e28 692c 2045 ertFunction(i, E │ │ │ │ -00018b70: 2929 2020 2020 2020 7c0a 7c20 2020 2020 )) |.| │ │ │ │ +00018b30: 2d2b 0a7c 6938 203a 2061 7070 6c79 2874 -+.|i8 : apply(t │ │ │ │ +00018b40: 6f4c 6973 7428 302e 2e31 3029 2c20 692d oList(0..10), i- │ │ │ │ +00018b50: 3e68 696c 6265 7274 4675 6e63 7469 6f6e >hilbertFunction │ │ │ │ +00018b60: 2869 2c20 4529 2920 2020 2020 207c 0a7c (i, E)) |.| │ │ │ │ +00018b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018bb0: 2020 2020 7c0a 7c6f 3820 3d20 7b31 2c20 |.|o8 = {1, │ │ │ │ -00018bc0: 332c 2035 2c20 372c 2039 2c20 3131 2c20 3, 5, 7, 9, 11, │ │ │ │ -00018bd0: 3133 2c20 3135 2c20 3137 2c20 3139 2c20 13, 15, 17, 19, │ │ │ │ -00018be0: 3231 7d20 2020 2020 2020 2020 2020 2020 21} │ │ │ │ -00018bf0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018ba0: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ +00018bb0: 207b 312c 2033 2c20 352c 2037 2c20 392c {1, 3, 5, 7, 9, │ │ │ │ +00018bc0: 2031 312c 2031 332c 2031 352c 2031 372c 11, 13, 15, 17, │ │ │ │ +00018bd0: 2031 392c 2032 317d 2020 2020 2020 2020 19, 21} │ │ │ │ +00018be0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c20: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00018c30: 3820 3a20 4c69 7374 2020 2020 2020 2020 8 : List │ │ │ │ +00018c20: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ +00018c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018c60: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00018c50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00018c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018ca0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4565 7665 ----+.|i9 : Eeve │ │ │ │ -00018cb0: 6e20 3d20 6576 656e 4578 744d 6f64 756c n = evenExtModul │ │ │ │ -00018cc0: 6520 4d32 2020 2020 2020 2020 2020 2020 e M2 │ │ │ │ -00018cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018ce0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018c90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ +00018ca0: 2045 6576 656e 203d 2065 7665 6e45 7874 Eeven = evenExt │ │ │ │ +00018cb0: 4d6f 6475 6c65 204d 3220 2020 2020 2020 Module M2 │ │ │ │ +00018cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018cd0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00018d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d30: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00018d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d50: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ -00018d60: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020 (kk[X ..X ]) │ │ │ │ +00018d10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018d20: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ +00018d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018d50: 6f39 203d 2028 6b6b 5b58 202e 2e58 205d o9 = (kk[X ..X ] │ │ │ │ +00018d60: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00018d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018d90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00018da0: 2030 2020 2031 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00018d80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018d90: 2020 2020 2020 3020 2020 3120 2020 2020 0 1 │ │ │ │ +00018da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00018dc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00018dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00018e10: 3920 3a20 6b6b 5b58 202e 2e58 205d 2d6d 9 : kk[X ..X ]-m │ │ │ │ -00018e20: 6f64 756c 652c 2066 7265 652c 2064 6567 odule, free, deg │ │ │ │ -00018e30: 7265 6573 207b 302e 2e31 2c20 323a 317d rees {0..1, 2:1} │ │ │ │ -00018e40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00018e50: 2020 2020 3020 2020 3120 2020 2020 2020 0 1 │ │ │ │ +00018e00: 207c 0a7c 6f39 203a 206b 6b5b 5820 2e2e |.|o9 : kk[X .. │ │ │ │ +00018e10: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565 X ]-module, free │ │ │ │ +00018e20: 2c20 6465 6772 6565 7320 7b30 2e2e 312c , degrees {0..1, │ │ │ │ +00018e30: 2032 3a31 7d20 2020 2020 2020 207c 0a7c 2:1} |.| │ │ │ │ +00018e40: 2020 2020 2020 2020 2030 2020 2031 2020 0 1 │ │ │ │ +00018e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018e80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00018e70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00018e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00018ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00018ec0: 2b0a 7c69 3130 203a 2061 7070 6c79 2874 +.|i10 : apply(t │ │ │ │ -00018ed0: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e oList(0..5), i-> │ │ │ │ -00018ee0: 6869 6c62 6572 7446 756e 6374 696f 6e28 hilbertFunction( │ │ │ │ -00018ef0: 692c 2045 6576 656e 2929 2020 7c0a 7c20 i, Eeven)) |.| │ │ │ │ +00018eb0: 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 6170 -----+.|i10 : ap │ │ │ │ +00018ec0: 706c 7928 746f 4c69 7374 2830 2e2e 3529 ply(toList(0..5) │ │ │ │ +00018ed0: 2c20 692d 3e68 696c 6265 7274 4675 6e63 , i->hilbertFunc │ │ │ │ +00018ee0: 7469 6f6e 2869 2c20 4565 7665 6e29 2920 tion(i, Eeven)) │ │ │ │ +00018ef0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00018f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018f30: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ -00018f40: 207b 312c 2035 2c20 392c 2031 332c 2031 {1, 5, 9, 13, 1 │ │ │ │ -00018f50: 372c 2032 317d 2020 2020 2020 2020 2020 7, 21} │ │ │ │ -00018f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018f70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00018f20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00018f30: 6f31 3020 3d20 7b31 2c20 352c 2039 2c20 o10 = {1, 5, 9, │ │ │ │ +00018f40: 3133 2c20 3137 2c20 3231 7d20 2020 2020 13, 17, 21} │ │ │ │ +00018f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00018f60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00018f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fb0: 7c0a 7c6f 3130 203a 204c 6973 7420 2020 |.|o10 : List │ │ │ │ +00018fa0: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69 |.|o10 : Li │ │ │ │ +00018fb0: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ 00018fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00018fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00018fe0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00018fe0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00018ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019020: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -00019030: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -00019040: 2a20 2a6e 6f74 6520 4578 744d 6f64 756c * *note ExtModul │ │ │ │ -00019050: 653a 2045 7874 4d6f 6475 6c65 2c20 2d2d e: ExtModule, -- │ │ │ │ -00019060: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ -00019070: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -00019080: 7273 6563 7469 6f6e 2061 730a 2020 2020 rsection as. │ │ │ │ -00019090: 6d6f 6475 6c65 206f 7665 7220 4349 206f module over CI o │ │ │ │ -000190a0: 7065 7261 746f 7220 7269 6e67 0a20 202a perator ring. * │ │ │ │ -000190b0: 202a 6e6f 7465 206f 6464 4578 744d 6f64 *note oddExtMod │ │ │ │ -000190c0: 756c 653a 206f 6464 4578 744d 6f64 756c ule: oddExtModul │ │ │ │ -000190d0: 652c 202d 2d20 6f64 6420 7061 7274 206f e, -- odd part o │ │ │ │ -000190e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -000190f0: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -00019100: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ -00019110: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -00019120: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -00019130: 2a20 2a6e 6f74 6520 4f75 7452 696e 673a * *note OutRing: │ │ │ │ -00019140: 204f 7574 5269 6e67 2c20 2d2d 204f 7074 OutRing, -- Opt │ │ │ │ -00019150: 696f 6e20 616c 6c6f 7769 6e67 2073 7065 ion allowing spe │ │ │ │ -00019160: 6369 6669 6361 7469 6f6e 206f 6620 7468 cification of th │ │ │ │ -00019170: 6520 7269 6e67 206f 7665 720a 2020 2020 e ring over. │ │ │ │ -00019180: 7768 6963 6820 7468 6520 6f75 7470 7574 which the output │ │ │ │ -00019190: 2069 7320 6465 6669 6e65 640a 0a57 6179 is defined..Way │ │ │ │ -000191a0: 7320 746f 2075 7365 2065 7665 6e45 7874 s to use evenExt │ │ │ │ -000191b0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d Module:.======== │ │ │ │ -000191c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000191d0: 3d3d 0a0a 2020 2a20 2265 7665 6e45 7874 ==.. * "evenExt │ │ │ │ -000191e0: 4d6f 6475 6c65 284d 6f64 756c 6529 220a Module(Module)". │ │ │ │ -000191f0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ -00019200: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ -00019210: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ -00019220: 6374 202a 6e6f 7465 2065 7665 6e45 7874 ct *note evenExt │ │ │ │ -00019230: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -00019240: 6f64 756c 652c 2069 7320 6120 2a6e 6f74 odule, is a *not │ │ │ │ -00019250: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -00019260: 6e20 7769 7468 0a6f 7074 696f 6e73 3a20 n with.options: │ │ │ │ -00019270: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00019280: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -00019290: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00019010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00019020: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +00019030: 3d0a 0a20 202a 202a 6e6f 7465 2045 7874 =.. * *note Ext │ │ │ │ +00019040: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c Module: ExtModul │ │ │ │ +00019050: 652c 202d 2d20 4578 745e 2a28 4d2c 6b29 e, -- Ext^*(M,k) │ │ │ │ +00019060: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +00019070: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ +00019080: 0a20 2020 206d 6f64 756c 6520 6f76 6572 . module over │ │ │ │ +00019090: 2043 4920 6f70 6572 6174 6f72 2072 696e CI operator rin │ │ │ │ +000190a0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445 g. * *note oddE │ │ │ │ +000190b0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874 xtModule: oddExt │ │ │ │ +000190c0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070 Module, -- odd p │ │ │ │ +000190d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +000190e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +000190f0: 650a 2020 2020 696e 7465 7273 6563 7469 e. intersecti │ │ │ │ +00019100: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ +00019110: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +00019120: 6e67 0a20 202a 202a 6e6f 7465 204f 7574 ng. * *note Out │ │ │ │ +00019130: 5269 6e67 3a20 4f75 7452 696e 672c 202d Ring: OutRing, - │ │ │ │ +00019140: 2d20 4f70 7469 6f6e 2061 6c6c 6f77 696e - Option allowin │ │ │ │ +00019150: 6720 7370 6563 6966 6963 6174 696f 6e20 g specification │ │ │ │ +00019160: 6f66 2074 6865 2072 696e 6720 6f76 6572 of the ring over │ │ │ │ +00019170: 0a20 2020 2077 6869 6368 2074 6865 206f . which the o │ │ │ │ +00019180: 7574 7075 7420 6973 2064 6566 696e 6564 utput is defined │ │ │ │ +00019190: 0a0a 5761 7973 2074 6f20 7573 6520 6576 ..Ways to use ev │ │ │ │ +000191a0: 656e 4578 744d 6f64 756c 653a 0a3d 3d3d enExtModule:.=== │ │ │ │ +000191b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000191c0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6576 =======.. * "ev │ │ │ │ +000191d0: 656e 4578 744d 6f64 756c 6528 4d6f 6475 enExtModule(Modu │ │ │ │ +000191e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072 le)"..For the pr │ │ │ │ +000191f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00019200: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00019210: 206f 626a 6563 7420 2a6e 6f74 6520 6576 object *note ev │ │ │ │ +00019220: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +00019230: 6e45 7874 4d6f 6475 6c65 2c20 6973 2061 nExtModule, is a │ │ │ │ +00019240: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ +00019250: 6e63 7469 6f6e 2077 6974 680a 6f70 7469 nction with.opti │ │ │ │ +00019260: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00019270: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00019280: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00019290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000192c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000192e0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -000192f0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00019300: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00019310: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00019320: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00019330: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -00019340: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00019350: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00019360: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00019370: 6e73 2e6d 323a 3336 3433 3a30 2e0a 1f0a ns.m2:3643:0.... │ │ │ │ -00019380: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00019390: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000193a0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -000193b0: 3a20 6578 706f 2c20 4e65 7874 3a20 6578 : expo, Next: ex │ │ │ │ -000193c0: 7465 7269 6f72 4578 744d 6f64 756c 652c teriorExtModule, │ │ │ │ -000193d0: 2050 7265 763a 2065 7665 6e45 7874 4d6f Prev: evenExtMo │ │ │ │ -000193e0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65 dule, Up: Top..e │ │ │ │ -000193f0: 7870 6f20 2d2d 2072 6574 7572 6e73 2061 xpo -- returns a │ │ │ │ -00019400: 2073 6574 2063 6f72 7265 7370 6f6e 6469 set correspondi │ │ │ │ -00019410: 6e67 2074 6f20 7468 6520 6261 7369 7320 ng to the basis │ │ │ │ -00019420: 6f66 2061 2064 6976 6964 6564 2070 6f77 of a divided pow │ │ │ │ -00019430: 6572 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a er.************* │ │ │ │ +000192d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +000192e0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +000192f0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00019300: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00019310: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00019320: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +00019330: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00019340: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00019350: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00019360: 6c75 7469 6f6e 732e 6d32 3a33 3634 333a lutions.m2:3643: │ │ │ │ +00019370: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00019380: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00019390: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +000193a0: 204e 6f64 653a 2065 7870 6f2c 204e 6578 Node: expo, Nex │ │ │ │ +000193b0: 743a 2065 7874 6572 696f 7245 7874 4d6f t: exteriorExtMo │ │ │ │ +000193c0: 6475 6c65 2c20 5072 6576 3a20 6576 656e dule, Prev: even │ │ │ │ +000193d0: 4578 744d 6f64 756c 652c 2055 703a 2054 ExtModule, Up: T │ │ │ │ +000193e0: 6f70 0a0a 6578 706f 202d 2d20 7265 7475 op..expo -- retu │ │ │ │ +000193f0: 726e 7320 6120 7365 7420 636f 7272 6573 rns a set corres │ │ │ │ +00019400: 706f 6e64 696e 6720 746f 2074 6865 2062 ponding to the b │ │ │ │ +00019410: 6173 6973 206f 6620 6120 6469 7669 6465 asis of a divide │ │ │ │ +00019420: 6420 706f 7765 720a 2a2a 2a2a 2a2a 2a2a d power.******** │ │ │ │ +00019430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00019440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00019450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00019470: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00019480: 653a 200a 2020 2020 2020 2020 4220 3d20 e: . B = │ │ │ │ -00019490: 6578 706f 2863 2c4e 290a 2020 2020 2020 expo(c,N). │ │ │ │ -000194a0: 2020 4220 3d20 6578 706f 2863 2c4c 290a B = expo(c,L). │ │ │ │ -000194b0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -000194c0: 2020 2a20 4e2c 2061 6e20 2a6e 6f74 6520 * N, an *note │ │ │ │ -000194d0: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -000194e0: 6179 3244 6f63 295a 5a2c 2c20 0a20 2020 ay2Doc)ZZ,, . │ │ │ │ -000194f0: 2020 202a 2063 2c20 616e 202a 6e6f 7465 * c, an *note │ │ │ │ -00019500: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -00019510: 6c61 7932 446f 6329 5a5a 2c2c 200a 2020 lay2Doc)ZZ,, . │ │ │ │ -00019520: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465 * L, a *note │ │ │ │ -00019530: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ -00019540: 3244 6f63 294c 6973 742c 2c20 6f66 2063 2Doc)List,, of c │ │ │ │ -00019550: 206e 6f6e 2d6e 6567 6174 6976 6520 696e non-negative in │ │ │ │ -00019560: 7465 6765 7273 0a20 202a 204f 7574 7075 tegers. * Outpu │ │ │ │ -00019570: 7473 3a0a 2020 2020 2020 2a20 422c 2061 ts:. * B, a │ │ │ │ -00019580: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ -00019590: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ -000195a0: 2c20 7061 7274 6974 696f 6e73 2077 6974 , partitions wit │ │ │ │ -000195b0: 6820 6320 6e6f 6e2d 6e65 6761 7469 7665 h c non-negative │ │ │ │ -000195c0: 0a20 2020 2020 2020 2070 6172 7473 0a0a . parts.. │ │ │ │ -000195d0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -000195e0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 2066 6f72 =======..The for │ │ │ │ -000195f0: 6d20 6578 706f 2863 2c4e 2920 7265 7475 m expo(c,N) retu │ │ │ │ -00019600: 726e 7320 7061 7274 6974 696f 6e73 206f rns partitions o │ │ │ │ -00019610: 6620 4e20 7769 7468 2063 206e 6f6e 2d6e f N with c non-n │ │ │ │ -00019620: 6567 6174 6976 6520 7061 7274 732e 2054 egative parts. T │ │ │ │ -00019630: 6865 2066 6f72 6d0a 6578 706f 2863 2c20 he form.expo(c, │ │ │ │ -00019640: 4c29 2072 6574 7572 6e73 2070 6172 7469 L) returns parti │ │ │ │ -00019650: 7469 6f6e 7320 7769 7468 206e 6f6e 2d6e tions with non-n │ │ │ │ -00019660: 6567 6174 6976 6520 7061 7274 7320 7468 egative parts th │ │ │ │ -00019670: 6174 2061 7265 2063 6f6d 706f 6e65 6e74 at are component │ │ │ │ -00019680: 7769 7365 203c 3d0a 4c20 2861 6e64 2061 wise <=.L (and a │ │ │ │ -00019690: 6e79 2073 756d 203c 3d20 7375 6d20 4c29 ny sum <= sum L) │ │ │ │ -000196a0: 2e0a 0a54 6865 206c 6973 7420 6578 706f ...The list expo │ │ │ │ -000196b0: 2863 2c4e 2920 206d 6179 2062 6520 7468 (c,N) may be th │ │ │ │ -000196c0: 6f75 6768 7420 6f66 2061 7320 7468 6520 ought of as the │ │ │ │ -000196d0: 6c69 7374 206f 6620 6578 706f 6e65 6e74 list of exponent │ │ │ │ -000196e0: 2076 6563 746f 7273 206f 6620 7468 650a vectors of the. │ │ │ │ -000196f0: 6d6f 6e6f 6d69 616c 7320 6f66 2064 6567 monomials of deg │ │ │ │ -00019700: 7265 6520 4e20 696e 2063 2076 6172 6961 ree N in c varia │ │ │ │ -00019710: 626c 6573 2e20 5468 6973 2069 7320 7573 bles. This is us │ │ │ │ -00019720: 6564 2069 6e20 7468 6520 636f 6e73 7472 ed in the constr │ │ │ │ -00019730: 7563 7469 6f6e 206f 6620 7468 650a 4569 uction of the.Ei │ │ │ │ -00019740: 7365 6e62 7564 2d53 6861 6d61 7368 2072 senbud-Shamash r │ │ │ │ -00019750: 6573 6f6c 7574 696f 6e2e 0a0a 5468 6520 esolution...The │ │ │ │ -00019760: 6c69 7374 2065 7870 6f28 632c 204c 292c list expo(c, L), │ │ │ │ -00019770: 206f 6e20 7468 6520 6f74 6865 7220 6861 on the other ha │ │ │ │ -00019780: 6e64 2c20 6d61 7920 6265 2074 686f 7567 nd, may be thoug │ │ │ │ -00019790: 6874 206f 6620 6173 2074 6865 206c 6973 ht of as the lis │ │ │ │ -000197a0: 7420 6f66 0a64 6976 6973 6f72 7320 6f66 t of.divisors of │ │ │ │ -000197b0: 2065 5e4c 203d 2065 5f30 5e7b 4c5f 307d e^L = e_0^{L_0} │ │ │ │ -000197c0: 202e 2e2e 2065 5f63 5e7b 4c5f 637d 2e20 ... e_c^{L_c}. │ │ │ │ -000197d0: 5468 6973 2069 7320 7573 6564 2069 6e20 This is used in │ │ │ │ -000197e0: 7468 6520 636f 6e73 7472 7563 7469 6f6e the construction │ │ │ │ -000197f0: 206f 660a 7468 6520 6869 6768 6572 2068 of.the higher h │ │ │ │ -00019800: 6f6d 6f74 6f70 6965 7320 6f6e 2061 2063 omotopies on a c │ │ │ │ -00019810: 6f6d 706c 6578 2e0a 0a2b 2d2d 2d2d 2d2d omplex...+------ │ │ │ │ +00019460: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00019470: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00019480: 2042 203d 2065 7870 6f28 632c 4e29 0a20 B = expo(c,N). │ │ │ │ +00019490: 2020 2020 2020 2042 203d 2065 7870 6f28 B = expo( │ │ │ │ +000194a0: 632c 4c29 0a20 202a 2049 6e70 7574 733a c,L). * Inputs: │ │ │ │ +000194b0: 0a20 2020 2020 202a 204e 2c20 616e 202a . * N, an * │ │ │ │ +000194c0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +000194d0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +000194e0: 200a 2020 2020 2020 2a20 632c 2061 6e20 . * c, an │ │ │ │ +000194f0: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ +00019500: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +00019510: 2c20 0a20 2020 2020 202a 204c 2c20 6120 , . * L, a │ │ │ │ +00019520: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00019530: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00019540: 206f 6620 6320 6e6f 6e2d 6e65 6761 7469 of c non-negati │ │ │ │ +00019550: 7665 2069 6e74 6567 6572 730a 2020 2a20 ve integers. * │ │ │ │ +00019560: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00019570: 2042 2c20 6120 2a6e 6f74 6520 6c69 7374 B, a *note list │ │ │ │ +00019580: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00019590: 4c69 7374 2c2c 2070 6172 7469 7469 6f6e List,, partition │ │ │ │ +000195a0: 7320 7769 7468 2063 206e 6f6e 2d6e 6567 s with c non-neg │ │ │ │ +000195b0: 6174 6976 650a 2020 2020 2020 2020 7061 ative. pa │ │ │ │ +000195c0: 7274 730a 0a44 6573 6372 6970 7469 6f6e rts..Description │ │ │ │ +000195d0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ +000195e0: 6520 666f 726d 2065 7870 6f28 632c 4e29 e form expo(c,N) │ │ │ │ +000195f0: 2072 6574 7572 6e73 2070 6172 7469 7469 returns partiti │ │ │ │ +00019600: 6f6e 7320 6f66 204e 2077 6974 6820 6320 ons of N with c │ │ │ │ +00019610: 6e6f 6e2d 6e65 6761 7469 7665 2070 6172 non-negative par │ │ │ │ +00019620: 7473 2e20 5468 6520 666f 726d 0a65 7870 ts. The form.exp │ │ │ │ +00019630: 6f28 632c 204c 2920 7265 7475 726e 7320 o(c, L) returns │ │ │ │ +00019640: 7061 7274 6974 696f 6e73 2077 6974 6820 partitions with │ │ │ │ +00019650: 6e6f 6e2d 6e65 6761 7469 7665 2070 6172 non-negative par │ │ │ │ +00019660: 7473 2074 6861 7420 6172 6520 636f 6d70 ts that are comp │ │ │ │ +00019670: 6f6e 656e 7477 6973 6520 3c3d 0a4c 2028 onentwise <=.L ( │ │ │ │ +00019680: 616e 6420 616e 7920 7375 6d20 3c3d 2073 and any sum <= s │ │ │ │ +00019690: 756d 204c 292e 0a0a 5468 6520 6c69 7374 um L)...The list │ │ │ │ +000196a0: 2065 7870 6f28 632c 4e29 2020 6d61 7920 expo(c,N) may │ │ │ │ +000196b0: 6265 2074 686f 7567 6874 206f 6620 6173 be thought of as │ │ │ │ +000196c0: 2074 6865 206c 6973 7420 6f66 2065 7870 the list of exp │ │ │ │ +000196d0: 6f6e 656e 7420 7665 6374 6f72 7320 6f66 onent vectors of │ │ │ │ +000196e0: 2074 6865 0a6d 6f6e 6f6d 6961 6c73 206f the.monomials o │ │ │ │ +000196f0: 6620 6465 6772 6565 204e 2069 6e20 6320 f degree N in c │ │ │ │ +00019700: 7661 7269 6162 6c65 732e 2054 6869 7320 variables. This │ │ │ │ +00019710: 6973 2075 7365 6420 696e 2074 6865 2063 is used in the c │ │ │ │ +00019720: 6f6e 7374 7275 6374 696f 6e20 6f66 2074 onstruction of t │ │ │ │ +00019730: 6865 0a45 6973 656e 6275 642d 5368 616d he.Eisenbud-Sham │ │ │ │ +00019740: 6173 6820 7265 736f 6c75 7469 6f6e 2e0a ash resolution.. │ │ │ │ +00019750: 0a54 6865 206c 6973 7420 6578 706f 2863 .The list expo(c │ │ │ │ +00019760: 2c20 4c29 2c20 6f6e 2074 6865 206f 7468 , L), on the oth │ │ │ │ +00019770: 6572 2068 616e 642c 206d 6179 2062 6520 er hand, may be │ │ │ │ +00019780: 7468 6f75 6768 7420 6f66 2061 7320 7468 thought of as th │ │ │ │ +00019790: 6520 6c69 7374 206f 660a 6469 7669 736f e list of.diviso │ │ │ │ +000197a0: 7273 206f 6620 655e 4c20 3d20 655f 305e rs of e^L = e_0^ │ │ │ │ +000197b0: 7b4c 5f30 7d20 2e2e 2e20 655f 635e 7b4c {L_0} ... e_c^{L │ │ │ │ +000197c0: 5f63 7d2e 2054 6869 7320 6973 2075 7365 _c}. This is use │ │ │ │ +000197d0: 6420 696e 2074 6865 2063 6f6e 7374 7275 d in the constru │ │ │ │ +000197e0: 6374 696f 6e20 6f66 0a74 6865 2068 6967 ction of.the hig │ │ │ │ +000197f0: 6865 7220 686f 6d6f 746f 7069 6573 206f her homotopies o │ │ │ │ +00019800: 6e20 6120 636f 6d70 6c65 782e 0a0a 2b2d n a complex...+- │ │ │ │ +00019810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019860: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2065 -------+.|i1 : e │ │ │ │ -00019870: 7870 6f28 332c 3529 2020 2020 2020 2020 xpo(3,5) │ │ │ │ +00019850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00019860: 3120 3a20 6578 706f 2833 2c35 2920 2020 1 : expo(3,5) │ │ │ │ +00019870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000198a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000198b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000198a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000198b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000198c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000198d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000198e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000198f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019900: 2020 2020 2020 207c 0a7c 6f31 203d 207b |.|o1 = { │ │ │ │ -00019910: 7b35 2c20 302c 2030 7d2c 207b 342c 2031 {5, 0, 0}, {4, 1 │ │ │ │ -00019920: 2c20 307d 2c20 7b34 2c20 302c 2031 7d2c , 0}, {4, 0, 1}, │ │ │ │ -00019930: 207b 332c 2032 2c20 307d 2c20 7b33 2c20 {3, 2, 0}, {3, │ │ │ │ -00019940: 312c 2031 7d2c 207b 332c 2030 2c20 327d 1, 1}, {3, 0, 2} │ │ │ │ -00019950: 2c20 7b32 2c20 207c 0a7c 2020 2020 202d , {2, |.| - │ │ │ │ +000198f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019900: 3120 3d20 7b7b 352c 2030 2c20 307d 2c20 1 = {{5, 0, 0}, │ │ │ │ +00019910: 7b34 2c20 312c 2030 7d2c 207b 342c 2030 {4, 1, 0}, {4, 0 │ │ │ │ +00019920: 2c20 317d 2c20 7b33 2c20 322c 2030 7d2c , 1}, {3, 2, 0}, │ │ │ │ +00019930: 207b 332c 2031 2c20 317d 2c20 7b33 2c20 {3, 1, 1}, {3, │ │ │ │ +00019940: 302c 2032 7d2c 207b 322c 2020 7c0a 7c20 0, 2}, {2, |.| │ │ │ │ +00019950: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000199a0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2033 -------|.| 3 │ │ │ │ -000199b0: 2c20 307d 2c20 7b32 2c20 322c 2031 7d2c , 0}, {2, 2, 1}, │ │ │ │ -000199c0: 207b 322c 2031 2c20 327d 2c20 7b32 2c20 {2, 1, 2}, {2, │ │ │ │ -000199d0: 302c 2033 7d2c 207b 312c 2034 2c20 307d 0, 3}, {1, 4, 0} │ │ │ │ -000199e0: 2c20 7b31 2c20 332c 2031 7d2c 207b 312c , {1, 3, 1}, {1, │ │ │ │ -000199f0: 2032 2c20 327d 2c7c 0a7c 2020 2020 202d 2, 2},|.| - │ │ │ │ +00019990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +000199a0: 2020 2020 332c 2030 7d2c 207b 322c 2032 3, 0}, {2, 2 │ │ │ │ +000199b0: 2c20 317d 2c20 7b32 2c20 312c 2032 7d2c , 1}, {2, 1, 2}, │ │ │ │ +000199c0: 207b 322c 2030 2c20 337d 2c20 7b31 2c20 {2, 0, 3}, {1, │ │ │ │ +000199d0: 342c 2030 7d2c 207b 312c 2033 2c20 317d 4, 0}, {1, 3, 1} │ │ │ │ +000199e0: 2c20 7b31 2c20 322c 2032 7d2c 7c0a 7c20 , {1, 2, 2},|.| │ │ │ │ +000199f0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019a40: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b -------|.| { │ │ │ │ -00019a50: 312c 2031 2c20 337d 2c20 7b31 2c20 302c 1, 1, 3}, {1, 0, │ │ │ │ -00019a60: 2034 7d2c 207b 302c 2035 2c20 307d 2c20 4}, {0, 5, 0}, │ │ │ │ -00019a70: 7b30 2c20 342c 2031 7d2c 207b 302c 2033 {0, 4, 1}, {0, 3 │ │ │ │ -00019a80: 2c20 327d 2c20 7b30 2c20 322c 2033 7d2c , 2}, {0, 2, 3}, │ │ │ │ -00019a90: 207b 302c 2031 2c7c 0a7c 2020 2020 202d {0, 1,|.| - │ │ │ │ +00019a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019a40: 2020 2020 7b31 2c20 312c 2033 7d2c 207b {1, 1, 3}, { │ │ │ │ +00019a50: 312c 2030 2c20 347d 2c20 7b30 2c20 352c 1, 0, 4}, {0, 5, │ │ │ │ +00019a60: 2030 7d2c 207b 302c 2034 2c20 317d 2c20 0}, {0, 4, 1}, │ │ │ │ +00019a70: 7b30 2c20 332c 2032 7d2c 207b 302c 2032 {0, 3, 2}, {0, 2 │ │ │ │ +00019a80: 2c20 337d 2c20 7b30 2c20 312c 7c0a 7c20 , 3}, {0, 1,|.| │ │ │ │ +00019a90: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019ae0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2034 -------|.| 4 │ │ │ │ -00019af0: 7d2c 207b 302c 2030 2c20 357d 7d20 2020 }, {0, 0, 5}} │ │ │ │ +00019ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019ae0: 2020 2020 347d 2c20 7b30 2c20 302c 2035 4}, {0, 0, 5 │ │ │ │ +00019af0: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ 00019b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019b80: 2020 2020 2020 207c 0a7c 6f31 203a 204c |.|o1 : L │ │ │ │ -00019b90: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00019b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019b80: 3120 3a20 4c69 7374 2020 2020 2020 2020 1 : List │ │ │ │ +00019b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00019bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00019bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2065 -------+.|i2 : e │ │ │ │ -00019c30: 7870 6f28 332c 207b 332c 322c 317d 2920 xpo(3, {3,2,1}) │ │ │ │ +00019c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00019c20: 3220 3a20 6578 706f 2833 2c20 7b33 2c32 2 : expo(3, {3,2 │ │ │ │ +00019c30: 2c31 7d29 2020 2020 2020 2020 2020 2020 ,1}) │ │ │ │ 00019c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019cc0: 2020 2020 2020 207c 0a7c 6f32 203d 207b |.|o2 = { │ │ │ │ -00019cd0: 7b30 2c20 302c 2030 7d2c 207b 312c 2030 {0, 0, 0}, {1, 0 │ │ │ │ -00019ce0: 2c20 307d 2c20 7b30 2c20 312c 2030 7d2c , 0}, {0, 1, 0}, │ │ │ │ -00019cf0: 207b 302c 2030 2c20 317d 2c20 7b32 2c20 {0, 0, 1}, {2, │ │ │ │ -00019d00: 302c 2030 7d2c 207b 312c 2031 2c20 307d 0, 0}, {1, 1, 0} │ │ │ │ -00019d10: 2c20 7b31 2c20 207c 0a7c 2020 2020 202d , {1, |.| - │ │ │ │ +00019cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019cc0: 3220 3d20 7b7b 302c 2030 2c20 307d 2c20 2 = {{0, 0, 0}, │ │ │ │ +00019cd0: 7b31 2c20 302c 2030 7d2c 207b 302c 2031 {1, 0, 0}, {0, 1 │ │ │ │ +00019ce0: 2c20 307d 2c20 7b30 2c20 302c 2031 7d2c , 0}, {0, 0, 1}, │ │ │ │ +00019cf0: 207b 322c 2030 2c20 307d 2c20 7b31 2c20 {2, 0, 0}, {1, │ │ │ │ +00019d00: 312c 2030 7d2c 207b 312c 2020 7c0a 7c20 1, 0}, {1, |.| │ │ │ │ +00019d10: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019d60: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2030 -------|.| 0 │ │ │ │ -00019d70: 2c20 317d 2c20 7b30 2c20 322c 2030 7d2c , 1}, {0, 2, 0}, │ │ │ │ -00019d80: 207b 302c 2031 2c20 317d 2c20 7b33 2c20 {0, 1, 1}, {3, │ │ │ │ -00019d90: 302c 2030 7d2c 207b 322c 2031 2c20 307d 0, 0}, {2, 1, 0} │ │ │ │ -00019da0: 2c20 7b32 2c20 302c 2031 7d2c 207b 312c , {2, 0, 1}, {1, │ │ │ │ -00019db0: 2032 2c20 307d 2c7c 0a7c 2020 2020 202d 2, 0},|.| - │ │ │ │ +00019d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019d60: 2020 2020 302c 2031 7d2c 207b 302c 2032 0, 1}, {0, 2 │ │ │ │ +00019d70: 2c20 307d 2c20 7b30 2c20 312c 2031 7d2c , 0}, {0, 1, 1}, │ │ │ │ +00019d80: 207b 332c 2030 2c20 307d 2c20 7b32 2c20 {3, 0, 0}, {2, │ │ │ │ +00019d90: 312c 2030 7d2c 207b 322c 2030 2c20 317d 1, 0}, {2, 0, 1} │ │ │ │ +00019da0: 2c20 7b31 2c20 322c 2030 7d2c 7c0a 7c20 , {1, 2, 0},|.| │ │ │ │ +00019db0: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019dd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019de0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019e00: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 207b -------|.| { │ │ │ │ -00019e10: 312c 2031 2c20 317d 2c20 7b30 2c20 322c 1, 1, 1}, {0, 2, │ │ │ │ -00019e20: 2031 7d2c 207b 332c 2031 2c20 307d 2c20 1}, {3, 1, 0}, │ │ │ │ -00019e30: 7b33 2c20 302c 2031 7d2c 207b 322c 2032 {3, 0, 1}, {2, 2 │ │ │ │ -00019e40: 2c20 307d 2c20 7b32 2c20 312c 2031 7d2c , 0}, {2, 1, 1}, │ │ │ │ -00019e50: 207b 312c 2032 2c7c 0a7c 2020 2020 202d {1, 2,|.| - │ │ │ │ +00019df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019e00: 2020 2020 7b31 2c20 312c 2031 7d2c 207b {1, 1, 1}, { │ │ │ │ +00019e10: 302c 2032 2c20 317d 2c20 7b33 2c20 312c 0, 2, 1}, {3, 1, │ │ │ │ +00019e20: 2030 7d2c 207b 332c 2030 2c20 317d 2c20 0}, {3, 0, 1}, │ │ │ │ +00019e30: 7b32 2c20 322c 2030 7d2c 207b 322c 2031 {2, 2, 0}, {2, 1 │ │ │ │ +00019e40: 2c20 317d 2c20 7b31 2c20 322c 7c0a 7c20 , 1}, {1, 2,|.| │ │ │ │ +00019e50: 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ------------ │ │ │ │ 00019e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019ea0: 2d2d 2d2d 2d2d 2d7c 0a7c 2020 2020 2031 -------|.| 1 │ │ │ │ -00019eb0: 7d2c 207b 332c 2032 2c20 307d 2c20 7b33 }, {3, 2, 0}, {3 │ │ │ │ -00019ec0: 2c20 312c 2031 7d2c 207b 322c 2032 2c20 , 1, 1}, {2, 2, │ │ │ │ -00019ed0: 317d 7d20 2020 2020 2020 2020 2020 2020 1}} │ │ │ │ -00019ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019ef0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00019e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +00019ea0: 2020 2020 317d 2c20 7b33 2c20 322c 2030 1}, {3, 2, 0 │ │ │ │ +00019eb0: 7d2c 207b 332c 2031 2c20 317d 2c20 7b32 }, {3, 1, 1}, {2 │ │ │ │ +00019ec0: 2c20 322c 2031 7d7d 2020 2020 2020 2020 , 2, 1}} │ │ │ │ +00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00019ee0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00019ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f40: 2020 2020 2020 207c 0a7c 6f32 203a 204c |.|o2 : L │ │ │ │ -00019f50: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +00019f30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00019f40: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ +00019f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019f90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00019f80: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00019f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00019fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00019fe0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00019ff0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -0001a000: 202a 6e6f 7465 2045 6973 656e 6275 6453 *note EisenbudS │ │ │ │ -0001a010: 6861 6d61 7368 3a20 4569 7365 6e62 7564 hamash: Eisenbud │ │ │ │ -0001a020: 5368 616d 6173 682c 202d 2d20 436f 6d70 Shamash, -- Comp │ │ │ │ -0001a030: 7574 6573 2074 6865 2045 6973 656e 6275 utes the Eisenbu │ │ │ │ -0001a040: 642d 5368 616d 6173 680a 2020 2020 436f d-Shamash. Co │ │ │ │ -0001a050: 6d70 6c65 780a 2020 2a20 2a6e 6f74 6520 mplex. * *note │ │ │ │ -0001a060: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -0001a070: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -0001a080: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ -0001a090: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ -0001a0a0: 2020 686f 6d6f 746f 7069 6573 0a0a 5761 homotopies..Wa │ │ │ │ -0001a0b0: 7973 2074 6f20 7573 6520 6578 706f 3a0a ys to use expo:. │ │ │ │ -0001a0c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001a0d0: 3d0a 0a20 202a 2022 6578 706f 285a 5a2c =.. * "expo(ZZ, │ │ │ │ -0001a0e0: 4c69 7374 2922 0a20 202a 2022 6578 706f List)". * "expo │ │ │ │ -0001a0f0: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ -0001a100: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0001a110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0001a120: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0001a130: 6520 6578 706f 3a20 6578 706f 2c20 6973 e expo: expo, is │ │ │ │ -0001a140: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -0001a150: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -0001a160: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -0001a170: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00019fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00019fe0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00019ff0: 0a0a 2020 2a20 2a6e 6f74 6520 4569 7365 .. * *note Eise │ │ │ │ +0001a000: 6e62 7564 5368 616d 6173 683a 2045 6973 nbudShamash: Eis │ │ │ │ +0001a010: 656e 6275 6453 6861 6d61 7368 2c20 2d2d enbudShamash, -- │ │ │ │ +0001a020: 2043 6f6d 7075 7465 7320 7468 6520 4569 Computes the Ei │ │ │ │ +0001a030: 7365 6e62 7564 2d53 6861 6d61 7368 0a20 senbud-Shamash. │ │ │ │ +0001a040: 2020 2043 6f6d 706c 6578 0a20 202a 202a Complex. * * │ │ │ │ +0001a050: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +0001a060: 6965 733a 206d 616b 6548 6f6d 6f74 6f70 ies: makeHomotop │ │ │ │ +0001a070: 6965 732c 202d 2d20 7265 7475 726e 7320 ies, -- returns │ │ │ │ +0001a080: 6120 7379 7374 656d 206f 6620 6869 6768 a system of high │ │ │ │ +0001a090: 6572 0a20 2020 2068 6f6d 6f74 6f70 6965 er. homotopie │ │ │ │ +0001a0a0: 730a 0a57 6179 7320 746f 2075 7365 2065 s..Ways to use e │ │ │ │ +0001a0b0: 7870 6f3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d xpo:.=========== │ │ │ │ +0001a0c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7870 ======.. * "exp │ │ │ │ +0001a0d0: 6f28 5a5a 2c4c 6973 7429 220a 2020 2a20 o(ZZ,List)". * │ │ │ │ +0001a0e0: 2265 7870 6f28 5a5a 2c5a 5a29 220a 0a46 "expo(ZZ,ZZ)"..F │ │ │ │ +0001a0f0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0001a100: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0001a110: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0001a120: 202a 6e6f 7465 2065 7870 6f3a 2065 7870 *note expo: exp │ │ │ │ +0001a130: 6f2c 2069 7320 6120 2a6e 6f74 6520 6d65 o, is a *note me │ │ │ │ +0001a140: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0001a150: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0001a160: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0001a170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a1c0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0001a1d0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0001a1e0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0001a1f0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0001a200: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0001a210: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -0001a220: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0001a230: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0001a240: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0001a250: 732e 6d32 3a35 3038 363a 302e 0a1f 0a46 s.m2:5086:0....F │ │ │ │ -0001a260: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0001a270: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0001a280: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0001a290: 2065 7874 6572 696f 7245 7874 4d6f 6475 exteriorExtModu │ │ │ │ -0001a2a0: 6c65 2c20 4e65 7874 3a20 6578 7465 7269 le, Next: exteri │ │ │ │ -0001a2b0: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ -0001a2c0: 2c20 5072 6576 3a20 6578 706f 2c20 5570 , Prev: expo, Up │ │ │ │ -0001a2d0: 3a20 546f 700a 0a65 7874 6572 696f 7245 : Top..exteriorE │ │ │ │ -0001a2e0: 7874 4d6f 6475 6c65 202d 2d20 4578 7428 xtModule -- Ext( │ │ │ │ -0001a2f0: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29 M,k) or Ext(M,N) │ │ │ │ -0001a300: 2061 7320 6120 6d6f 6475 6c65 206f 7665 as a module ove │ │ │ │ -0001a310: 7220 616e 2065 7874 6572 696f 7220 616c r an exterior al │ │ │ │ -0001a320: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a gebra.********** │ │ │ │ +0001a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0001a1c0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0001a1d0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0001a1e0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0001a1f0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0001a200: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +0001a210: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0001a220: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +0001a230: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0001a240: 7574 696f 6e73 2e6d 323a 3530 3836 3a30 utions.m2:5086:0 │ │ │ │ +0001a250: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +0001a260: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0001a270: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +0001a280: 4e6f 6465 3a20 6578 7465 7269 6f72 4578 Node: exteriorEx │ │ │ │ +0001a290: 744d 6f64 756c 652c 204e 6578 743a 2065 tModule, Next: e │ │ │ │ +0001a2a0: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d xteriorHomologyM │ │ │ │ +0001a2b0: 6f64 756c 652c 2050 7265 763a 2065 7870 odule, Prev: exp │ │ │ │ +0001a2c0: 6f2c 2055 703a 2054 6f70 0a0a 6578 7465 o, Up: Top..exte │ │ │ │ +0001a2d0: 7269 6f72 4578 744d 6f64 756c 6520 2d2d riorExtModule -- │ │ │ │ +0001a2e0: 2045 7874 284d 2c6b 2920 6f72 2045 7874 Ext(M,k) or Ext │ │ │ │ +0001a2f0: 284d 2c4e 2920 6173 2061 206d 6f64 756c (M,N) as a modul │ │ │ │ +0001a300: 6520 6f76 6572 2061 6e20 6578 7465 7269 e over an exteri │ │ │ │ +0001a310: 6f72 2061 6c67 6562 7261 0a2a 2a2a 2a2a or algebra.***** │ │ │ │ +0001a320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001a350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001a360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001a370: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0001a380: 200a 2020 2020 2020 2020 4520 3d20 6578 . E = ex │ │ │ │ -0001a390: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001a3a0: 662c 4d29 0a20 202a 2049 6e70 7574 733a f,M). * Inputs: │ │ │ │ -0001a3b0: 0a20 2020 2020 202a 2066 2c20 6120 2a6e . * f, a *n │ │ │ │ -0001a3c0: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -0001a3d0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -0001a3e0: 2c2c 2031 2078 2063 2c20 656e 7472 6965 ,, 1 x c, entrie │ │ │ │ -0001a3f0: 7320 6d75 7374 2062 650a 2020 2020 2020 s must be. │ │ │ │ -0001a400: 2020 686f 6d6f 746f 7069 6320 746f 2030 homotopic to 0 │ │ │ │ -0001a410: 206f 6e20 460a 2020 2020 2020 2a20 4d2c on F. * M, │ │ │ │ -0001a420: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ -0001a430: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0001a440: 6f64 756c 652c 2c20 616e 6e69 6869 6c61 odule,, annihila │ │ │ │ -0001a450: 7465 6420 6279 2074 6865 2065 6c65 6d65 ted by the eleme │ │ │ │ -0001a460: 6e74 730a 2020 2020 2020 2020 6f66 2066 nts. of f │ │ │ │ -0001a470: 660a 2020 2020 2020 2a20 4e2c 2061 202a f. * N, a * │ │ │ │ -0001a480: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0001a490: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0001a4a0: 652c 2c20 616e 6e69 6869 6c61 7465 6420 e,, annihilated │ │ │ │ -0001a4b0: 6279 2074 6865 2065 6c65 6d65 6e74 730a by the elements. │ │ │ │ -0001a4c0: 2020 2020 2020 2020 6f66 2066 660a 2020 of ff. │ │ │ │ -0001a4d0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -0001a4e0: 202a 2045 2c20 6120 2a6e 6f74 6520 6d6f * E, a *note mo │ │ │ │ -0001a4f0: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ -0001a500: 446f 6329 4d6f 6475 6c65 2c2c 204d 6f64 Doc)Module,, Mod │ │ │ │ -0001a510: 756c 6520 6f76 6572 2061 6e20 6578 7465 ule over an exte │ │ │ │ -0001a520: 7269 6f72 0a20 2020 2020 2020 2061 6c67 rior. alg │ │ │ │ -0001a530: 6562 7261 2077 6974 6820 7661 7269 6162 ebra with variab │ │ │ │ -0001a540: 6c65 7320 636f 7272 6573 706f 6e64 696e les correspondin │ │ │ │ -0001a550: 6720 746f 2065 6c65 6d65 6e74 7320 6f66 g to elements of │ │ │ │ -0001a560: 2066 0a0a 4465 7363 7269 7074 696f 6e0a f..Description. │ │ │ │ -0001a570: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ -0001a580: 4d2c 4e20 6172 6520 532d 6d6f 6475 6c65 M,N are S-module │ │ │ │ -0001a590: 7320 616e 6e69 6869 6c61 7465 6420 6279 s annihilated by │ │ │ │ -0001a5a0: 2074 6865 2065 6c65 6d65 6e74 7320 6f66 the elements of │ │ │ │ -0001a5b0: 2074 6865 206d 6174 7269 7820 6666 203d the matrix ff = │ │ │ │ -0001a5c0: 2028 665f 312e 2e66 5f63 292c 0a61 6e64 (f_1..f_c),.and │ │ │ │ -0001a5d0: 206b 2069 7320 7468 6520 7265 7369 6475 k is the residu │ │ │ │ -0001a5e0: 6520 6669 656c 6420 6f66 2053 2c20 7468 e field of S, th │ │ │ │ -0001a5f0: 656e 2074 6865 2073 6372 6970 7420 6578 en the script ex │ │ │ │ -0001a600: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001a610: 662c 4d29 2072 6574 7572 6e73 0a45 7874 f,M) returns.Ext │ │ │ │ -0001a620: 5f53 284d 2c20 6b29 2061 7320 6120 6d6f _S(M, k) as a mo │ │ │ │ -0001a630: 6475 6c65 206f 7665 7220 616e 2065 7874 dule over an ext │ │ │ │ -0001a640: 6572 696f 7220 616c 6765 6272 6120 4520 erior algebra E │ │ │ │ -0001a650: 3d20 6b3c 655f 312c 2e2e 2e2c 655f 633e = k │ │ │ │ -0001a660: 2c20 7768 6572 6520 7468 650a 655f 6920 , where the.e_i │ │ │ │ -0001a670: 6861 7665 2064 6567 7265 6520 312e 2049 have degree 1. I │ │ │ │ -0001a680: 7420 6973 2063 6f6d 7075 7465 6420 6173 t is computed as │ │ │ │ -0001a690: 2074 6865 2045 2d64 7561 6c20 6f66 2065 the E-dual of e │ │ │ │ -0001a6a0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001a6b0: 2e0a 0a54 6865 2073 6372 6970 7420 6578 ...The script ex │ │ │ │ -0001a6c0: 7465 7269 6f72 546f 724d 6f64 756c 6528 teriorTorModule( │ │ │ │ -0001a6d0: 662c 4d2c 4e29 2072 6574 7572 6e73 2045 f,M,N) returns E │ │ │ │ -0001a6e0: 7874 5f53 284d 2c4e 2920 6173 2061 206d xt_S(M,N) as a m │ │ │ │ -0001a6f0: 6f64 756c 6520 6f76 6572 2061 0a62 6967 odule over a.big │ │ │ │ -0001a700: 7261 6465 6420 7269 6e67 2053 4520 3d20 raded ring SE = │ │ │ │ -0001a710: 533c 655f 312c 2e2e 2c65 5f63 3e2c 2077 S, w │ │ │ │ -0001a720: 6865 7265 2074 6865 2065 5f69 2068 6176 here the e_i hav │ │ │ │ -0001a730: 6520 6465 6772 6565 7320 7b64 5f69 2c31 e degrees {d_i,1 │ │ │ │ -0001a740: 7d2c 2077 6865 7265 2064 5f69 0a69 7320 }, where d_i.is │ │ │ │ -0001a750: 7468 6520 6465 6772 6565 206f 6620 665f the degree of f_ │ │ │ │ -0001a760: 692e 2054 6865 206d 6f64 756c 6520 7374 i. The module st │ │ │ │ -0001a770: 7275 6374 7572 652c 2069 6e20 6569 7468 ructure, in eith │ │ │ │ -0001a780: 6572 2063 6173 652c 2069 7320 6465 6669 er case, is defi │ │ │ │ -0001a790: 6e65 6420 6279 2074 6865 0a68 6f6d 6f74 ned by the.homot │ │ │ │ -0001a7a0: 6f70 6965 7320 666f 7220 7468 6520 665f opies for the f_ │ │ │ │ -0001a7b0: 6920 6f6e 2074 6865 2072 6573 6f6c 7574 i on the resolut │ │ │ │ -0001a7c0: 696f 6e20 6f66 204d 2c20 636f 6d70 7574 ion of M, comput │ │ │ │ -0001a7d0: 6564 2062 7920 7468 6520 7363 7269 7074 ed by the script │ │ │ │ -0001a7e0: 0a6d 616b 6548 6f6d 6f74 6f70 6965 7331 .makeHomotopies1 │ │ │ │ -0001a7f0: 2e54 6865 2073 6372 6970 7420 6361 6c6c .The script call │ │ │ │ -0001a800: 7320 6d61 6b65 4d6f 6475 6c65 2074 6f20 s makeModule to │ │ │ │ -0001a810: 636f 6d70 7574 6520 6120 286e 6f6e 2d6d compute a (non-m │ │ │ │ -0001a820: 696e 696d 616c 290a 7072 6573 656e 7461 inimal).presenta │ │ │ │ -0001a830: 7469 6f6e 206f 6620 7468 6973 206d 6f64 tion of this mod │ │ │ │ -0001a840: 756c 652e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ule...+--------- │ │ │ │ +0001a360: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0001a370: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ +0001a380: 203d 2065 7874 6572 696f 7245 7874 4d6f = exteriorExtMo │ │ │ │ +0001a390: 6475 6c65 2866 2c4d 290a 2020 2a20 496e dule(f,M). * In │ │ │ │ +0001a3a0: 7075 7473 3a0a 2020 2020 2020 2a20 662c puts:. * f, │ │ │ │ +0001a3b0: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +0001a3c0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0001a3d0: 6174 7269 782c 2c20 3120 7820 632c 2065 atrix,, 1 x c, e │ │ │ │ +0001a3e0: 6e74 7269 6573 206d 7573 7420 6265 0a20 ntries must be. │ │ │ │ +0001a3f0: 2020 2020 2020 2068 6f6d 6f74 6f70 6963 homotopic │ │ │ │ +0001a400: 2074 6f20 3020 6f6e 2046 0a20 2020 2020 to 0 on F. │ │ │ │ +0001a410: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f * M, a *note mo │ │ │ │ +0001a420: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ +0001a430: 446f 6329 4d6f 6475 6c65 2c2c 2061 6e6e Doc)Module,, ann │ │ │ │ +0001a440: 6968 696c 6174 6564 2062 7920 7468 6520 ihilated by the │ │ │ │ +0001a450: 656c 656d 656e 7473 0a20 2020 2020 2020 elements. │ │ │ │ +0001a460: 206f 6620 6666 0a20 2020 2020 202a 204e of ff. * N │ │ │ │ +0001a470: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +0001a480: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0001a490: 4d6f 6475 6c65 2c2c 2061 6e6e 6968 696c Module,, annihil │ │ │ │ +0001a4a0: 6174 6564 2062 7920 7468 6520 656c 656d ated by the elem │ │ │ │ +0001a4b0: 656e 7473 0a20 2020 2020 2020 206f 6620 ents. of │ │ │ │ +0001a4c0: 6666 0a20 202a 204f 7574 7075 7473 3a0a ff. * Outputs:. │ │ │ │ +0001a4d0: 2020 2020 2020 2a20 452c 2061 202a 6e6f * E, a *no │ │ │ │ +0001a4e0: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ +0001a4f0: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0001a500: 2c20 4d6f 6475 6c65 206f 7665 7220 616e , Module over an │ │ │ │ +0001a510: 2065 7874 6572 696f 720a 2020 2020 2020 exterior. │ │ │ │ +0001a520: 2020 616c 6765 6272 6120 7769 7468 2076 algebra with v │ │ │ │ +0001a530: 6172 6961 626c 6573 2063 6f72 7265 7370 ariables corresp │ │ │ │ +0001a540: 6f6e 6469 6e67 2074 6f20 656c 656d 656e onding to elemen │ │ │ │ +0001a550: 7473 206f 6620 660a 0a44 6573 6372 6970 ts of f..Descrip │ │ │ │ +0001a560: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +0001a570: 0a0a 4966 204d 2c4e 2061 7265 2053 2d6d ..If M,N are S-m │ │ │ │ +0001a580: 6f64 756c 6573 2061 6e6e 6968 696c 6174 odules annihilat │ │ │ │ +0001a590: 6564 2062 7920 7468 6520 656c 656d 656e ed by the elemen │ │ │ │ +0001a5a0: 7473 206f 6620 7468 6520 6d61 7472 6978 ts of the matrix │ │ │ │ +0001a5b0: 2066 6620 3d20 2866 5f31 2e2e 665f 6329 ff = (f_1..f_c) │ │ │ │ +0001a5c0: 2c0a 616e 6420 6b20 6973 2074 6865 2072 ,.and k is the r │ │ │ │ +0001a5d0: 6573 6964 7565 2066 6965 6c64 206f 6620 esidue field of │ │ │ │ +0001a5e0: 532c 2074 6865 6e20 7468 6520 7363 7269 S, then the scri │ │ │ │ +0001a5f0: 7074 2065 7874 6572 696f 7245 7874 4d6f pt exteriorExtMo │ │ │ │ +0001a600: 6475 6c65 2866 2c4d 2920 7265 7475 726e dule(f,M) return │ │ │ │ +0001a610: 730a 4578 745f 5328 4d2c 206b 2920 6173 s.Ext_S(M, k) as │ │ │ │ +0001a620: 2061 206d 6f64 756c 6520 6f76 6572 2061 a module over a │ │ │ │ +0001a630: 6e20 6578 7465 7269 6f72 2061 6c67 6562 n exterior algeb │ │ │ │ +0001a640: 7261 2045 203d 206b 3c65 5f31 2c2e 2e2e ra E = k, where the │ │ │ │ +0001a660: 0a65 5f69 2068 6176 6520 6465 6772 6565 .e_i have degree │ │ │ │ +0001a670: 2031 2e20 4974 2069 7320 636f 6d70 7574 1. It is comput │ │ │ │ +0001a680: 6564 2061 7320 7468 6520 452d 6475 616c ed as the E-dual │ │ │ │ +0001a690: 206f 6620 6578 7465 7269 6f72 546f 724d of exteriorTorM │ │ │ │ +0001a6a0: 6f64 756c 652e 0a0a 5468 6520 7363 7269 odule...The scri │ │ │ │ +0001a6b0: 7074 2065 7874 6572 696f 7254 6f72 4d6f pt exteriorTorMo │ │ │ │ +0001a6c0: 6475 6c65 2866 2c4d 2c4e 2920 7265 7475 dule(f,M,N) retu │ │ │ │ +0001a6d0: 726e 7320 4578 745f 5328 4d2c 4e29 2061 rns Ext_S(M,N) a │ │ │ │ +0001a6e0: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ +0001a6f0: 610a 6269 6772 6164 6564 2072 696e 6720 a.bigraded ring │ │ │ │ +0001a700: 5345 203d 2053 3c65 5f31 2c2e 2e2c 655f SE = S, where the e_ │ │ │ │ +0001a720: 6920 6861 7665 2064 6567 7265 6573 207b i have degrees { │ │ │ │ +0001a730: 645f 692c 317d 2c20 7768 6572 6520 645f d_i,1}, where d_ │ │ │ │ +0001a740: 690a 6973 2074 6865 2064 6567 7265 6520 i.is the degree │ │ │ │ +0001a750: 6f66 2066 5f69 2e20 5468 6520 6d6f 6475 of f_i. The modu │ │ │ │ +0001a760: 6c65 2073 7472 7563 7475 7265 2c20 696e le structure, in │ │ │ │ +0001a770: 2065 6974 6865 7220 6361 7365 2c20 6973 either case, is │ │ │ │ +0001a780: 2064 6566 696e 6564 2062 7920 7468 650a defined by the. │ │ │ │ +0001a790: 686f 6d6f 746f 7069 6573 2066 6f72 2074 homotopies for t │ │ │ │ +0001a7a0: 6865 2066 5f69 206f 6e20 7468 6520 7265 he f_i on the re │ │ │ │ +0001a7b0: 736f 6c75 7469 6f6e 206f 6620 4d2c 2063 solution of M, c │ │ │ │ +0001a7c0: 6f6d 7075 7465 6420 6279 2074 6865 2073 omputed by the s │ │ │ │ +0001a7d0: 6372 6970 740a 6d61 6b65 486f 6d6f 746f cript.makeHomoto │ │ │ │ +0001a7e0: 7069 6573 312e 5468 6520 7363 7269 7074 pies1.The script │ │ │ │ +0001a7f0: 2063 616c 6c73 206d 616b 654d 6f64 756c calls makeModul │ │ │ │ +0001a800: 6520 746f 2063 6f6d 7075 7465 2061 2028 e to compute a ( │ │ │ │ +0001a810: 6e6f 6e2d 6d69 6e69 6d61 6c29 0a70 7265 non-minimal).pre │ │ │ │ +0001a820: 7365 6e74 6174 696f 6e20 6f66 2074 6869 sentation of thi │ │ │ │ +0001a830: 7320 6d6f 6475 6c65 2e0a 0a2b 2d2d 2d2d s module...+---- │ │ │ │ +0001a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a880: 2d2d 2b0a 7c69 3120 3a20 6b6b 203d 205a --+.|i1 : kk = Z │ │ │ │ -0001a890: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +0001a870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ +0001a880: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020 k = ZZ/101 │ │ │ │ +0001a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001a8b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001a900: 7c6f 3120 3d20 6b6b 2020 2020 2020 2020 |o1 = kk │ │ │ │ +0001a8f0: 2020 207c 0a7c 6f31 203d 206b 6b20 2020 |.|o1 = kk │ │ │ │ +0001a900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a930: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001a930: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001a940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a970: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -0001a980: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0001a960: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001a970: 0a7c 6f31 203a 2051 756f 7469 656e 7452 .|o1 : QuotientR │ │ │ │ +0001a980: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 0001a990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a9b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001a9a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001a9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001a9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001a9f0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 ------+.|i2 : S │ │ │ │ -0001aa00: 3d20 6b6b 5b61 2c62 2c63 5d20 2020 2020 = kk[a,b,c] │ │ │ │ +0001a9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0001a9f0: 203a 2053 203d 206b 6b5b 612c 622c 635d : S = kk[a,b,c] │ │ │ │ +0001aa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001aa20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001aa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aa70: 2020 7c0a 7c6f 3220 3d20 5320 2020 2020 |.|o2 = S │ │ │ │ +0001aa60: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +0001aa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aab0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001aaa0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001aab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aae0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001aaf0: 7c6f 3220 3a20 506f 6c79 6e6f 6d69 616c |o2 : Polynomial │ │ │ │ -0001ab00: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ +0001aae0: 2020 207c 0a7c 6f32 203a 2050 6f6c 796e |.|o2 : Polyn │ │ │ │ +0001aaf0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ +0001ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ab10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ab20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001ab20: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001ab30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ab40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ab60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -0001ab70: 3a20 6620 3d20 6d61 7472 6978 2261 342c : f = matrix"a4, │ │ │ │ -0001ab80: 6234 2c63 3422 2020 2020 2020 2020 2020 b4,c4" │ │ │ │ -0001ab90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aba0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001ab50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ab60: 0a7c 6933 203a 2066 203d 206d 6174 7269 .|i3 : f = matri │ │ │ │ +0001ab70: 7822 6134 2c62 342c 6334 2220 2020 2020 x"a4,b4,c4" │ │ │ │ +0001ab80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ab90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001aba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001abb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001abc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001abd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001abe0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20 |.|o3 = | │ │ │ │ -0001abf0: 6134 2062 3420 6334 207c 2020 2020 2020 a4 b4 c4 | │ │ │ │ +0001abd0: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0001abe0: 203d 207c 2061 3420 6234 2063 3420 7c20 = | a4 b4 c4 | │ │ │ │ +0001abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001ac10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ac20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac60: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001ac70: 2020 3120 2020 2020 2033 2020 2020 2020 1 3 │ │ │ │ +0001ac50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001ac60: 2020 2020 2020 2031 2020 2020 2020 3320 1 3 │ │ │ │ +0001ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ac80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ac90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aca0: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053 |.|o3 : Matrix S │ │ │ │ -0001acb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ +0001ac90: 2020 2020 207c 0a7c 6f33 203a 204d 6174 |.|o3 : Mat │ │ │ │ +0001aca0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +0001acb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001acd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ace0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001acd0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001ace0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001acf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ad00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ad10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001ad20: 3420 3a20 5220 3d20 532f 6964 6561 6c20 4 : R = S/ideal │ │ │ │ -0001ad30: 6620 2020 2020 2020 2020 2020 2020 2020 f │ │ │ │ -0001ad40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ad50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001ad10: 2d2b 0a7c 6934 203a 2052 203d 2053 2f69 -+.|i4 : R = S/i │ │ │ │ +0001ad20: 6465 616c 2066 2020 2020 2020 2020 2020 deal f │ │ │ │ +0001ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ad40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ad50: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ad90: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -0001ada0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0001ad80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001ad90: 6f34 203d 2052 2020 2020 2020 2020 2020 o4 = R │ │ │ │ +0001ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001add0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001adc0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ade0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae10: 2020 2020 7c0a 7c6f 3420 3a20 5175 6f74 |.|o4 : Quot │ │ │ │ -0001ae20: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0001ae00: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ +0001ae10: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +0001ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ae50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001ae40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001ae50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ae60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ae70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ae90: 2b0a 7c69 3520 3a20 7020 3d20 6d61 7028 +.|i5 : p = map( │ │ │ │ -0001aea0: 522c 5329 2020 2020 2020 2020 2020 2020 R,S) │ │ │ │ +0001ae80: 2d2d 2d2d 2d2b 0a7c 6935 203a 2070 203d -----+.|i5 : p = │ │ │ │ +0001ae90: 206d 6170 2852 2c53 2920 2020 2020 2020 map(R,S) │ │ │ │ +0001aea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001aec0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001aed0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001aec0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001aed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001aef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001af10: 3520 3d20 6d61 7020 2852 2c20 532c 207b 5 = map (R, S, { │ │ │ │ -0001af20: 612c 2062 2c20 637d 2920 2020 2020 2020 a, b, c}) │ │ │ │ -0001af30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001af00: 207c 0a7c 6f35 203d 206d 6170 2028 522c |.|o5 = map (R, │ │ │ │ +0001af10: 2053 2c20 7b61 2c20 622c 2063 7d29 2020 S, {a, b, c}) │ │ │ │ +0001af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001af30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001af40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001af80: 2020 2020 2020 2020 7c0a 7c6f 3520 3a20 |.|o5 : │ │ │ │ -0001af90: 5269 6e67 4d61 7020 5220 3c2d 2d20 5320 RingMap R <-- S │ │ │ │ +0001af70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001af80: 6f35 203a 2052 696e 674d 6170 2052 203c o5 : RingMap R < │ │ │ │ +0001af90: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ 0001afa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001afc0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001afb0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001afc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001afd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b000: 2d2d 2d2d 2b0a 7c69 3620 3a20 4d20 3d20 ----+.|i6 : M = │ │ │ │ -0001b010: 636f 6b65 7220 6d61 7028 525e 322c 2052 coker map(R^2, R │ │ │ │ -0001b020: 5e7b 333a 2d31 7d2c 207b 7b61 2c62 2c63 ^{3:-1}, {{a,b,c │ │ │ │ -0001b030: 7d2c 7b62 2c63 2c61 7d7d 2920 2020 2020 },{b,c,a}}) │ │ │ │ -0001b040: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001aff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ +0001b000: 204d 203d 2063 6f6b 6572 206d 6170 2852 M = coker map(R │ │ │ │ +0001b010: 5e32 2c20 525e 7b33 3a2d 317d 2c20 7b7b ^2, R^{3:-1}, {{ │ │ │ │ +0001b020: 612c 622c 637d 2c7b 622c 632c 617d 7d29 a,b,c},{b,c,a}}) │ │ │ │ +0001b030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b080: 7c0a 7c6f 3620 3d20 636f 6b65 726e 656c |.|o6 = cokernel │ │ │ │ -0001b090: 207c 2061 2062 2063 207c 2020 2020 2020 | a b c | │ │ │ │ +0001b070: 2020 2020 207c 0a7c 6f36 203d 2063 6f6b |.|o6 = cok │ │ │ │ +0001b080: 6572 6e65 6c20 7c20 6120 6220 6320 7c20 ernel | a b c | │ │ │ │ +0001b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b0c0: 7c20 2020 2020 2020 2020 2020 2020 207c | | │ │ │ │ -0001b0d0: 2062 2063 2061 207c 2020 2020 2020 2020 b c a | │ │ │ │ +0001b0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b0c0: 2020 2020 7c20 6220 6320 6120 7c20 2020 | b c a | │ │ │ │ +0001b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b0f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001b0f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b130: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b150: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ -0001b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b170: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ -0001b180: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965 R-module, quotie │ │ │ │ -0001b190: 6e74 206f 6620 5220 2020 2020 2020 2020 nt of R │ │ │ │ -0001b1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b1b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001b120: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b130: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001b140: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +0001b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b170: 6f36 203a 2052 2d6d 6f64 756c 652c 2071 o6 : R-module, q │ │ │ │ +0001b180: 756f 7469 656e 7420 6f66 2052 2020 2020 uotient of R │ │ │ │ +0001b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b1a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b1f0: 2d2d 2d2d 2b0a 7c69 3720 3a20 6265 7474 ----+.|i7 : bett │ │ │ │ -0001b200: 6920 2846 4620 3d66 7265 6552 6573 6f6c i (FF =freeResol │ │ │ │ -0001b210: 7574 696f 6e28 204d 2c20 4c65 6e67 7468 ution( M, Length │ │ │ │ -0001b220: 4c69 6d69 7420 3d3e 3629 2920 2020 2020 Limit =>6)) │ │ │ │ -0001b230: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b1e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a ---------+.|i7 : │ │ │ │ +0001b1f0: 2062 6574 7469 2028 4646 203d 6672 6565 betti (FF =free │ │ │ │ +0001b200: 5265 736f 6c75 7469 6f6e 2820 4d2c 204c Resolution( M, L │ │ │ │ +0001b210: 656e 6774 684c 696d 6974 203d 3e36 2929 engthLimit =>6)) │ │ │ │ +0001b220: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b270: 7c0a 7c20 2020 2020 2020 2020 2020 2030 |.| 0 │ │ │ │ -0001b280: 2031 2032 2033 2034 2020 3520 2036 2020 1 2 3 4 5 6 │ │ │ │ +0001b260: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b270: 2020 2020 3020 3120 3220 3320 3420 2035 0 1 2 3 4 5 │ │ │ │ +0001b280: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 0001b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b2a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b2b0: 7c6f 3720 3d20 746f 7461 6c3a 2032 2033 |o7 = total: 2 3 │ │ │ │ -0001b2c0: 2034 2036 2039 2031 3320 3138 2020 2020 4 6 9 13 18 │ │ │ │ +0001b2a0: 2020 207c 0a7c 6f37 203d 2074 6f74 616c |.|o7 = total │ │ │ │ +0001b2b0: 3a20 3220 3320 3420 3620 3920 3133 2031 : 2 3 4 6 9 13 1 │ │ │ │ +0001b2c0: 3820 2020 2020 2020 2020 2020 2020 2020 8 │ │ │ │ 0001b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001b2f0: 2020 2020 2020 2020 303a 2032 2033 202e 0: 2 3 . │ │ │ │ -0001b300: 202e 202e 2020 2e20 202e 2020 2020 2020 . . . . │ │ │ │ -0001b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001b330: 2020 2020 2020 313a 202e 202e 2031 202e 1: . . 1 . │ │ │ │ -0001b340: 202e 2020 2e20 202e 2020 2020 2020 2020 . . . │ │ │ │ -0001b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b360: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001b370: 2020 2020 323a 202e 202e 2033 2033 202e 2: . . 3 3 . │ │ │ │ -0001b380: 2020 2e20 202e 2020 2020 2020 2020 2020 . . │ │ │ │ -0001b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b3a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001b3b0: 2020 333a 202e 202e 202e 2033 2033 2020 3: . . . 3 3 │ │ │ │ -0001b3c0: 2e20 202e 2020 2020 2020 2020 2020 2020 . . │ │ │ │ -0001b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b3e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001b3f0: 343a 202e 202e 202e 202e 2033 2020 3320 4: . . . . 3 3 │ │ │ │ -0001b400: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b420: 2020 7c0a 7c20 2020 2020 2020 2020 353a |.| 5: │ │ │ │ -0001b430: 202e 202e 202e 202e 2033 2020 3920 2036 . . . . 3 9 6 │ │ │ │ +0001b2e0: 207c 0a7c 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ +0001b2f0: 3220 3320 2e20 2e20 2e20 202e 2020 2e20 2 3 . . . . . │ │ │ │ +0001b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b320: 0a7c 2020 2020 2020 2020 2031 3a20 2e20 .| 1: . │ │ │ │ +0001b330: 2e20 3120 2e20 2e20 202e 2020 2e20 2020 . 1 . . . . │ │ │ │ +0001b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b350: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b360: 2020 2020 2020 2020 2032 3a20 2e20 2e20 2: . . │ │ │ │ +0001b370: 3320 3320 2e20 202e 2020 2e20 2020 2020 3 3 . . . │ │ │ │ +0001b380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b390: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001b3a0: 2020 2020 2020 2033 3a20 2e20 2e20 2e20 3: . . . │ │ │ │ +0001b3b0: 3320 3320 202e 2020 2e20 2020 2020 2020 3 3 . . │ │ │ │ +0001b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001b3e0: 2020 2020 2034 3a20 2e20 2e20 2e20 2e20 4: . . . . │ │ │ │ +0001b3f0: 3320 2033 2020 2e20 2020 2020 2020 2020 3 3 . │ │ │ │ +0001b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b410: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b420: 2020 2035 3a20 2e20 2e20 2e20 2e20 3320 5: . . . . 3 │ │ │ │ +0001b430: 2039 2020 3620 2020 2020 2020 2020 2020 9 6 │ │ │ │ 0001b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b460: 7c0a 7c20 2020 2020 2020 2020 363a 202e |.| 6: . │ │ │ │ -0001b470: 202e 202e 202e 202e 2020 2e20 2033 2020 . . . . . 3 │ │ │ │ +0001b450: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b460: 2036 3a20 2e20 2e20 2e20 2e20 2e20 202e 6: . . . . . . │ │ │ │ +0001b470: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b490: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b4a0: 7c20 2020 2020 2020 2020 373a 202e 202e | 7: . . │ │ │ │ -0001b4b0: 202e 202e 202e 2020 3120 2039 2020 2020 . . . 1 9 │ │ │ │ +0001b490: 2020 207c 0a7c 2020 2020 2020 2020 2037 |.| 7 │ │ │ │ +0001b4a0: 3a20 2e20 2e20 2e20 2e20 2e20 2031 2020 : . . . . . 1 │ │ │ │ +0001b4b0: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ 0001b4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b4d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001b4d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b510: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0001b520: 3a20 4265 7474 6954 616c 6c79 2020 2020 : BettiTally │ │ │ │ +0001b500: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001b510: 0a7c 6f37 203a 2042 6574 7469 5461 6c6c .|o7 : BettiTall │ │ │ │ +0001b520: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ 0001b530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b550: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001b540: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b590: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 ------+.|i8 : MS │ │ │ │ -0001b5a0: 203d 2070 7275 6e65 2070 7573 6846 6f72 = prune pushFor │ │ │ │ -0001b5b0: 7761 7264 2870 2c20 636f 6b65 7220 4646 ward(p, coker FF │ │ │ │ -0001b5c0: 2e64 645f 3629 3b20 2020 2020 2020 2020 .dd_6); │ │ │ │ -0001b5d0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001b580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +0001b590: 203a 204d 5320 3d20 7072 756e 6520 7075 : MS = prune pu │ │ │ │ +0001b5a0: 7368 466f 7277 6172 6428 702c 2063 6f6b shForward(p, cok │ │ │ │ +0001b5b0: 6572 2046 462e 6464 5f36 293b 2020 2020 er FF.dd_6); │ │ │ │ +0001b5c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b610: 2d2d 2b0a 7c69 3920 3a20 7265 7346 6c64 --+.|i9 : resFld │ │ │ │ -0001b620: 203a 3d20 7075 7368 466f 7277 6172 6428 := pushForward( │ │ │ │ -0001b630: 702c 2063 6f6b 6572 2076 6172 7320 5229 p, coker vars R) │ │ │ │ -0001b640: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0001b650: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001b600: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2072 -------+.|i9 : r │ │ │ │ +0001b610: 6573 466c 6420 3a3d 2070 7573 6846 6f72 esFld := pushFor │ │ │ │ +0001b620: 7761 7264 2870 2c20 636f 6b65 7220 7661 ward(p, coker va │ │ │ │ +0001b630: 7273 2052 293b 2020 2020 2020 2020 2020 rs R); │ │ │ │ +0001b640: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001b650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001b690: 7c69 3130 203a 2054 203d 2065 7874 6572 |i10 : T = exter │ │ │ │ -0001b6a0: 696f 7254 6f72 4d6f 6475 6c65 2866 2c4d iorTorModule(f,M │ │ │ │ -0001b6b0: 5329 3b20 2020 2020 2020 2020 2020 2020 S); │ │ │ │ -0001b6c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001b680: 2d2d 2d2b 0a7c 6931 3020 3a20 5420 3d20 ---+.|i10 : T = │ │ │ │ +0001b690: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001b6a0: 6528 662c 4d53 293b 2020 2020 2020 2020 e(f,MS); │ │ │ │ +0001b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b6c0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001b6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b700: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ -0001b710: 203a 2045 203d 2065 7874 6572 696f 7245 : E = exteriorE │ │ │ │ -0001b720: 7874 4d6f 6475 6c65 2866 2c4d 5329 3b20 xtModule(f,MS); │ │ │ │ -0001b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b740: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001b6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001b700: 0a7c 6931 3120 3a20 4520 3d20 6578 7465 .|i11 : E = exte │ │ │ │ +0001b710: 7269 6f72 4578 744d 6f64 756c 6528 662c riorExtModule(f, │ │ │ │ +0001b720: 4d53 293b 2020 2020 2020 2020 2020 2020 MS); │ │ │ │ +0001b730: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b780: 2d2d 2d2d 2d2d 2b0a 7c69 3132 203a 2068 ------+.|i12 : h │ │ │ │ -0001b790: 6628 2d34 2e2e 302c 4529 2020 2020 2020 f(-4..0,E) │ │ │ │ +0001b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +0001b780: 3220 3a20 6866 282d 342e 2e30 2c45 2920 2 : hf(-4..0,E) │ │ │ │ +0001b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001b7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b800: 2020 7c0a 7c6f 3132 203d 207b 302c 2039 |.|o12 = {0, 9 │ │ │ │ -0001b810: 2c20 3239 2c20 3333 2c20 3133 7d20 2020 , 29, 33, 13} │ │ │ │ +0001b7f0: 2020 2020 2020 207c 0a7c 6f31 3220 3d20 |.|o12 = │ │ │ │ +0001b800: 7b30 2c20 392c 2032 392c 2033 332c 2031 {0, 9, 29, 33, 1 │ │ │ │ +0001b810: 337d 2020 2020 2020 2020 2020 2020 2020 3} │ │ │ │ 0001b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b840: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001b830: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001b840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001b880: 7c6f 3132 203a 204c 6973 7420 2020 2020 |o12 : List │ │ │ │ +0001b870: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374 |.|o12 : List │ │ │ │ +0001b880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b8b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001b8b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001b8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001b8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ -0001b900: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ -0001b910: 6f6c 7574 696f 6e20 4d53 2020 2020 2020 olution MS │ │ │ │ -0001b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b930: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001b8f0: 0a7c 6931 3320 3a20 6265 7474 6920 6672 .|i13 : betti fr │ │ │ │ +0001b900: 6565 5265 736f 6c75 7469 6f6e 204d 5320 eeResolution MS │ │ │ │ +0001b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001b920: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001b930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001b950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b970: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001b980: 2020 2020 2020 2030 2020 3120 2032 2033 0 1 2 3 │ │ │ │ +0001b960: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001b970: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ +0001b980: 2020 3220 3320 2020 2020 2020 2020 2020 2 3 │ │ │ │ 0001b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9b0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74 |.|o13 = tot │ │ │ │ -0001b9c0: 616c 3a20 3133 2033 3320 3239 2039 2020 al: 13 33 29 9 │ │ │ │ +0001b9a0: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0001b9b0: 3d20 746f 7461 6c3a 2031 3320 3333 2032 = total: 13 33 2 │ │ │ │ +0001b9c0: 3920 3920 2020 2020 2020 2020 2020 2020 9 9 │ │ │ │ 0001b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001b9f0: 2020 7c0a 7c20 2020 2020 2020 2020 2039 |.| 9 │ │ │ │ -0001ba00: 3a20 2033 2020 2e20 202e 202e 2020 2020 : 3 . . . │ │ │ │ +0001b9e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001b9f0: 2020 2020 393a 2020 3320 202e 2020 2e20 9: 3 . . │ │ │ │ +0001ba00: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ 0001ba10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba30: 7c0a 7c20 2020 2020 2020 2020 3130 3a20 |.| 10: │ │ │ │ -0001ba40: 2039 2020 3620 202e 202e 2020 2020 2020 9 6 . . │ │ │ │ +0001ba20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ba30: 2031 303a 2020 3920 2036 2020 2e20 2e20 10: 9 6 . . │ │ │ │ +0001ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ba60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ba70: 7c20 2020 2020 2020 2020 3131 3a20 202e | 11: . │ │ │ │ -0001ba80: 2020 3320 202e 202e 2020 2020 2020 2020 3 . . │ │ │ │ +0001ba60: 2020 207c 0a7c 2020 2020 2020 2020 2031 |.| 1 │ │ │ │ +0001ba70: 313a 2020 2e20 2033 2020 2e20 2e20 2020 1: . 3 . . │ │ │ │ +0001ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ba90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001baa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001bab0: 2020 2020 2020 2020 3132 3a20 2031 2031 12: 1 1 │ │ │ │ -0001bac0: 3520 202e 202e 2020 2020 2020 2020 2020 5 . . │ │ │ │ -0001bad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bae0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001baf0: 2020 2020 2020 3133 3a20 202e 2020 3920 13: . 9 │ │ │ │ -0001bb00: 2038 202e 2020 2020 2020 2020 2020 2020 8 . │ │ │ │ -0001bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001bb30: 2020 2020 3134 3a20 202e 2020 2e20 2036 14: . . 6 │ │ │ │ -0001bb40: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001bb70: 2020 3135 3a20 202e 2020 2e20 3132 202e 15: . . 12 . │ │ │ │ +0001baa0: 207c 0a7c 2020 2020 2020 2020 2031 323a |.| 12: │ │ │ │ +0001bab0: 2020 3120 3135 2020 2e20 2e20 2020 2020 1 15 . . │ │ │ │ +0001bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bad0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bae0: 0a7c 2020 2020 2020 2020 2031 333a 2020 .| 13: │ │ │ │ +0001baf0: 2e20 2039 2020 3820 2e20 2020 2020 2020 . 9 8 . │ │ │ │ +0001bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001bb20: 2020 2020 2020 2020 2031 343a 2020 2e20 14: . │ │ │ │ +0001bb30: 202e 2020 3620 2e20 2020 2020 2020 2020 . 6 . │ │ │ │ +0001bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bb50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001bb60: 2020 2020 2020 2031 353a 2020 2e20 202e 15: . . │ │ │ │ +0001bb70: 2031 3220 2e20 2020 2020 2020 2020 2020 12 . │ │ │ │ 0001bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bba0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001bbb0: 3136 3a20 202e 2020 2e20 2033 2033 2020 16: . . 3 3 │ │ │ │ +0001bb90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001bba0: 2020 2020 2031 363a 2020 2e20 202e 2020 16: . . │ │ │ │ +0001bbb0: 3320 3320 2020 2020 2020 2020 2020 2020 3 3 │ │ │ │ 0001bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bbe0: 2020 7c0a 7c20 2020 2020 2020 2020 3137 |.| 17 │ │ │ │ -0001bbf0: 3a20 202e 2020 2e20 202e 2033 2020 2020 : . . . 3 │ │ │ │ +0001bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001bbe0: 2020 2031 373a 2020 2e20 202e 2020 2e20 17: . . . │ │ │ │ +0001bbf0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0001bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc20: 7c0a 7c20 2020 2020 2020 2020 3138 3a20 |.| 18: │ │ │ │ -0001bc30: 202e 2020 2e20 202e 2033 2020 2020 2020 . . . 3 │ │ │ │ +0001bc10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001bc20: 2031 383a 2020 2e20 202e 2020 2e20 3320 18: . . . 3 │ │ │ │ +0001bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001bc60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001bc50: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bc90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001bca0: 3133 203a 2042 6574 7469 5461 6c6c 7920 13 : BettiTally │ │ │ │ +0001bc90: 207c 0a7c 6f31 3320 3a20 4265 7474 6954 |.|o13 : BettiT │ │ │ │ +0001bca0: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 0001bcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bcd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001bcc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bcd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001bce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bd10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ -0001bd20: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ -0001bd30: 7574 696f 6e20 2850 4520 3d20 7072 756e ution (PE = prun │ │ │ │ -0001bd40: 6520 452c 204c 656e 6774 684c 696d 6974 e E, LengthLimit │ │ │ │ -0001bd50: 203d 3e20 3629 7c0a 7c20 2020 2020 2020 => 6)|.| │ │ │ │ +0001bd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001bd10: 6931 3420 3a20 6265 7474 6920 6672 6565 i14 : betti free │ │ │ │ +0001bd20: 5265 736f 6c75 7469 6f6e 2028 5045 203d Resolution (PE = │ │ │ │ +0001bd30: 2070 7275 6e65 2045 2c20 4c65 6e67 7468 prune E, Length │ │ │ │ +0001bd40: 4c69 6d69 7420 3d3e 2036 297c 0a7c 2020 Limit => 6)|.| │ │ │ │ +0001bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bd90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001bda0: 2020 2020 2030 2020 3120 2032 2020 3320 0 1 2 3 │ │ │ │ -0001bdb0: 2034 2020 2035 2020 2036 2020 2020 2020 4 5 6 │ │ │ │ -0001bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bdd0: 2020 7c0a 7c6f 3134 203d 2074 6f74 616c |.|o14 = total │ │ │ │ -0001bde0: 3a20 3136 2031 3320 3235 2034 3920 3831 : 16 13 25 49 81 │ │ │ │ -0001bdf0: 2031 3231 2031 3639 2020 2020 2020 2020 121 169 │ │ │ │ -0001be00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001be10: 7c0a 7c20 2020 2020 2020 2020 2d33 3a20 |.| -3: │ │ │ │ -0001be20: 2039 2020 3420 2033 2020 3320 2033 2020 9 4 3 3 3 │ │ │ │ -0001be30: 2033 2020 2033 2020 2020 2020 2020 2020 3 3 │ │ │ │ -0001be40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001be50: 7c20 2020 2020 2020 2020 2d32 3a20 2036 | -2: 6 │ │ │ │ -0001be60: 2020 3320 202e 2020 2e20 202e 2020 202e 3 . . . . │ │ │ │ -0001be70: 2020 202e 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001be80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001be90: 2020 2020 2020 2020 2d31 3a20 202e 2020 -1: . │ │ │ │ -0001bea0: 2e20 2037 2031 3820 3333 2020 3532 2020 . 7 18 33 52 │ │ │ │ -0001beb0: 3735 2020 2020 2020 2020 2020 2020 2020 75 │ │ │ │ -0001bec0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001bed0: 2020 2020 2020 2030 3a20 2031 2020 3620 0: 1 6 │ │ │ │ -0001bee0: 3135 2032 3820 3435 2020 3636 2020 3931 15 28 45 66 91 │ │ │ │ -0001bef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001bd80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001bd90: 2020 2020 2020 2020 2020 3020 2031 2020 0 1 │ │ │ │ +0001bda0: 3220 2033 2020 3420 2020 3520 2020 3620 2 3 4 5 6 │ │ │ │ +0001bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001bdc0: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ +0001bdd0: 746f 7461 6c3a 2031 3620 3133 2032 3520 total: 16 13 25 │ │ │ │ +0001bde0: 3439 2038 3120 3132 3120 3136 3920 2020 49 81 121 169 │ │ │ │ +0001bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001be10: 202d 333a 2020 3920 2034 2020 3320 2033 -3: 9 4 3 3 │ │ │ │ +0001be20: 2020 3320 2020 3320 2020 3320 2020 2020 3 3 3 │ │ │ │ +0001be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be40: 2020 207c 0a7c 2020 2020 2020 2020 202d |.| - │ │ │ │ +0001be50: 323a 2020 3620 2033 2020 2e20 202e 2020 2: 6 3 . . │ │ │ │ +0001be60: 2e20 2020 2e20 2020 2e20 2020 2020 2020 . . . │ │ │ │ +0001be70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001be80: 207c 0a7c 2020 2020 2020 2020 202d 313a |.| -1: │ │ │ │ +0001be90: 2020 2e20 202e 2020 3720 3138 2033 3320 . . 7 18 33 │ │ │ │ +0001bea0: 2035 3220 2037 3520 2020 2020 2020 2020 52 75 │ │ │ │ +0001beb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001bec0: 0a7c 2020 2020 2020 2020 2020 303a 2020 .| 0: │ │ │ │ +0001bed0: 3120 2036 2031 3520 3238 2034 3520 2036 1 6 15 28 45 6 │ │ │ │ +0001bee0: 3620 2039 3120 2020 2020 2020 2020 2020 6 91 │ │ │ │ +0001bef0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf40: 2020 2020 2020 7c0a 7c6f 3134 203a 2042 |.|o14 : B │ │ │ │ -0001bf50: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +0001bf30: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001bf40: 3420 3a20 4265 7474 6954 616c 6c79 2020 4 : BettiTally │ │ │ │ +0001bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001bf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001bf80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001bf70: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001bf80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bf90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001bfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001bfc0: 2d2d 2b0a 7c69 3135 203a 2062 6574 7469 --+.|i15 : betti │ │ │ │ -0001bfd0: 2066 7265 6552 6573 6f6c 7574 696f 6e20 freeResolution │ │ │ │ -0001bfe0: 2850 5420 3d20 7072 756e 6520 542c 204c (PT = prune T, L │ │ │ │ -0001bff0: 656e 6774 684c 696d 6974 203d 3e20 3629 engthLimit => 6) │ │ │ │ -0001c000: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001bfb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 3a20 -------+.|i15 : │ │ │ │ +0001bfc0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ +0001bfd0: 7469 6f6e 2028 5054 203d 2070 7275 6e65 tion (PT = prune │ │ │ │ +0001bfe0: 2054 2c20 4c65 6e67 7468 4c69 6d69 7420 T, LengthLimit │ │ │ │ +0001bff0: 3d3e 2036 297c 0a7c 2020 2020 2020 2020 => 6)|.| │ │ │ │ +0001c000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001c040: 7c20 2020 2020 2020 2020 2020 2020 2030 | 0 │ │ │ │ -0001c050: 2020 3120 2032 2020 2033 2020 2034 2020 1 2 3 4 │ │ │ │ -0001c060: 2035 2020 2036 2020 2020 2020 2020 2020 5 6 │ │ │ │ -0001c070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001c080: 3135 203d 2074 6f74 616c 3a20 3331 2035 15 = total: 31 5 │ │ │ │ -0001c090: 3520 3837 2031 3237 2031 3735 2032 3331 5 87 127 175 231 │ │ │ │ -0001c0a0: 2032 3935 2020 2020 2020 2020 2020 2020 295 │ │ │ │ -0001c0b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001c0c0: 2020 2020 2020 2030 3a20 3133 2032 3420 0: 13 24 │ │ │ │ -0001c0d0: 3339 2020 3538 2020 3831 2031 3038 2031 39 58 81 108 1 │ │ │ │ -0001c0e0: 3339 2020 2020 2020 2020 2020 2020 2020 39 │ │ │ │ -0001c0f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001c100: 2020 2020 2031 3a20 3138 2033 3120 3438 1: 18 31 48 │ │ │ │ -0001c110: 2020 3639 2020 3934 2031 3233 2031 3536 69 94 123 156 │ │ │ │ -0001c120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c130: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001c030: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c040: 2020 2020 3020 2031 2020 3220 2020 3320 0 1 2 3 │ │ │ │ +0001c050: 2020 3420 2020 3520 2020 3620 2020 2020 4 5 6 │ │ │ │ +0001c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c070: 207c 0a7c 6f31 3520 3d20 746f 7461 6c3a |.|o15 = total: │ │ │ │ +0001c080: 2033 3120 3535 2038 3720 3132 3720 3137 31 55 87 127 17 │ │ │ │ +0001c090: 3520 3233 3120 3239 3520 2020 2020 2020 5 231 295 │ │ │ │ +0001c0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c0b0: 0a7c 2020 2020 2020 2020 2020 303a 2031 .| 0: 1 │ │ │ │ +0001c0c0: 3320 3234 2033 3920 2035 3820 2038 3120 3 24 39 58 81 │ │ │ │ +0001c0d0: 3130 3820 3133 3920 2020 2020 2020 2020 108 139 │ │ │ │ +0001c0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c0f0: 2020 2020 2020 2020 2020 313a 2031 3820 1: 18 │ │ │ │ +0001c100: 3331 2034 3820 2036 3920 2039 3420 3132 31 48 69 94 12 │ │ │ │ +0001c110: 3320 3135 3620 2020 2020 2020 2020 2020 3 156 │ │ │ │ +0001c120: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c170: 2020 2020 7c0a 7c6f 3135 203a 2042 6574 |.|o15 : Bet │ │ │ │ -0001c180: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +0001c160: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +0001c170: 3a20 4265 7474 6954 616c 6c79 2020 2020 : BettiTally │ │ │ │ +0001c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c1b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001c1a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001c1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c1f0: 2b0a 7c69 3136 203a 2045 3120 3d20 7072 +.|i16 : E1 = pr │ │ │ │ -0001c200: 756e 6520 6578 7465 7269 6f72 4578 744d une exteriorExtM │ │ │ │ -0001c210: 6f64 756c 6528 662c 204d 532c 2072 6573 odule(f, MS, res │ │ │ │ -0001c220: 466c 6429 3b20 2020 2020 2020 2020 7c0a Fld); |. │ │ │ │ -0001c230: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001c1e0: 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 4531 -----+.|i16 : E1 │ │ │ │ +0001c1f0: 203d 2070 7275 6e65 2065 7874 6572 696f = prune exterio │ │ │ │ +0001c200: 7245 7874 4d6f 6475 6c65 2866 2c20 4d53 rExtModule(f, MS │ │ │ │ +0001c210: 2c20 7265 7346 6c64 293b 2020 2020 2020 , resFld); │ │ │ │ +0001c220: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001c230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001c270: 3137 203a 2072 696e 6720 4531 2020 2020 17 : ring E1 │ │ │ │ +0001c260: 2d2b 0a7c 6931 3720 3a20 7269 6e67 2045 -+.|i17 : ring E │ │ │ │ +0001c270: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001c290: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c2a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001c2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c2e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -0001c2f0: 206b 6b5b 5820 2e2e 5820 2c20 6520 2e2e kk[X ..X , e .. │ │ │ │ -0001c300: 6520 5d20 2020 2020 2020 2020 2020 2020 e ] │ │ │ │ -0001c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c320: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001c330: 2020 2030 2020 2032 2020 2030 2020 2032 0 2 0 2 │ │ │ │ +0001c2d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c2e0: 6f31 3720 3d20 6b6b 5b58 202e 2e58 202c o17 = kk[X ..X , │ │ │ │ +0001c2f0: 2065 202e 2e65 205d 2020 2020 2020 2020 e ..e ] │ │ │ │ +0001c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c310: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001c320: 2020 2020 2020 2020 3020 2020 3220 2020 0 2 │ │ │ │ +0001c330: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ 0001c340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c360: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001c350: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c3a0: 2020 7c0a 7c6f 3137 203a 2050 6f6c 796e |.|o17 : Polyn │ │ │ │ -0001c3b0: 6f6d 6961 6c52 696e 672c 2033 2073 6b65 omialRing, 3 ske │ │ │ │ -0001c3c0: 7720 636f 6d6d 7574 6174 6976 6520 7661 w commutative va │ │ │ │ -0001c3d0: 7269 6162 6c65 2873 2920 2020 2020 2020 riable(s) │ │ │ │ -0001c3e0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001c390: 2020 2020 2020 207c 0a7c 6f31 3720 3a20 |.|o17 : │ │ │ │ +0001c3a0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2c20 PolynomialRing, │ │ │ │ +0001c3b0: 3320 736b 6577 2063 6f6d 6d75 7461 7469 3 skew commutati │ │ │ │ +0001c3c0: 7665 2076 6172 6961 626c 6528 7329 2020 ve variable(s) │ │ │ │ +0001c3d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001c3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001c420: 7c69 3138 203a 2065 7852 696e 6720 3d20 |i18 : exRing = │ │ │ │ -0001c430: 6b6b 5b65 5f30 2c65 5f31 2c65 5f32 2c20 kk[e_0,e_1,e_2, │ │ │ │ -0001c440: 536b 6577 436f 6d6d 7574 6174 6976 6520 SkewCommutative │ │ │ │ -0001c450: 3d3e 7472 7565 5d20 2020 2020 7c0a 7c20 =>true] |.| │ │ │ │ +0001c410: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 5269 ---+.|i18 : exRi │ │ │ │ +0001c420: 6e67 203d 206b 6b5b 655f 302c 655f 312c ng = kk[e_0,e_1, │ │ │ │ +0001c430: 655f 322c 2053 6b65 7743 6f6d 6d75 7461 e_2, SkewCommuta │ │ │ │ +0001c440: 7469 7665 203d 3e74 7275 655d 2020 2020 tive =>true] │ │ │ │ +0001c450: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c490: 2020 2020 2020 2020 2020 7c0a 7c6f 3138 |.|o18 │ │ │ │ -0001c4a0: 203d 2065 7852 696e 6720 2020 2020 2020 = exRing │ │ │ │ +0001c480: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c490: 0a7c 6f31 3820 3d20 6578 5269 6e67 2020 .|o18 = exRing │ │ │ │ +0001c4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c4d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001c4c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c510: 2020 2020 2020 7c0a 7c6f 3138 203a 2050 |.|o18 : P │ │ │ │ -0001c520: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033 olynomialRing, 3 │ │ │ │ -0001c530: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -0001c540: 6520 7661 7269 6162 6c65 2873 2920 2020 e variable(s) │ │ │ │ -0001c550: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001c500: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0001c510: 3820 3a20 506f 6c79 6e6f 6d69 616c 5269 8 : PolynomialRi │ │ │ │ +0001c520: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75 ng, 3 skew commu │ │ │ │ +0001c530: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +0001c540: 7329 2020 2020 2020 207c 0a2b 2d2d 2d2d s) |.+---- │ │ │ │ +0001c550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c590: 2d2d 2b0a 0a57 6520 6361 6e20 616c 736f --+..We can also │ │ │ │ -0001c5a0: 2063 6f6e 7374 7275 6374 2074 6865 2065 construct the e │ │ │ │ -0001c5b0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ -0001c5c0: 2061 7320 6120 6269 6772 6164 6564 206d as a bigraded m │ │ │ │ -0001c5d0: 6f64 756c 652c 206f 7665 7220 6120 7269 odule, over a ri │ │ │ │ -0001c5e0: 6e67 0a53 4520 7468 6174 2068 6173 2062 ng.SE that has b │ │ │ │ -0001c5f0: 6f74 6820 706f 6c79 6e6f 6d69 616c 2076 oth polynomial v │ │ │ │ -0001c600: 6172 6961 626c 6573 206c 696b 6520 5320 ariables like S │ │ │ │ -0001c610: 616e 6420 6578 7465 7269 6f72 2076 6172 and exterior var │ │ │ │ -0001c620: 6961 626c 6573 206c 696b 6520 452e 2054 iables like E. T │ │ │ │ -0001c630: 6865 0a70 6f6c 796e 6f6d 6961 6c20 7661 he.polynomial va │ │ │ │ -0001c640: 7269 6162 6c65 7320 6861 7665 2064 6567 riables have deg │ │ │ │ -0001c650: 7265 6573 207b 312c 307d 2e20 5468 6520 rees {1,0}. The │ │ │ │ -0001c660: 6578 7465 7269 6f72 2076 6172 6961 626c exterior variabl │ │ │ │ -0001c670: 6573 2068 6176 6520 6465 6772 6565 730a es have degrees. │ │ │ │ -0001c680: 7b64 6567 2066 665f 692c 2031 7d2e 0a0a {deg ff_i, 1}... │ │ │ │ -0001c690: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001c580: 2d2d 2d2d 2d2d 2d2b 0a0a 5765 2063 616e -------+..We can │ │ │ │ +0001c590: 2061 6c73 6f20 636f 6e73 7472 7563 7420 also construct │ │ │ │ +0001c5a0: 7468 6520 6578 7465 7269 6f72 4578 744d the exteriorExtM │ │ │ │ +0001c5b0: 6f64 756c 6520 6173 2061 2062 6967 7261 odule as a bigra │ │ │ │ +0001c5c0: 6465 6420 6d6f 6475 6c65 2c20 6f76 6572 ded module, over │ │ │ │ +0001c5d0: 2061 2072 696e 670a 5345 2074 6861 7420 a ring.SE that │ │ │ │ +0001c5e0: 6861 7320 626f 7468 2070 6f6c 796e 6f6d has both polynom │ │ │ │ +0001c5f0: 6961 6c20 7661 7269 6162 6c65 7320 6c69 ial variables li │ │ │ │ +0001c600: 6b65 2053 2061 6e64 2065 7874 6572 696f ke S and exterio │ │ │ │ +0001c610: 7220 7661 7269 6162 6c65 7320 6c69 6b65 r variables like │ │ │ │ +0001c620: 2045 2e20 5468 650a 706f 6c79 6e6f 6d69 E. The.polynomi │ │ │ │ +0001c630: 616c 2076 6172 6961 626c 6573 2068 6176 al variables hav │ │ │ │ +0001c640: 6520 6465 6772 6565 7320 7b31 2c30 7d2e e degrees {1,0}. │ │ │ │ +0001c650: 2054 6865 2065 7874 6572 696f 7220 7661 The exterior va │ │ │ │ +0001c660: 7269 6162 6c65 7320 6861 7665 2064 6567 riables have deg │ │ │ │ +0001c670: 7265 6573 0a7b 6465 6720 6666 5f69 2c20 rees.{deg ff_i, │ │ │ │ +0001c680: 317d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 1}...+---------- │ │ │ │ +0001c690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c6c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 3a20 -------+.|i19 : │ │ │ │ -0001c6d0: 4531 203d 2070 7275 6e65 2065 7874 6572 E1 = prune exter │ │ │ │ -0001c6e0: 696f 7245 7874 4d6f 6475 6c65 2866 2c20 iorExtModule(f, │ │ │ │ -0001c6f0: 4d53 2c20 7265 7346 6c64 293b 2020 2020 MS, resFld); │ │ │ │ -0001c700: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001c6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0001c6c0: 3139 203a 2045 3120 3d20 7072 756e 6520 19 : E1 = prune │ │ │ │ +0001c6d0: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ +0001c6e0: 6528 662c 204d 532c 2072 6573 466c 6429 e(f, MS, resFld) │ │ │ │ +0001c6f0: 3b20 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d ; |.+-------- │ │ │ │ +0001c700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c730: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 ---------+.|i20 │ │ │ │ -0001c740: 3a20 7269 6e67 2045 3120 2020 2020 2020 : ring E1 │ │ │ │ +0001c720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0001c730: 7c69 3230 203a 2072 696e 6720 4531 2020 |i20 : ring E1 │ │ │ │ +0001c740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c770: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c760: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001c770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7a0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0001c7b0: 3020 3d20 6b6b 5b58 202e 2e58 202c 2065 0 = kk[X ..X , e │ │ │ │ -0001c7c0: 202e 2e65 205d 2020 2020 2020 2020 2020 ..e ] │ │ │ │ -0001c7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c7e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001c7f0: 2030 2020 2032 2020 2030 2020 2032 2020 0 2 0 2 │ │ │ │ +0001c7a0: 7c0a 7c6f 3230 203d 206b 6b5b 5820 2e2e |.|o20 = kk[X .. │ │ │ │ +0001c7b0: 5820 2c20 6520 2e2e 6520 5d20 2020 2020 X , e ..e ] │ │ │ │ +0001c7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001c7d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001c7e0: 2020 2020 2020 3020 2020 3220 2020 3020 0 2 0 │ │ │ │ +0001c7f0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001c800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c810: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001c810: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001c820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c850: 2020 2020 2020 7c0a 7c6f 3230 203a 2050 |.|o20 : P │ │ │ │ -0001c860: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033 olynomialRing, 3 │ │ │ │ -0001c870: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -0001c880: 6520 7661 7269 6162 6c65 2873 2920 207c e variable(s) | │ │ │ │ -0001c890: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0001c840: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ +0001c850: 3020 3a20 506f 6c79 6e6f 6d69 616c 5269 0 : PolynomialRi │ │ │ │ +0001c860: 6e67 2c20 3320 736b 6577 2063 6f6d 6d75 ng, 3 skew commu │ │ │ │ +0001c870: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +0001c880: 7329 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d s) |.+--------- │ │ │ │ +0001c890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001c8c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3231 203a --------+.|i21 : │ │ │ │ -0001c8d0: 2065 7852 696e 6720 3d20 6b6b 5b65 5f30 exRing = kk[e_0 │ │ │ │ -0001c8e0: 2c65 5f31 2c65 5f32 2c20 536b 6577 436f ,e_1,e_2, SkewCo │ │ │ │ -0001c8f0: 6d6d 7574 6174 6976 6520 3d3e 7472 7565 mmutative =>true │ │ │ │ -0001c900: 5d7c 0a7c 2020 2020 2020 2020 2020 2020 ]|.| │ │ │ │ +0001c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001c8c0: 6932 3120 3a20 6578 5269 6e67 203d 206b i21 : exRing = k │ │ │ │ +0001c8d0: 6b5b 655f 302c 655f 312c 655f 322c 2053 k[e_0,e_1,e_2, S │ │ │ │ +0001c8e0: 6b65 7743 6f6d 6d75 7461 7469 7665 203d kewCommutative = │ │ │ │ +0001c8f0: 3e74 7275 655d 7c0a 7c20 2020 2020 2020 >true]|.| │ │ │ │ +0001c900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c930: 2020 2020 2020 2020 2020 7c0a 7c6f 3231 |.|o21 │ │ │ │ -0001c940: 203d 2065 7852 696e 6720 2020 2020 2020 = exRing │ │ │ │ +0001c920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001c930: 0a7c 6f32 3120 3d20 6578 5269 6e67 2020 .|o21 = exRing │ │ │ │ +0001c940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c970: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001c960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001c970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001c990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001c9a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001c9b0: 3231 203a 2050 6f6c 796e 6f6d 6961 6c52 21 : PolynomialR │ │ │ │ -0001c9c0: 696e 672c 2033 2073 6b65 7720 636f 6d6d ing, 3 skew comm │ │ │ │ -0001c9d0: 7574 6174 6976 6520 7661 7269 6162 6c65 utative variable │ │ │ │ -0001c9e0: 2873 2920 207c 0a2b 2d2d 2d2d 2d2d 2d2d (s) |.+-------- │ │ │ │ +0001c9a0: 207c 0a7c 6f32 3120 3a20 506f 6c79 6e6f |.|o21 : Polyno │ │ │ │ +0001c9b0: 6d69 616c 5269 6e67 2c20 3320 736b 6577 mialRing, 3 skew │ │ │ │ +0001c9c0: 2063 6f6d 6d75 7461 7469 7665 2076 6172 commutative var │ │ │ │ +0001c9d0: 6961 626c 6528 7329 2020 7c0a 2b2d 2d2d iable(s) |.+--- │ │ │ │ +0001c9e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001c9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ca00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001ca20: 0a54 6f20 7365 6520 7468 6174 2074 6869 .To see that thi │ │ │ │ -0001ca30: 7320 6973 2072 6561 6c6c 7920 7468 6520 s is really the │ │ │ │ -0001ca40: 7361 6d65 206d 6f64 756c 652c 2077 6974 same module, wit │ │ │ │ -0001ca50: 6820 6120 6d6f 7265 2063 6f6d 706c 6578 h a more complex │ │ │ │ -0001ca60: 2067 7261 6469 6e67 2c20 7765 2063 616e grading, we can │ │ │ │ -0001ca70: 0a62 7269 6e67 2069 7420 6f76 6572 2074 .bring it over t │ │ │ │ -0001ca80: 6f20 6120 7075 7265 2065 7874 6572 696f o a pure exterio │ │ │ │ -0001ca90: 7220 616c 6765 6272 612e 204e 6f74 6520 r algebra. Note │ │ │ │ -0001caa0: 7468 6174 2074 6865 206e 6563 6573 7361 that the necessa │ │ │ │ -0001cab0: 7279 206d 6170 206f 6620 7269 6e67 730a ry map of rings. │ │ │ │ -0001cac0: 6d75 7374 2063 6f6e 7461 696e 2061 2044 must contain a D │ │ │ │ -0001cad0: 6567 7265 654d 6170 206f 7074 696f 6e2e egreeMap option. │ │ │ │ -0001cae0: 2049 6e20 6765 6e65 7261 6c20 7765 2063 In general we c │ │ │ │ -0001caf0: 6f75 6c64 206f 6e6c 7920 7461 6b65 2074 ould only take t │ │ │ │ -0001cb00: 6865 2064 6567 7265 6573 206f 660a 7468 he degrees of.th │ │ │ │ -0001cb10: 6520 6765 6e65 7261 746f 7273 206f 6620 e generators of │ │ │ │ -0001cb20: 7468 6520 6578 7465 7269 6f72 2061 6c67 the exterior alg │ │ │ │ -0001cb30: 6562 7261 2074 6f20 6265 2074 6865 2067 ebra to be the g │ │ │ │ -0001cb40: 6364 206f 6620 2074 6865 2064 6567 2066 cd of the deg f │ │ │ │ -0001cb50: 665f 6920 3b20 696e 2074 6865 0a65 7861 f_i ; in the.exa │ │ │ │ -0001cb60: 6d70 6c65 2061 626f 7665 2074 6869 7320 mple above this │ │ │ │ -0001cb70: 6973 2031 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d is 1...+-------- │ │ │ │ +0001ca10: 2d2d 2d2b 0a0a 546f 2073 6565 2074 6861 ---+..To see tha │ │ │ │ +0001ca20: 7420 7468 6973 2069 7320 7265 616c 6c79 t this is really │ │ │ │ +0001ca30: 2074 6865 2073 616d 6520 6d6f 6475 6c65 the same module │ │ │ │ +0001ca40: 2c20 7769 7468 2061 206d 6f72 6520 636f , with a more co │ │ │ │ +0001ca50: 6d70 6c65 7820 6772 6164 696e 672c 2077 mplex grading, w │ │ │ │ +0001ca60: 6520 6361 6e0a 6272 696e 6720 6974 206f e can.bring it o │ │ │ │ +0001ca70: 7665 7220 746f 2061 2070 7572 6520 6578 ver to a pure ex │ │ │ │ +0001ca80: 7465 7269 6f72 2061 6c67 6562 7261 2e20 terior algebra. │ │ │ │ +0001ca90: 4e6f 7465 2074 6861 7420 7468 6520 6e65 Note that the ne │ │ │ │ +0001caa0: 6365 7373 6172 7920 6d61 7020 6f66 2072 cessary map of r │ │ │ │ +0001cab0: 696e 6773 0a6d 7573 7420 636f 6e74 6169 ings.must contai │ │ │ │ +0001cac0: 6e20 6120 4465 6772 6565 4d61 7020 6f70 n a DegreeMap op │ │ │ │ +0001cad0: 7469 6f6e 2e20 496e 2067 656e 6572 616c tion. In general │ │ │ │ +0001cae0: 2077 6520 636f 756c 6420 6f6e 6c79 2074 we could only t │ │ │ │ +0001caf0: 616b 6520 7468 6520 6465 6772 6565 7320 ake the degrees │ │ │ │ +0001cb00: 6f66 0a74 6865 2067 656e 6572 6174 6f72 of.the generator │ │ │ │ +0001cb10: 7320 6f66 2074 6865 2065 7874 6572 696f s of the exterio │ │ │ │ +0001cb20: 7220 616c 6765 6272 6120 746f 2062 6520 r algebra to be │ │ │ │ +0001cb30: 7468 6520 6763 6420 6f66 2020 7468 6520 the gcd of the │ │ │ │ +0001cb40: 6465 6720 6666 5f69 203b 2069 6e20 7468 deg ff_i ; in th │ │ │ │ +0001cb50: 650a 6578 616d 706c 6520 6162 6f76 6520 e.example above │ │ │ │ +0001cb60: 7468 6973 2069 7320 312e 0a0a 2b2d 2d2d this is 1...+--- │ │ │ │ +0001cb70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cb80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cb90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cbc0: 2d2d 2b0a 7c69 3232 203a 2071 203d 206d --+.|i22 : q = m │ │ │ │ -0001cbd0: 6170 2865 7852 696e 672c 2072 696e 6720 ap(exRing, ring │ │ │ │ -0001cbe0: 4531 2c20 7b33 3a30 2c65 5f30 2c65 5f31 E1, {3:0,e_0,e_1 │ │ │ │ -0001cbf0: 2c65 5f32 7d2c 2044 6567 7265 654d 6170 ,e_2}, DegreeMap │ │ │ │ -0001cc00: 203d 3e20 6420 2d3e 207b 645f 317d 297c => d -> {d_1})| │ │ │ │ -0001cc10: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001cbb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 -------+.|i22 : │ │ │ │ +0001cbc0: 7120 3d20 6d61 7028 6578 5269 6e67 2c20 q = map(exRing, │ │ │ │ +0001cbd0: 7269 6e67 2045 312c 207b 333a 302c 655f ring E1, {3:0,e_ │ │ │ │ +0001cbe0: 302c 655f 312c 655f 327d 2c20 4465 6772 0,e_1,e_2}, Degr │ │ │ │ +0001cbf0: 6565 4d61 7020 3d3e 2064 202d 3e20 7b64 eeMap => d -> {d │ │ │ │ +0001cc00: 5f31 7d29 7c0a 7c20 2020 2020 2020 2020 _1})|.| │ │ │ │ +0001cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001cc60: 3232 203d 206d 6170 2028 6578 5269 6e67 22 = map (exRing │ │ │ │ -0001cc70: 2c20 6b6b 5b58 202e 2e58 202c 2065 202e , kk[X ..X , e . │ │ │ │ -0001cc80: 2e65 205d 2c20 7b30 2c20 302c 2030 2c20 .e ], {0, 0, 0, │ │ │ │ -0001cc90: 6520 2c20 6520 2c20 6520 7d29 2020 2020 e , e , e }) │ │ │ │ -0001cca0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0001ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ccc0: 2020 2030 2020 2032 2020 2030 2020 2032 0 2 0 2 │ │ │ │ -0001ccd0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ -0001cce0: 2020 3120 2020 3220 2020 2020 2020 2020 1 2 │ │ │ │ -0001ccf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001cc50: 207c 0a7c 6f32 3220 3d20 6d61 7020 2865 |.|o22 = map (e │ │ │ │ +0001cc60: 7852 696e 672c 206b 6b5b 5820 2e2e 5820 xRing, kk[X ..X │ │ │ │ +0001cc70: 2c20 6520 2e2e 6520 5d2c 207b 302c 2030 , e ..e ], {0, 0 │ │ │ │ +0001cc80: 2c20 302c 2065 202c 2065 202c 2065 207d , 0, e , e , e } │ │ │ │ +0001cc90: 2920 2020 2020 2020 2020 2020 2020 7c0a ) |. │ │ │ │ +0001cca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001ccb0: 2020 2020 2020 2020 3020 2020 3220 2020 0 2 │ │ │ │ +0001ccc0: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ +0001ccd0: 2020 2030 2020 2031 2020 2032 2020 2020 0 1 2 │ │ │ │ +0001cce0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd40: 2020 207c 0a7c 6f32 3220 3a20 5269 6e67 |.|o22 : Ring │ │ │ │ -0001cd50: 4d61 7020 6578 5269 6e67 203c 2d2d 206b Map exRing <-- k │ │ │ │ -0001cd60: 6b5b 5820 2e2e 5820 2c20 6520 2e2e 6520 k[X ..X , e ..e │ │ │ │ -0001cd70: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ -0001cd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cd90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001cda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdb0: 3020 2020 3220 2020 3020 2020 3220 2020 0 2 0 2 │ │ │ │ +0001cd30: 2020 2020 2020 2020 7c0a 7c6f 3232 203a |.|o22 : │ │ │ │ +0001cd40: 2052 696e 674d 6170 2065 7852 696e 6720 RingMap exRing │ │ │ │ +0001cd50: 3c2d 2d20 6b6b 5b58 202e 2e58 202c 2065 <-- kk[X ..X , e │ │ │ │ +0001cd60: 202e 2e65 205d 2020 2020 2020 2020 2020 ..e ] │ │ │ │ +0001cd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cd80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001cd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001cda0: 2020 2020 2030 2020 2032 2020 2030 2020 0 2 0 │ │ │ │ +0001cdb0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0001cdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cdd0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001cdd0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0001cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ce20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 ----------+.|i23 │ │ │ │ -0001ce30: 203a 2045 3220 3d20 636f 6b65 7220 7120 : E2 = coker q │ │ │ │ -0001ce40: 7072 6573 656e 7461 7469 6f6e 2045 313b presentation E1; │ │ │ │ +0001ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ce20: 0a7c 6932 3320 3a20 4532 203d 2063 6f6b .|i23 : E2 = cok │ │ │ │ +0001ce30: 6572 2071 2070 7265 7365 6e74 6174 696f er q presentatio │ │ │ │ +0001ce40: 6e20 4531 3b20 2020 2020 2020 2020 2020 n E1; │ │ │ │ 0001ce50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ce70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001ce60: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001ce70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ce80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ce90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cec0: 2d2d 2d2d 2b0a 7c69 3234 203a 2068 6628 ----+.|i24 : hf( │ │ │ │ -0001ced0: 2d35 2e2e 352c 4532 2920 3d3d 2068 6628 -5..5,E2) == hf( │ │ │ │ -0001cee0: 2d35 2e2e 352c 4529 2020 2020 2020 2020 -5..5,E) │ │ │ │ +0001ceb0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3420 ---------+.|i24 │ │ │ │ +0001cec0: 3a20 6866 282d 352e 2e35 2c45 3229 203d : hf(-5..5,E2) = │ │ │ │ +0001ced0: 3d20 6866 282d 352e 2e35 2c45 2920 2020 = hf(-5..5,E) │ │ │ │ +0001cee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001cf00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001cf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cf50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001cf60: 7c6f 3234 203d 2074 7275 6520 2020 2020 |o24 = true │ │ │ │ +0001cf50: 2020 207c 0a7c 6f32 3420 3d20 7472 7565 |.|o24 = true │ │ │ │ +0001cf60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001cf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001cfa0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001cfa0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0001cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001cfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001cff0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -0001d000: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0001d010: 2a20 2a6e 6f74 6520 6578 7465 7269 6f72 * *note exterior │ │ │ │ -0001d020: 546f 724d 6f64 756c 653a 2065 7874 6572 TorModule: exter │ │ │ │ -0001d030: 696f 7254 6f72 4d6f 6475 6c65 2c20 2d2d iorTorModule, -- │ │ │ │ -0001d040: 2054 6f72 2061 7320 6120 6d6f 6475 6c65 Tor as a module │ │ │ │ -0001d050: 206f 7665 7220 616e 0a20 2020 2065 7874 over an. ext │ │ │ │ -0001d060: 6572 696f 7220 616c 6765 6272 6120 6f72 erior algebra or │ │ │ │ -0001d070: 2062 6967 7261 6465 6420 616c 6765 6272 bigraded algebr │ │ │ │ -0001d080: 610a 2020 2a20 2a6e 6f74 6520 6d61 6b65 a. * *note make │ │ │ │ -0001d090: 4d6f 6475 6c65 3a20 6d61 6b65 4d6f 6475 Module: makeModu │ │ │ │ -0001d0a0: 6c65 2c20 2d2d 206d 616b 6573 2061 204d le, -- makes a M │ │ │ │ -0001d0b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063 odule out of a c │ │ │ │ -0001d0c0: 6f6c 6c65 6374 696f 6e20 6f66 0a20 2020 ollection of. │ │ │ │ -0001d0d0: 206d 6f64 756c 6573 2061 6e64 206d 6170 modules and map │ │ │ │ -0001d0e0: 730a 0a57 6179 7320 746f 2075 7365 2065 s..Ways to use e │ │ │ │ -0001d0f0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ -0001d100: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -0001d110: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d120: 0a0a 2020 2a20 2265 7874 6572 696f 7245 .. * "exteriorE │ │ │ │ -0001d130: 7874 4d6f 6475 6c65 284d 6174 7269 782c xtModule(Matrix, │ │ │ │ -0001d140: 4d6f 6475 6c65 2922 0a20 202a 2022 6578 Module)". * "ex │ │ │ │ -0001d150: 7465 7269 6f72 4578 744d 6f64 756c 6528 teriorExtModule( │ │ │ │ -0001d160: 4d61 7472 6978 2c4d 6f64 756c 652c 4d6f Matrix,Module,Mo │ │ │ │ -0001d170: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ -0001d180: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0001d190: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0001d1a0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0001d1b0: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ -0001d1c0: 653a 2065 7874 6572 696f 7245 7874 4d6f e: exteriorExtMo │ │ │ │ -0001d1d0: 6475 6c65 2c20 6973 2061 202a 6e6f 7465 dule, is a *note │ │ │ │ -0001d1e0: 206d 6574 686f 640a 6675 6e63 7469 6f6e method.function │ │ │ │ -0001d1f0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0001d200: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ -0001d210: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0001cfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0001cff0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0001d000: 3d0a 0a20 202a 202a 6e6f 7465 2065 7874 =.. * *note ext │ │ │ │ +0001d010: 6572 696f 7254 6f72 4d6f 6475 6c65 3a20 eriorTorModule: │ │ │ │ +0001d020: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001d030: 652c 202d 2d20 546f 7220 6173 2061 206d e, -- Tor as a m │ │ │ │ +0001d040: 6f64 756c 6520 6f76 6572 2061 6e0a 2020 odule over an. │ │ │ │ +0001d050: 2020 6578 7465 7269 6f72 2061 6c67 6562 exterior algeb │ │ │ │ +0001d060: 7261 206f 7220 6269 6772 6164 6564 2061 ra or bigraded a │ │ │ │ +0001d070: 6c67 6562 7261 0a20 202a 202a 6e6f 7465 lgebra. * *note │ │ │ │ +0001d080: 206d 616b 654d 6f64 756c 653a 206d 616b makeModule: mak │ │ │ │ +0001d090: 654d 6f64 756c 652c 202d 2d20 6d61 6b65 eModule, -- make │ │ │ │ +0001d0a0: 7320 6120 4d6f 6475 6c65 206f 7574 206f s a Module out o │ │ │ │ +0001d0b0: 6620 6120 636f 6c6c 6563 7469 6f6e 206f f a collection o │ │ │ │ +0001d0c0: 660a 2020 2020 6d6f 6475 6c65 7320 616e f. modules an │ │ │ │ +0001d0d0: 6420 6d61 7073 0a0a 5761 7973 2074 6f20 d maps..Ways to │ │ │ │ +0001d0e0: 7573 6520 6578 7465 7269 6f72 4578 744d use exteriorExtM │ │ │ │ +0001d0f0: 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d 3d3d odule:.========= │ │ │ │ +0001d100: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d110: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465 =====.. * "exte │ │ │ │ +0001d120: 7269 6f72 4578 744d 6f64 756c 6528 4d61 riorExtModule(Ma │ │ │ │ +0001d130: 7472 6978 2c4d 6f64 756c 6529 220a 2020 trix,Module)". │ │ │ │ +0001d140: 2a20 2265 7874 6572 696f 7245 7874 4d6f * "exteriorExtMo │ │ │ │ +0001d150: 6475 6c65 284d 6174 7269 782c 4d6f 6475 dule(Matrix,Modu │ │ │ │ +0001d160: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72 le,Module)"..For │ │ │ │ +0001d170: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0001d180: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0001d190: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0001d1a0: 6e6f 7465 2065 7874 6572 696f 7245 7874 note exteriorExt │ │ │ │ +0001d1b0: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72 Module: exterior │ │ │ │ +0001d1c0: 4578 744d 6f64 756c 652c 2069 7320 6120 ExtModule, is a │ │ │ │ +0001d1d0: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ +0001d1e0: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ +0001d1f0: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ +0001d200: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +0001d210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d260: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -0001d270: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -0001d280: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -0001d290: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -0001d2a0: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -0001d2b0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -0001d2c0: 7061 636b 6167 6573 2f0a 436f 6d70 6c65 packages/.Comple │ │ │ │ -0001d2d0: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -0001d2e0: 736f 6c75 7469 6f6e 732e 6d32 3a34 3236 solutions.m2:426 │ │ │ │ -0001d2f0: 333a 302e 0a1f 0a46 696c 653a 2043 6f6d 3:0....File: Com │ │ │ │ -0001d300: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -0001d310: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66 nResolutions.inf │ │ │ │ -0001d320: 6f2c 204e 6f64 653a 2065 7874 6572 696f o, Node: exterio │ │ │ │ -0001d330: 7248 6f6d 6f6c 6f67 794d 6f64 756c 652c rHomologyModule, │ │ │ │ -0001d340: 204e 6578 743a 2065 7874 6572 696f 7254 Next: exteriorT │ │ │ │ -0001d350: 6f72 4d6f 6475 6c65 2c20 5072 6576 3a20 orModule, Prev: │ │ │ │ -0001d360: 6578 7465 7269 6f72 4578 744d 6f64 756c exteriorExtModul │ │ │ │ -0001d370: 652c 2055 703a 2054 6f70 0a0a 6578 7465 e, Up: Top..exte │ │ │ │ -0001d380: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ -0001d390: 6c65 202d 2d20 4d61 6b65 2074 6865 2068 le -- Make the h │ │ │ │ -0001d3a0: 6f6d 6f6c 6f67 7920 6f66 2061 2063 6f6d omology of a com │ │ │ │ -0001d3b0: 706c 6578 2069 6e74 6f20 6120 6d6f 6475 plex into a modu │ │ │ │ -0001d3c0: 6c65 206f 7665 7220 616e 2065 7874 6572 le over an exter │ │ │ │ -0001d3d0: 696f 7220 616c 6765 6272 610a 2a2a 2a2a ior algebra.**** │ │ │ │ +0001d250: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +0001d260: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +0001d270: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +0001d280: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +0001d290: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +0001d2a0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +0001d2b0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ +0001d2c0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0001d2d0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ +0001d2e0: 323a 3432 3633 3a30 2e0a 1f0a 4669 6c65 2:4263:0....File │ │ │ │ +0001d2f0: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ +0001d300: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +0001d310: 732e 696e 666f 2c20 4e6f 6465 3a20 6578 s.info, Node: ex │ │ │ │ +0001d320: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f teriorHomologyMo │ │ │ │ +0001d330: 6475 6c65 2c20 4e65 7874 3a20 6578 7465 dule, Next: exte │ │ │ │ +0001d340: 7269 6f72 546f 724d 6f64 756c 652c 2050 riorTorModule, P │ │ │ │ +0001d350: 7265 763a 2065 7874 6572 696f 7245 7874 rev: exteriorExt │ │ │ │ +0001d360: 4d6f 6475 6c65 2c20 5570 3a20 546f 700a Module, Up: Top. │ │ │ │ +0001d370: 0a65 7874 6572 696f 7248 6f6d 6f6c 6f67 .exteriorHomolog │ │ │ │ +0001d380: 794d 6f64 756c 6520 2d2d 204d 616b 6520 yModule -- Make │ │ │ │ +0001d390: 7468 6520 686f 6d6f 6c6f 6779 206f 6620 the homology of │ │ │ │ +0001d3a0: 6120 636f 6d70 6c65 7820 696e 746f 2061 a complex into a │ │ │ │ +0001d3b0: 206d 6f64 756c 6520 6f76 6572 2061 6e20 module over an │ │ │ │ +0001d3c0: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ +0001d3d0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0001d3e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d3f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d400: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001d420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001d430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0001d440: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0001d450: 204d 203d 2065 7874 6572 696f 7248 6f6d M = exteriorHom │ │ │ │ -0001d460: 6f6c 6f67 794d 6f64 756c 6528 6666 2c20 ologyModule(ff, │ │ │ │ -0001d470: 4329 0a20 202a 2049 6e70 7574 733a 0a20 C). * Inputs:. │ │ │ │ -0001d480: 2020 2020 202a 2066 662c 2061 202a 6e6f * ff, a *no │ │ │ │ -0001d490: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -0001d4a0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -0001d4b0: 2c20 4d61 7472 6978 206f 6620 656c 656d , Matrix of elem │ │ │ │ -0001d4c0: 656e 7473 2074 6861 7420 6172 650a 2020 ents that are. │ │ │ │ -0001d4d0: 2020 2020 2020 686f 6d6f 746f 7069 6320 homotopic │ │ │ │ -0001d4e0: 746f 2030 206f 6e20 430a 2020 2020 2020 to 0 on C. │ │ │ │ -0001d4f0: 2a20 432c 2061 202a 6e6f 7465 2063 6f6d * C, a *note com │ │ │ │ -0001d500: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -0001d510: 2943 6f6d 706c 6578 2c2c 200a 2020 2a20 )Complex,, . * │ │ │ │ -0001d520: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0001d530: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0001d540: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0001d550: 6329 4d6f 6475 6c65 2c2c 200a 0a44 6573 c)Module,, ..Des │ │ │ │ -0001d560: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0001d570: 3d3d 3d3d 0a0a 4173 7375 6d69 6e67 2074 ====..Assuming t │ │ │ │ -0001d580: 6861 7420 7468 6520 656c 656d 656e 7473 hat the elements │ │ │ │ -0001d590: 206f 6620 7468 6520 3178 6320 6d61 7472 of the 1xc matr │ │ │ │ -0001d5a0: 6978 2066 6620 6172 6520 6e75 6c6c 2d68 ix ff are null-h │ │ │ │ -0001d5b0: 6f6d 6f74 6f70 6963 206f 6e20 432c 2074 omotopic on C, t │ │ │ │ -0001d5c0: 6865 0a73 6372 6970 7420 7265 7475 726e he.script return │ │ │ │ -0001d5d0: 7320 7468 6520 6469 7265 6374 2073 756d s the direct sum │ │ │ │ -0001d5e0: 206f 6620 7468 6520 686f 6d6f 6c6f 6779 of the homology │ │ │ │ -0001d5f0: 206f 6620 4320 6173 2061 206d 6f64 756c of C as a modul │ │ │ │ -0001d600: 6520 6f76 6572 2061 206e 6577 2072 696e e over a new rin │ │ │ │ -0001d610: 672c 0a63 6f6e 7369 7374 696e 6720 6f66 g,.consisting of │ │ │ │ -0001d620: 2072 696e 6720 4320 7769 7468 2063 2065 ring C with c e │ │ │ │ -0001d630: 7874 6572 696f 7220 7661 7269 6162 6c65 xterior variable │ │ │ │ -0001d640: 7320 6164 6a6f 696e 6564 2e20 5468 6520 s adjoined. The │ │ │ │ -0001d650: 7363 7269 7074 2069 7320 7468 6520 6d61 script is the ma │ │ │ │ -0001d660: 696e 0a63 6f6d 706f 6e65 6e74 206f 6620 in.component of │ │ │ │ -0001d670: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ -0001d680: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d e..See also.==== │ │ │ │ -0001d690: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0001d6a0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ -0001d6b0: 653a 2065 7874 6572 696f 7254 6f72 4d6f e: exteriorTorMo │ │ │ │ -0001d6c0: 6475 6c65 2c20 2d2d 2054 6f72 2061 7320 dule, -- Tor as │ │ │ │ -0001d6d0: 6120 6d6f 6475 6c65 206f 7665 7220 616e a module over an │ │ │ │ -0001d6e0: 0a20 2020 2065 7874 6572 696f 7220 616c . exterior al │ │ │ │ -0001d6f0: 6765 6272 6120 6f72 2062 6967 7261 6465 gebra or bigrade │ │ │ │ -0001d700: 6420 616c 6765 6272 610a 2020 2a20 2a6e d algebra. * *n │ │ │ │ -0001d710: 6f74 6520 6d61 6b65 486f 6d6f 746f 7069 ote makeHomotopi │ │ │ │ -0001d720: 6573 4f6e 486f 6d6f 6c6f 6779 3a20 6d61 esOnHomology: ma │ │ │ │ -0001d730: 6b65 486f 6d6f 746f 7069 6573 4f6e 486f keHomotopiesOnHo │ │ │ │ -0001d740: 6d6f 6c6f 6779 2c20 2d2d 2048 6f6d 6f6c mology, -- Homol │ │ │ │ -0001d750: 6f67 7920 6f66 2061 0a20 2020 2063 6f6d ogy of a. com │ │ │ │ -0001d760: 706c 6578 2061 7320 6578 7465 7269 6f72 plex as exterior │ │ │ │ -0001d770: 206d 6f64 756c 650a 0a57 6179 7320 746f module..Ways to │ │ │ │ -0001d780: 2075 7365 2065 7874 6572 696f 7248 6f6d use exteriorHom │ │ │ │ -0001d790: 6f6c 6f67 794d 6f64 756c 653a 0a3d 3d3d ologyModule:.=== │ │ │ │ +0001d430: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0001d440: 2020 2020 2020 4d20 3d20 6578 7465 7269 M = exteri │ │ │ │ +0001d450: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ +0001d460: 2866 662c 2043 290a 2020 2a20 496e 7075 (ff, C). * Inpu │ │ │ │ +0001d470: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ +0001d480: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +0001d490: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +0001d4a0: 7472 6978 2c2c 204d 6174 7269 7820 6f66 trix,, Matrix of │ │ │ │ +0001d4b0: 2065 6c65 6d65 6e74 7320 7468 6174 2061 elements that a │ │ │ │ +0001d4c0: 7265 0a20 2020 2020 2020 2068 6f6d 6f74 re. homot │ │ │ │ +0001d4d0: 6f70 6963 2074 6f20 3020 6f6e 2043 0a20 opic to 0 on C. │ │ │ │ +0001d4e0: 2020 2020 202a 2043 2c20 6120 2a6e 6f74 * C, a *not │ │ │ │ +0001d4f0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +0001d500: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +0001d510: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0001d520: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0001d530: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0001d540: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0001d550: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0001d560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a41 7373 756d =========..Assum │ │ │ │ +0001d570: 696e 6720 7468 6174 2074 6865 2065 6c65 ing that the ele │ │ │ │ +0001d580: 6d65 6e74 7320 6f66 2074 6865 2031 7863 ments of the 1xc │ │ │ │ +0001d590: 206d 6174 7269 7820 6666 2061 7265 206e matrix ff are n │ │ │ │ +0001d5a0: 756c 6c2d 686f 6d6f 746f 7069 6320 6f6e ull-homotopic on │ │ │ │ +0001d5b0: 2043 2c20 7468 650a 7363 7269 7074 2072 C, the.script r │ │ │ │ +0001d5c0: 6574 7572 6e73 2074 6865 2064 6972 6563 eturns the direc │ │ │ │ +0001d5d0: 7420 7375 6d20 6f66 2074 6865 2068 6f6d t sum of the hom │ │ │ │ +0001d5e0: 6f6c 6f67 7920 6f66 2043 2061 7320 6120 ology of C as a │ │ │ │ +0001d5f0: 6d6f 6475 6c65 206f 7665 7220 6120 6e65 module over a ne │ │ │ │ +0001d600: 7720 7269 6e67 2c0a 636f 6e73 6973 7469 w ring,.consisti │ │ │ │ +0001d610: 6e67 206f 6620 7269 6e67 2043 2077 6974 ng of ring C wit │ │ │ │ +0001d620: 6820 6320 6578 7465 7269 6f72 2076 6172 h c exterior var │ │ │ │ +0001d630: 6961 626c 6573 2061 646a 6f69 6e65 642e iables adjoined. │ │ │ │ +0001d640: 2054 6865 2073 6372 6970 7420 6973 2074 The script is t │ │ │ │ +0001d650: 6865 206d 6169 6e0a 636f 6d70 6f6e 656e he main.componen │ │ │ │ +0001d660: 7420 6f66 2065 7874 6572 696f 7254 6f72 t of exteriorTor │ │ │ │ +0001d670: 4d6f 6475 6c65 0a0a 5365 6520 616c 736f Module..See also │ │ │ │ +0001d680: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0001d690: 6e6f 7465 2065 7874 6572 696f 7254 6f72 note exteriorTor │ │ │ │ +0001d6a0: 4d6f 6475 6c65 3a20 6578 7465 7269 6f72 Module: exterior │ │ │ │ +0001d6b0: 546f 724d 6f64 756c 652c 202d 2d20 546f TorModule, -- To │ │ │ │ +0001d6c0: 7220 6173 2061 206d 6f64 756c 6520 6f76 r as a module ov │ │ │ │ +0001d6d0: 6572 2061 6e0a 2020 2020 6578 7465 7269 er an. exteri │ │ │ │ +0001d6e0: 6f72 2061 6c67 6562 7261 206f 7220 6269 or algebra or bi │ │ │ │ +0001d6f0: 6772 6164 6564 2061 6c67 6562 7261 0a20 graded algebra. │ │ │ │ +0001d700: 202a 202a 6e6f 7465 206d 616b 6548 6f6d * *note makeHom │ │ │ │ +0001d710: 6f74 6f70 6965 734f 6e48 6f6d 6f6c 6f67 otopiesOnHomolog │ │ │ │ +0001d720: 793a 206d 616b 6548 6f6d 6f74 6f70 6965 y: makeHomotopie │ │ │ │ +0001d730: 734f 6e48 6f6d 6f6c 6f67 792c 202d 2d20 sOnHomology, -- │ │ │ │ +0001d740: 486f 6d6f 6c6f 6779 206f 6620 610a 2020 Homology of a. │ │ │ │ +0001d750: 2020 636f 6d70 6c65 7820 6173 2065 7874 complex as ext │ │ │ │ +0001d760: 6572 696f 7220 6d6f 6475 6c65 0a0a 5761 erior module..Wa │ │ │ │ +0001d770: 7973 2074 6f20 7573 6520 6578 7465 7269 ys to use exteri │ │ │ │ +0001d780: 6f72 486f 6d6f 6c6f 6779 4d6f 6475 6c65 orHomologyModule │ │ │ │ +0001d790: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ 0001d7a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d7b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0001d7c0: 0a0a 2020 2a20 2265 7874 6572 696f 7248 .. * "exteriorH │ │ │ │ -0001d7d0: 6f6d 6f6c 6f67 794d 6f64 756c 6528 4d61 omologyModule(Ma │ │ │ │ -0001d7e0: 7472 6978 2c43 6f6d 706c 6578 2922 0a0a trix,Complex)".. │ │ │ │ -0001d7f0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0001d800: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0001d810: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0001d820: 7420 2a6e 6f74 6520 6578 7465 7269 6f72 t *note exterior │ │ │ │ -0001d830: 486f 6d6f 6c6f 6779 4d6f 6475 6c65 3a20 HomologyModule: │ │ │ │ -0001d840: 6578 7465 7269 6f72 486f 6d6f 6c6f 6779 exteriorHomology │ │ │ │ -0001d850: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f Module, is a *no │ │ │ │ -0001d860: 7465 0a6d 6574 686f 6420 6675 6e63 7469 te.method functi │ │ │ │ -0001d870: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -0001d880: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0001d890: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0001d7b0: 3d3d 3d3d 3d0a 0a20 202a 2022 6578 7465 =====.. * "exte │ │ │ │ +0001d7c0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ +0001d7d0: 6c65 284d 6174 7269 782c 436f 6d70 6c65 le(Matrix,Comple │ │ │ │ +0001d7e0: 7829 220a 0a46 6f72 2074 6865 2070 726f x)"..For the pro │ │ │ │ +0001d7f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +0001d800: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +0001d810: 6f62 6a65 6374 202a 6e6f 7465 2065 7874 object *note ext │ │ │ │ +0001d820: 6572 696f 7248 6f6d 6f6c 6f67 794d 6f64 eriorHomologyMod │ │ │ │ +0001d830: 756c 653a 2065 7874 6572 696f 7248 6f6d ule: exteriorHom │ │ │ │ +0001d840: 6f6c 6f67 794d 6f64 756c 652c 2069 7320 ologyModule, is │ │ │ │ +0001d850: 6120 2a6e 6f74 650a 6d65 7468 6f64 2066 a *note.method f │ │ │ │ +0001d860: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +0001d870: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0001d880: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0001d890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001d8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001d8e0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0001d8f0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0001d900: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0001d910: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0001d920: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0001d930: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0001d940: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -0001d950: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0001d960: 5265 736f 6c75 7469 6f6e 732e 6d32 3a32 Resolutions.m2:2 │ │ │ │ -0001d970: 3738 353a 302e 0a1f 0a46 696c 653a 2043 785:0....File: C │ │ │ │ -0001d980: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0001d990: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -0001d9a0: 6e66 6f2c 204e 6f64 653a 2065 7874 6572 nfo, Node: exter │ │ │ │ -0001d9b0: 696f 7254 6f72 4d6f 6475 6c65 2c20 4e65 iorTorModule, Ne │ │ │ │ -0001d9c0: 7874 3a20 6578 7449 734f 6e65 506f 6c79 xt: extIsOnePoly │ │ │ │ -0001d9d0: 6e6f 6d69 616c 2c20 5072 6576 3a20 6578 nomial, Prev: ex │ │ │ │ -0001d9e0: 7465 7269 6f72 486f 6d6f 6c6f 6779 4d6f teriorHomologyMo │ │ │ │ -0001d9f0: 6475 6c65 2c20 5570 3a20 546f 700a 0a65 dule, Up: Top..e │ │ │ │ -0001da00: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001da10: 202d 2d20 546f 7220 6173 2061 206d 6f64 -- Tor as a mod │ │ │ │ -0001da20: 756c 6520 6f76 6572 2061 6e20 6578 7465 ule over an exte │ │ │ │ -0001da30: 7269 6f72 2061 6c67 6562 7261 206f 7220 rior algebra or │ │ │ │ -0001da40: 6269 6772 6164 6564 2061 6c67 6562 7261 bigraded algebra │ │ │ │ -0001da50: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0001d8d0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0001d8e0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0001d8f0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0001d900: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0001d910: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0001d920: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0001d930: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0001d940: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +0001d950: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0001d960: 2e6d 323a 3237 3835 3a30 2e0a 1f0a 4669 .m2:2785:0....Fi │ │ │ │ +0001d970: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +0001d980: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0001d990: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +0001d9a0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +0001d9b0: 652c 204e 6578 743a 2065 7874 4973 4f6e e, Next: extIsOn │ │ │ │ +0001d9c0: 6550 6f6c 796e 6f6d 6961 6c2c 2050 7265 ePolynomial, Pre │ │ │ │ +0001d9d0: 763a 2065 7874 6572 696f 7248 6f6d 6f6c v: exteriorHomol │ │ │ │ +0001d9e0: 6f67 794d 6f64 756c 652c 2055 703a 2054 ogyModule, Up: T │ │ │ │ +0001d9f0: 6f70 0a0a 6578 7465 7269 6f72 546f 724d op..exteriorTorM │ │ │ │ +0001da00: 6f64 756c 6520 2d2d 2054 6f72 2061 7320 odule -- Tor as │ │ │ │ +0001da10: 6120 6d6f 6475 6c65 206f 7665 7220 616e a module over an │ │ │ │ +0001da20: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ +0001da30: 6120 6f72 2062 6967 7261 6465 6420 616c a or bigraded al │ │ │ │ +0001da40: 6765 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a gebra.********** │ │ │ │ +0001da50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001da60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001da70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0001da80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001da90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0001daa0: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ -0001dab0: 2020 2020 2020 2020 5420 3d20 6578 7465 T = exte │ │ │ │ -0001dac0: 7269 6f72 546f 724d 6f64 756c 6528 662c riorTorModule(f, │ │ │ │ -0001dad0: 4629 0a20 2020 2020 2020 2054 203d 2065 F). T = e │ │ │ │ -0001dae0: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001daf0: 2866 2c4d 2c4e 290a 2020 2a20 496e 7075 (f,M,N). * Inpu │ │ │ │ -0001db00: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ -0001db10: 202a 6e6f 7465 206d 6174 7269 783a 2028 *note matrix: ( │ │ │ │ -0001db20: 4d61 6361 756c 6179 3244 6f63 294d 6174 Macaulay2Doc)Mat │ │ │ │ -0001db30: 7269 782c 2c20 3120 7820 632c 2065 6e74 rix,, 1 x c, ent │ │ │ │ -0001db40: 7269 6573 206d 7573 7420 6265 0a20 2020 ries must be. │ │ │ │ -0001db50: 2020 2020 2068 6f6d 6f74 6f70 6963 2074 homotopic t │ │ │ │ -0001db60: 6f20 3020 6f6e 2046 0a20 2020 2020 202a o 0 on F. * │ │ │ │ -0001db70: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0001db80: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0001db90: 6329 4d6f 6475 6c65 2c2c 2053 2d6d 6f64 c)Module,, S-mod │ │ │ │ -0001dba0: 756c 6520 616e 6e69 6869 6c61 7465 6420 ule annihilated │ │ │ │ -0001dbb0: 6279 2069 6465 616c 0a20 2020 2020 2020 by ideal. │ │ │ │ -0001dbc0: 2066 0a20 2020 2020 202a 204e 2c20 6120 f. * N, a │ │ │ │ -0001dbd0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0001dbe0: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0001dbf0: 6c65 2c2c 2053 2d6d 6f64 756c 6520 616e le,, S-module an │ │ │ │ -0001dc00: 6e69 6869 6c61 7465 6420 6279 2069 6465 nihilated by ide │ │ │ │ -0001dc10: 616c 0a20 2020 2020 2020 2066 0a20 202a al. f. * │ │ │ │ -0001dc20: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0001dc30: 2a20 542c 2061 202a 6e6f 7465 206d 6f64 * T, a *note mod │ │ │ │ -0001dc40: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -0001dc50: 6f63 294d 6f64 756c 652c 2c20 546f 725e oc)Module,, Tor^ │ │ │ │ -0001dc60: 5328 4d2c 4e29 2061 7320 6120 4d6f 6475 S(M,N) as a Modu │ │ │ │ -0001dc70: 6c65 206f 7665 720a 2020 2020 2020 2020 le over. │ │ │ │ -0001dc80: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ -0001dc90: 6272 610a 0a44 6573 6372 6970 7469 6f6e bra..Description │ │ │ │ -0001dca0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 .===========..If │ │ │ │ -0001dcb0: 204d 2c4e 2061 7265 2053 2d6d 6f64 756c M,N are S-modul │ │ │ │ -0001dcc0: 6573 2061 6e6e 6968 696c 6174 6564 2062 es annihilated b │ │ │ │ -0001dcd0: 7920 7468 6520 656c 656d 656e 7473 206f y the elements o │ │ │ │ -0001dce0: 6620 7468 6520 6d61 7472 6978 2066 6620 f the matrix ff │ │ │ │ -0001dcf0: 3d20 2866 5f31 2e2e 665f 6329 2c0a 616e = (f_1..f_c),.an │ │ │ │ -0001dd00: 6420 6b20 6973 2074 6865 2072 6573 6964 d k is the resid │ │ │ │ -0001dd10: 7565 2066 6965 6c64 206f 6620 532c 2074 ue field of S, t │ │ │ │ -0001dd20: 6865 6e20 7468 6520 7363 7269 7074 2065 hen the script e │ │ │ │ -0001dd30: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ -0001dd40: 2866 2c4d 2920 7265 7475 726e 730a 546f (f,M) returns.To │ │ │ │ -0001dd50: 725e 5328 4d2c 206b 2920 6173 2061 206d r^S(M, k) as a m │ │ │ │ -0001dd60: 6f64 756c 6520 6f76 6572 2061 6e20 6578 odule over an ex │ │ │ │ -0001dd70: 7465 7269 6f72 2061 6c67 6562 7261 206b terior algebra k │ │ │ │ -0001dd80: 3c65 5f31 2c2e 2e2e 2c65 5f63 3e2c 2077 , w │ │ │ │ -0001dd90: 6865 7265 2074 6865 2065 5f69 0a68 6176 here the e_i.hav │ │ │ │ -0001dda0: 6520 6465 6772 6565 2031 2c20 7768 696c e degree 1, whil │ │ │ │ -0001ddb0: 6520 6578 7465 7269 6f72 546f 724d 6f64 e exteriorTorMod │ │ │ │ -0001ddc0: 756c 6528 662c 4d2c 4e29 2072 6574 7572 ule(f,M,N) retur │ │ │ │ -0001ddd0: 6e73 2054 6f72 5e53 284d 2c4e 2920 6173 ns Tor^S(M,N) as │ │ │ │ -0001dde0: 2061 206d 6f64 756c 650a 6f76 6572 2061 a module.over a │ │ │ │ -0001ddf0: 2062 6967 7261 6465 6420 7269 6e67 2053 bigraded ring S │ │ │ │ -0001de00: 4520 3d20 533c 655f 312c 2e2e 2c65 5f63 E = S, where the e_i │ │ │ │ -0001de20: 2068 6176 6520 6465 6772 6565 7320 7b64 have degrees {d │ │ │ │ -0001de30: 5f69 2c31 7d2c 0a77 6865 7265 2064 5f69 _i,1},.where d_i │ │ │ │ -0001de40: 2069 7320 7468 6520 6465 6772 6565 206f is the degree o │ │ │ │ -0001de50: 6620 665f 692e 2054 6865 206d 6f64 756c f f_i. The modul │ │ │ │ -0001de60: 6520 7374 7275 6374 7572 652c 2069 6e20 e structure, in │ │ │ │ -0001de70: 6569 7468 6572 2063 6173 652c 2069 730a either case, is. │ │ │ │ -0001de80: 6465 6669 6e65 6420 6279 2074 6865 2068 defined by the h │ │ │ │ -0001de90: 6f6d 6f74 6f70 6965 7320 666f 7220 7468 omotopies for th │ │ │ │ -0001dea0: 6520 665f 6920 6f6e 2074 6865 2072 6573 e f_i on the res │ │ │ │ -0001deb0: 6f6c 7574 696f 6e20 6f66 204d 2c20 636f olution of M, co │ │ │ │ -0001dec0: 6d70 7574 6564 2062 7920 7468 650a 7363 mputed by the.sc │ │ │ │ -0001ded0: 7269 7074 206d 616b 6548 6f6d 6f74 6f70 ript makeHomotop │ │ │ │ -0001dee0: 6965 7331 2e0a 0a54 6865 2073 6372 6970 ies1...The scrip │ │ │ │ -0001def0: 7473 2063 616c 6c20 6d61 6b65 4d6f 6475 ts call makeModu │ │ │ │ -0001df00: 6c65 2074 6f20 636f 6d70 7574 6520 6120 le to compute a │ │ │ │ -0001df10: 286e 6f6e 2d6d 696e 696d 616c 2920 7072 (non-minimal) pr │ │ │ │ -0001df20: 6573 656e 7461 7469 6f6e 206f 6620 7468 esentation of th │ │ │ │ -0001df30: 6973 0a6d 6f64 756c 652e 0a0a 4672 6f6d is.module...From │ │ │ │ -0001df40: 2074 6865 2064 6573 6372 6970 7469 6f6e the description │ │ │ │ -0001df50: 2062 7920 6d61 7472 6978 2066 6163 746f by matrix facto │ │ │ │ -0001df60: 7269 7a61 7469 6f6e 7320 616e 6420 7468 rizations and th │ │ │ │ -0001df70: 6520 7061 7065 7220 2254 6f72 2061 7320 e paper "Tor as │ │ │ │ -0001df80: 6120 6d6f 6475 6c65 0a6f 7665 7220 616e a module.over an │ │ │ │ -0001df90: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ -0001dfa0: 6122 206f 6620 4569 7365 6e62 7564 2c20 a" of Eisenbud, │ │ │ │ -0001dfb0: 5065 6576 6120 616e 6420 5363 6872 6579 Peeva and Schrey │ │ │ │ -0001dfc0: 6572 2069 7420 666f 6c6c 6f77 7320 7468 er it follows th │ │ │ │ -0001dfd0: 6174 2077 6865 6e0a 4d20 6973 2061 2068 at when.M is a h │ │ │ │ -0001dfe0: 6967 6820 7379 7a79 6779 2061 6e64 2046 igh syzygy and F │ │ │ │ -0001dff0: 2069 7320 6974 7320 7265 736f 6c75 7469 is its resoluti │ │ │ │ -0001e000: 6f6e 2c20 7468 656e 2074 6865 2070 7265 on, then the pre │ │ │ │ -0001e010: 7365 6e74 6174 696f 6e20 6f66 0a54 6f72 sentation of.Tor │ │ │ │ -0001e020: 284d 2c53 5e31 2f6d 6d29 2061 6c77 6179 (M,S^1/mm) alway │ │ │ │ -0001e030: 7320 6861 7320 6765 6e65 7261 746f 7273 s has generators │ │ │ │ -0001e040: 2069 6e20 6465 6772 6565 7320 302c 312c in degrees 0,1, │ │ │ │ -0001e050: 2063 6f72 7265 7370 6f6e 6469 6e67 2074 corresponding t │ │ │ │ -0001e060: 6f20 7468 650a 7461 7267 6574 7320 616e o the.targets an │ │ │ │ -0001e070: 6420 736f 7572 6365 7320 6f66 2074 6865 d sources of the │ │ │ │ -0001e080: 2073 7461 636b 206f 6620 6d61 7073 2042 stack of maps B │ │ │ │ -0001e090: 2869 292c 2061 6e64 2074 6861 7420 7468 (i), and that th │ │ │ │ -0001e0a0: 6520 7265 736f 6c75 7469 6f6e 2069 730a e resolution is. │ │ │ │ -0001e0b0: 636f 6d70 6f6e 656e 7477 6973 6520 6c69 componentwise li │ │ │ │ -0001e0c0: 6e65 6172 2069 6e20 6120 7375 6974 6162 near in a suitab │ │ │ │ -0001e0d0: 6c65 2073 656e 7365 2e20 496e 2074 6865 le sense. In the │ │ │ │ -0001e0e0: 2066 6f6c 6c6f 7769 6e67 2065 7861 6d70 following examp │ │ │ │ -0001e0f0: 6c65 2c20 7468 6573 6520 6661 6374 730a le, these facts. │ │ │ │ -0001e100: 6172 6520 7665 7269 6669 6564 2e20 5468 are verified. Th │ │ │ │ -0001e110: 6520 546f 7220 6d6f 6475 6c65 2064 6f65 e Tor module doe │ │ │ │ -0001e120: 7320 4e4f 5420 7370 6c69 7420 696e 746f s NOT split into │ │ │ │ -0001e130: 2074 6865 2064 6972 6563 7420 7375 6d20 the direct sum │ │ │ │ -0001e140: 6f66 2074 6865 0a73 7562 6d6f 6475 6c65 of the.submodule │ │ │ │ -0001e150: 7320 6765 6e65 7261 7465 6420 696e 2064 s generated in d │ │ │ │ -0001e160: 6567 7265 6573 2030 2061 6e64 2031 2c20 egrees 0 and 1, │ │ │ │ -0001e170: 686f 7765 7665 722e 0a0a 0a0a 2b2d 2d2d however.....+--- │ │ │ │ +0001da90: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +0001daa0: 6765 3a20 0a20 2020 2020 2020 2054 203d ge: . T = │ │ │ │ +0001dab0: 2065 7874 6572 696f 7254 6f72 4d6f 6475 exteriorTorModu │ │ │ │ +0001dac0: 6c65 2866 2c46 290a 2020 2020 2020 2020 le(f,F). │ │ │ │ +0001dad0: 5420 3d20 6578 7465 7269 6f72 546f 724d T = exteriorTorM │ │ │ │ +0001dae0: 6f64 756c 6528 662c 4d2c 4e29 0a20 202a odule(f,M,N). * │ │ │ │ +0001daf0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +0001db00: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472 f, a *note matr │ │ │ │ +0001db10: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ +0001db20: 6329 4d61 7472 6978 2c2c 2031 2078 2063 c)Matrix,, 1 x c │ │ │ │ +0001db30: 2c20 656e 7472 6965 7320 6d75 7374 2062 , entries must b │ │ │ │ +0001db40: 650a 2020 2020 2020 2020 686f 6d6f 746f e. homoto │ │ │ │ +0001db50: 7069 6320 746f 2030 206f 6e20 460a 2020 pic to 0 on F. │ │ │ │ +0001db60: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0001db70: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0001db80: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0001db90: 532d 6d6f 6475 6c65 2061 6e6e 6968 696c S-module annihil │ │ │ │ +0001dba0: 6174 6564 2062 7920 6964 6561 6c0a 2020 ated by ideal. │ │ │ │ +0001dbb0: 2020 2020 2020 660a 2020 2020 2020 2a20 f. * │ │ │ │ +0001dbc0: 4e2c 2061 202a 6e6f 7465 206d 6f64 756c N, a *note modul │ │ │ │ +0001dbd0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +0001dbe0: 294d 6f64 756c 652c 2c20 532d 6d6f 6475 )Module,, S-modu │ │ │ │ +0001dbf0: 6c65 2061 6e6e 6968 696c 6174 6564 2062 le annihilated b │ │ │ │ +0001dc00: 7920 6964 6561 6c0a 2020 2020 2020 2020 y ideal. │ │ │ │ +0001dc10: 660a 2020 2a20 4f75 7470 7574 733a 0a20 f. * Outputs:. │ │ │ │ +0001dc20: 2020 2020 202a 2054 2c20 6120 2a6e 6f74 * T, a *not │ │ │ │ +0001dc30: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0001dc40: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0001dc50: 2054 6f72 5e53 284d 2c4e 2920 6173 2061 Tor^S(M,N) as a │ │ │ │ +0001dc60: 204d 6f64 756c 6520 6f76 6572 0a20 2020 Module over. │ │ │ │ +0001dc70: 2020 2020 2061 6e20 6578 7465 7269 6f72 an exterior │ │ │ │ +0001dc80: 2061 6c67 6562 7261 0a0a 4465 7363 7269 algebra..Descri │ │ │ │ +0001dc90: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +0001dca0: 3d0a 0a49 6620 4d2c 4e20 6172 6520 532d =..If M,N are S- │ │ │ │ +0001dcb0: 6d6f 6475 6c65 7320 616e 6e69 6869 6c61 modules annihila │ │ │ │ +0001dcc0: 7465 6420 6279 2074 6865 2065 6c65 6d65 ted by the eleme │ │ │ │ +0001dcd0: 6e74 7320 6f66 2074 6865 206d 6174 7269 nts of the matri │ │ │ │ +0001dce0: 7820 6666 203d 2028 665f 312e 2e66 5f63 x ff = (f_1..f_c │ │ │ │ +0001dcf0: 292c 0a61 6e64 206b 2069 7320 7468 6520 ),.and k is the │ │ │ │ +0001dd00: 7265 7369 6475 6520 6669 656c 6420 6f66 residue field of │ │ │ │ +0001dd10: 2053 2c20 7468 656e 2074 6865 2073 6372 S, then the scr │ │ │ │ +0001dd20: 6970 7420 6578 7465 7269 6f72 546f 724d ipt exteriorTorM │ │ │ │ +0001dd30: 6f64 756c 6528 662c 4d29 2072 6574 7572 odule(f,M) retur │ │ │ │ +0001dd40: 6e73 0a54 6f72 5e53 284d 2c20 6b29 2061 ns.Tor^S(M, k) a │ │ │ │ +0001dd50: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ +0001dd60: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ +0001dd70: 6272 6120 6b3c 655f 312c 2e2e 2e2c 655f bra k, where the e_ │ │ │ │ +0001dd90: 690a 6861 7665 2064 6567 7265 6520 312c i.have degree 1, │ │ │ │ +0001dda0: 2077 6869 6c65 2065 7874 6572 696f 7254 while exteriorT │ │ │ │ +0001ddb0: 6f72 4d6f 6475 6c65 2866 2c4d 2c4e 2920 orModule(f,M,N) │ │ │ │ +0001ddc0: 7265 7475 726e 7320 546f 725e 5328 4d2c returns Tor^S(M, │ │ │ │ +0001ddd0: 4e29 2061 7320 6120 6d6f 6475 6c65 0a6f N) as a module.o │ │ │ │ +0001dde0: 7665 7220 6120 6269 6772 6164 6564 2072 ver a bigraded r │ │ │ │ +0001ddf0: 696e 6720 5345 203d 2053 3c65 5f31 2c2e ing SE = S, where th │ │ │ │ +0001de10: 6520 655f 6920 6861 7665 2064 6567 7265 e e_i have degre │ │ │ │ +0001de20: 6573 207b 645f 692c 317d 2c0a 7768 6572 es {d_i,1},.wher │ │ │ │ +0001de30: 6520 645f 6920 6973 2074 6865 2064 6567 e d_i is the deg │ │ │ │ +0001de40: 7265 6520 6f66 2066 5f69 2e20 5468 6520 ree of f_i. The │ │ │ │ +0001de50: 6d6f 6475 6c65 2073 7472 7563 7475 7265 module structure │ │ │ │ +0001de60: 2c20 696e 2065 6974 6865 7220 6361 7365 , in either case │ │ │ │ +0001de70: 2c20 6973 0a64 6566 696e 6564 2062 7920 , is.defined by │ │ │ │ +0001de80: 7468 6520 686f 6d6f 746f 7069 6573 2066 the homotopies f │ │ │ │ +0001de90: 6f72 2074 6865 2066 5f69 206f 6e20 7468 or the f_i on th │ │ │ │ +0001dea0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +0001deb0: 4d2c 2063 6f6d 7075 7465 6420 6279 2074 M, computed by t │ │ │ │ +0001dec0: 6865 0a73 6372 6970 7420 6d61 6b65 486f he.script makeHo │ │ │ │ +0001ded0: 6d6f 746f 7069 6573 312e 0a0a 5468 6520 motopies1...The │ │ │ │ +0001dee0: 7363 7269 7074 7320 6361 6c6c 206d 616b scripts call mak │ │ │ │ +0001def0: 654d 6f64 756c 6520 746f 2063 6f6d 7075 eModule to compu │ │ │ │ +0001df00: 7465 2061 2028 6e6f 6e2d 6d69 6e69 6d61 te a (non-minima │ │ │ │ +0001df10: 6c29 2070 7265 7365 6e74 6174 696f 6e20 l) presentation │ │ │ │ +0001df20: 6f66 2074 6869 730a 6d6f 6475 6c65 2e0a of this.module.. │ │ │ │ +0001df30: 0a46 726f 6d20 7468 6520 6465 7363 7269 .From the descri │ │ │ │ +0001df40: 7074 696f 6e20 6279 206d 6174 7269 7820 ption by matrix │ │ │ │ +0001df50: 6661 6374 6f72 697a 6174 696f 6e73 2061 factorizations a │ │ │ │ +0001df60: 6e64 2074 6865 2070 6170 6572 2022 546f nd the paper "To │ │ │ │ +0001df70: 7220 6173 2061 206d 6f64 756c 650a 6f76 r as a module.ov │ │ │ │ +0001df80: 6572 2061 6e20 6578 7465 7269 6f72 2061 er an exterior a │ │ │ │ +0001df90: 6c67 6562 7261 2220 6f66 2045 6973 656e lgebra" of Eisen │ │ │ │ +0001dfa0: 6275 642c 2050 6565 7661 2061 6e64 2053 bud, Peeva and S │ │ │ │ +0001dfb0: 6368 7265 7965 7220 6974 2066 6f6c 6c6f chreyer it follo │ │ │ │ +0001dfc0: 7773 2074 6861 7420 7768 656e 0a4d 2069 ws that when.M i │ │ │ │ +0001dfd0: 7320 6120 6869 6768 2073 797a 7967 7920 s a high syzygy │ │ │ │ +0001dfe0: 616e 6420 4620 6973 2069 7473 2072 6573 and F is its res │ │ │ │ +0001dff0: 6f6c 7574 696f 6e2c 2074 6865 6e20 7468 olution, then th │ │ │ │ +0001e000: 6520 7072 6573 656e 7461 7469 6f6e 206f e presentation o │ │ │ │ +0001e010: 660a 546f 7228 4d2c 535e 312f 6d6d 2920 f.Tor(M,S^1/mm) │ │ │ │ +0001e020: 616c 7761 7973 2068 6173 2067 656e 6572 always has gener │ │ │ │ +0001e030: 6174 6f72 7320 696e 2064 6567 7265 6573 ators in degrees │ │ │ │ +0001e040: 2030 2c31 2c20 636f 7272 6573 706f 6e64 0,1, correspond │ │ │ │ +0001e050: 696e 6720 746f 2074 6865 0a74 6172 6765 ing to the.targe │ │ │ │ +0001e060: 7473 2061 6e64 2073 6f75 7263 6573 206f ts and sources o │ │ │ │ +0001e070: 6620 7468 6520 7374 6163 6b20 6f66 206d f the stack of m │ │ │ │ +0001e080: 6170 7320 4228 6929 2c20 616e 6420 7468 aps B(i), and th │ │ │ │ +0001e090: 6174 2074 6865 2072 6573 6f6c 7574 696f at the resolutio │ │ │ │ +0001e0a0: 6e20 6973 0a63 6f6d 706f 6e65 6e74 7769 n is.componentwi │ │ │ │ +0001e0b0: 7365 206c 696e 6561 7220 696e 2061 2073 se linear in a s │ │ │ │ +0001e0c0: 7569 7461 626c 6520 7365 6e73 652e 2049 uitable sense. I │ │ │ │ +0001e0d0: 6e20 7468 6520 666f 6c6c 6f77 696e 6720 n the following │ │ │ │ +0001e0e0: 6578 616d 706c 652c 2074 6865 7365 2066 example, these f │ │ │ │ +0001e0f0: 6163 7473 0a61 7265 2076 6572 6966 6965 acts.are verifie │ │ │ │ +0001e100: 642e 2054 6865 2054 6f72 206d 6f64 756c d. The Tor modul │ │ │ │ +0001e110: 6520 646f 6573 204e 4f54 2073 706c 6974 e does NOT split │ │ │ │ +0001e120: 2069 6e74 6f20 7468 6520 6469 7265 6374 into the direct │ │ │ │ +0001e130: 2073 756d 206f 6620 7468 650a 7375 626d sum of the.subm │ │ │ │ +0001e140: 6f64 756c 6573 2067 656e 6572 6174 6564 odules generated │ │ │ │ +0001e150: 2069 6e20 6465 6772 6565 7320 3020 616e in degrees 0 an │ │ │ │ +0001e160: 6420 312c 2068 6f77 6576 6572 2e0a 0a0a d 1, however.... │ │ │ │ +0001e170: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e1c0: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f +.|i1 : kk = ZZ/ │ │ │ │ -0001e1d0: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ +0001e1b0: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20 -----+.|i1 : kk │ │ │ │ +0001e1c0: 3d20 5a5a 2f31 3031 2020 2020 2020 2020 = ZZ/101 │ │ │ │ +0001e1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e200: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001e1f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e240: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001e250: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +0001e240: 207c 0a7c 6f31 203d 206b 6b20 2020 2020 |.|o1 = kk │ │ │ │ +0001e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e290: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e2d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -0001e2e0: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0001e2c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e2d0: 6f31 203a 2051 756f 7469 656e 7452 696e o1 : QuotientRin │ │ │ │ +0001e2e0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 0001e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e310: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e320: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001e310: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e360: 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 3d20 ----+.|i2 : S = │ │ │ │ -0001e370: 6b6b 5b61 2c62 2c63 5d20 2020 2020 2020 kk[a,b,c] │ │ │ │ +0001e350: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ +0001e360: 2053 203d 206b 6b5b 612c 622c 635d 2020 S = kk[a,b,c] │ │ │ │ +0001e370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001e390: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e3f0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020 |.|o2 = S │ │ │ │ +0001e3e0: 2020 2020 207c 0a7c 6f32 203d 2053 2020 |.|o2 = S │ │ │ │ +0001e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001e420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e470: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001e480: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ -0001e490: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ +0001e470: 207c 0a7c 6f32 203a 2050 6f6c 796e 6f6d |.|o2 : Polynom │ │ │ │ +0001e480: 6961 6c52 696e 6720 2020 2020 2020 2020 ialRing │ │ │ │ +0001e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e4c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0001e4b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0001e4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e500: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -0001e510: 6620 3d20 6d61 7472 6978 2261 342c 6234 f = matrix"a4,b4 │ │ │ │ -0001e520: 2c63 3422 2020 2020 2020 2020 2020 2020 ,c4" │ │ │ │ +0001e4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0001e500: 6933 203a 2066 203d 206d 6174 7269 7822 i3 : f = matrix" │ │ │ │ +0001e510: 6134 2c62 342c 6334 2220 2020 2020 2020 a4,b4,c4" │ │ │ │ +0001e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e540: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e550: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0001e540: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e590: 2020 2020 7c0a 7c6f 3320 3d20 7c20 6134 |.|o3 = | a4 │ │ │ │ -0001e5a0: 2062 3420 6334 207c 2020 2020 2020 2020 b4 c4 | │ │ │ │ +0001e580: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ +0001e590: 207c 2061 3420 6234 2063 3420 7c20 2020 | a4 b4 c4 | │ │ │ │ +0001e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e5d0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001e5c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e5d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e620: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0001e630: 3120 2020 2020 2033 2020 2020 2020 2020 1 3 │ │ │ │ +0001e610: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001e620: 2020 2020 2031 2020 2020 2020 3320 2020 1 3 │ │ │ │ +0001e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e660: 2020 2020 2020 7c0a 7c6f 3320 3a20 4d61 |.|o3 : Ma │ │ │ │ -0001e670: 7472 6978 2053 2020 3c2d 2d20 5320 2020 trix S <-- S │ │ │ │ +0001e650: 2020 2020 2020 2020 2020 207c 0a7c 6f33 |.|o3 │ │ │ │ +0001e660: 203a 204d 6174 7269 7820 5320 203c 2d2d : Matrix S <-- │ │ │ │ +0001e670: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0001e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e6a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0001e6a0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0001e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e6f0: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f --+.|i4 : R = S/ │ │ │ │ -0001e700: 6964 6561 6c20 6620 2020 2020 2020 2020 ideal f │ │ │ │ +0001e6e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ +0001e6f0: 203d 2053 2f69 6465 616c 2066 2020 2020 = S/ideal f │ │ │ │ +0001e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e730: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001e720: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e770: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e780: 7c6f 3420 3d20 5220 2020 2020 2020 2020 |o4 = R │ │ │ │ +0001e770: 2020 207c 0a7c 6f34 203d 2052 2020 2020 |.|o4 = R │ │ │ │ +0001e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001e7b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e800: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0001e810: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0001e7f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001e800: 0a7c 6f34 203a 2051 756f 7469 656e 7452 .|o4 : QuotientR │ │ │ │ +0001e810: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 0001e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e850: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0001e840: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0001e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001e870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001e890: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 7020 ------+.|i5 : p │ │ │ │ -0001e8a0: 3d20 6d61 7028 522c 5329 2020 2020 2020 = map(R,S) │ │ │ │ +0001e880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +0001e890: 203a 2070 203d 206d 6170 2852 2c53 2920 : p = map(R,S) │ │ │ │ +0001e8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001e8d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001e8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e920: 2020 7c0a 7c6f 3520 3d20 6d61 7020 2852 |.|o5 = map (R │ │ │ │ -0001e930: 2c20 532c 207b 612c 2062 2c20 637d 2920 , S, {a, b, c}) │ │ │ │ +0001e910: 2020 2020 2020 207c 0a7c 6f35 203d 206d |.|o5 = m │ │ │ │ +0001e920: 6170 2028 522c 2053 2c20 7b61 2c20 622c ap (R, S, {a, b, │ │ │ │ +0001e930: 2063 7d29 2020 2020 2020 2020 2020 2020 c}) │ │ │ │ 0001e940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0001e950: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001e9b0: 7c6f 3520 3a20 5269 6e67 4d61 7020 5220 |o5 : RingMap R │ │ │ │ -0001e9c0: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0001e9a0: 2020 207c 0a7c 6f35 203a 2052 696e 674d |.|o5 : RingM │ │ │ │ +0001e9b0: 6170 2052 203c 2d2d 2053 2020 2020 2020 ap R <-- S │ │ │ │ +0001e9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001e9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001e9f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001e9e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001e9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ea10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ea30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0001ea40: 3a20 4d20 3d20 636f 6b65 7220 6d61 7028 : M = coker map( │ │ │ │ -0001ea50: 525e 322c 2052 5e7b 333a 2d31 7d2c 207b R^2, R^{3:-1}, { │ │ │ │ -0001ea60: 7b61 2c62 2c63 7d2c 7b62 2c63 2c61 7d7d {a,b,c},{b,c,a}} │ │ │ │ -0001ea70: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001ea80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ea20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ea30: 0a7c 6936 203a 204d 203d 2063 6f6b 6572 .|i6 : M = coker │ │ │ │ +0001ea40: 206d 6170 2852 5e32 2c20 525e 7b33 3a2d map(R^2, R^{3:- │ │ │ │ +0001ea50: 317d 2c20 7b7b 612c 622c 637d 2c7b 622c 1}, {{a,b,c},{b, │ │ │ │ +0001ea60: 632c 617d 7d29 2020 2020 2020 2020 2020 c,a}}) │ │ │ │ +0001ea70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ea80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ea90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eaa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eac0: 2020 2020 2020 7c0a 7c6f 3620 3d20 636f |.|o6 = co │ │ │ │ -0001ead0: 6b65 726e 656c 207c 2061 2062 2063 207c kernel | a b c | │ │ │ │ +0001eab0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0001eac0: 203d 2063 6f6b 6572 6e65 6c20 7c20 6120 = cokernel | a │ │ │ │ +0001ead0: 6220 6320 7c20 2020 2020 2020 2020 2020 b c | │ │ │ │ 0001eae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001eb10: 2020 2020 2020 2020 2020 2020 207c 2062 | b │ │ │ │ -0001eb20: 2063 2061 207c 2020 2020 2020 2020 2020 c a | │ │ │ │ +0001eb00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001eb10: 2020 7c20 6220 6320 6120 7c20 2020 2020 | b c a | │ │ │ │ +0001eb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb50: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001eb40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001eb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eb90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001eba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ebb0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ +0001eb80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001eb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001eba0: 2020 2020 2020 2020 2020 2020 3220 2020 2 │ │ │ │ +0001ebb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ebc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ebd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ebe0: 7c6f 3620 3a20 522d 6d6f 6475 6c65 2c20 |o6 : R-module, │ │ │ │ -0001ebf0: 7175 6f74 6965 6e74 206f 6620 5220 2020 quotient of R │ │ │ │ +0001ebd0: 2020 207c 0a7c 6f36 203a 2052 2d6d 6f64 |.|o6 : R-mod │ │ │ │ +0001ebe0: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ +0001ebf0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0001ec00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ec20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0001ec10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0001ec20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001ec40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001ec60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3720 ----------+.|i7 │ │ │ │ -0001ec70: 3a20 6265 7474 6920 2846 4620 3d66 7265 : betti (FF =fre │ │ │ │ -0001ec80: 6552 6573 6f6c 7574 696f 6e28 204d 2c20 eResolution( M, │ │ │ │ -0001ec90: 4c65 6e67 7468 4c69 6d69 7420 3d3e 3629 LengthLimit =>6) │ │ │ │ -0001eca0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ -0001ecb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001ec50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0001ec60: 0a7c 6937 203a 2062 6574 7469 2028 4646 .|i7 : betti (FF │ │ │ │ +0001ec70: 203d 6672 6565 5265 736f 6c75 7469 6f6e =freeResolution │ │ │ │ +0001ec80: 2820 4d2c 204c 656e 6774 684c 696d 6974 ( M, LengthLimit │ │ │ │ +0001ec90: 203d 3e36 2929 2020 2020 2020 2020 2020 =>6)) │ │ │ │ +0001eca0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001ecb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ecd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ece0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ecf0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ed00: 2020 2020 2030 2031 2032 2033 2034 2020 0 1 2 3 4 │ │ │ │ -0001ed10: 3520 2036 2020 2020 2020 2020 2020 2020 5 6 │ │ │ │ +0001ece0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001ecf0: 2020 2020 2020 2020 2020 3020 3120 3220 0 1 2 │ │ │ │ +0001ed00: 3320 3420 2035 2020 3620 2020 2020 2020 3 4 5 6 │ │ │ │ +0001ed10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0001ed40: 3720 3d20 746f 7461 6c3a 2032 2033 2034 7 = total: 2 3 4 │ │ │ │ -0001ed50: 2036 2039 2031 3320 3138 2020 2020 2020 6 9 13 18 │ │ │ │ +0001ed30: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20 |.|o7 = total: │ │ │ │ +0001ed40: 3220 3320 3420 3620 3920 3133 2031 3820 2 3 4 6 9 13 18 │ │ │ │ +0001ed50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ed80: 2020 7c0a 7c20 2020 2020 2020 2020 303a |.| 0: │ │ │ │ -0001ed90: 2032 2033 202e 202e 202e 2020 2e20 202e 2 3 . . . . . │ │ │ │ +0001ed70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001ed80: 2020 2030 3a20 3220 3320 2e20 2e20 2e20 0: 2 3 . . . │ │ │ │ +0001ed90: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ 0001eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001edc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001edd0: 2020 2020 313a 202e 202e 2031 202e 202e 1: . . 1 . . │ │ │ │ -0001ede0: 2020 2e20 202e 2020 2020 2020 2020 2020 . . │ │ │ │ +0001edb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001edc0: 2020 2020 2020 2020 2031 3a20 2e20 2e20 1: . . │ │ │ │ +0001edd0: 3120 2e20 2e20 202e 2020 2e20 2020 2020 1 . . . . │ │ │ │ +0001ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ee10: 7c20 2020 2020 2020 2020 323a 202e 202e | 2: . . │ │ │ │ -0001ee20: 2033 2033 202e 2020 2e20 202e 2020 2020 3 3 . . . │ │ │ │ +0001ee00: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +0001ee10: 3a20 2e20 2e20 3320 3320 2e20 202e 2020 : . . 3 3 . . │ │ │ │ +0001ee20: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ 0001ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001ee60: 333a 202e 202e 202e 2033 2033 2020 2e20 3: . . . 3 3 . │ │ │ │ -0001ee70: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -0001ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ee90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001eea0: 2020 2020 2020 343a 202e 202e 202e 202e 4: . . . . │ │ │ │ -0001eeb0: 2033 2020 3320 202e 2020 2020 2020 2020 3 3 . │ │ │ │ +0001ee40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ee50: 2020 2020 2033 3a20 2e20 2e20 2e20 3320 3: . . . 3 │ │ │ │ +0001ee60: 3320 202e 2020 2e20 2020 2020 2020 2020 3 . . │ │ │ │ +0001ee70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001ee80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001ee90: 0a7c 2020 2020 2020 2020 2034 3a20 2e20 .| 4: . │ │ │ │ +0001eea0: 2e20 2e20 2e20 3320 2033 2020 2e20 2020 . . . 3 3 . │ │ │ │ +0001eeb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001eec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eee0: 7c0a 7c20 2020 2020 2020 2020 353a 202e |.| 5: . │ │ │ │ -0001eef0: 202e 202e 202e 2033 2020 3920 2036 2020 . . . 3 9 6 │ │ │ │ +0001eed0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001eee0: 2035 3a20 2e20 2e20 2e20 2e20 3320 2039 5: . . . . 3 9 │ │ │ │ +0001eef0: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ 0001ef00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001ef30: 2020 363a 202e 202e 202e 202e 202e 2020 6: . . . . . │ │ │ │ -0001ef40: 2e20 2033 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ +0001ef10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001ef20: 2020 2020 2020 2036 3a20 2e20 2e20 2e20 6: . . . │ │ │ │ +0001ef30: 2e20 2e20 202e 2020 3320 2020 2020 2020 . . . 3 │ │ │ │ +0001ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ef60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001ef70: 2020 2020 2020 2020 373a 202e 202e 202e 7: . . . │ │ │ │ -0001ef80: 202e 202e 2020 3120 2039 2020 2020 2020 . . 1 9 │ │ │ │ +0001ef60: 207c 0a7c 2020 2020 2020 2020 2037 3a20 |.| 7: │ │ │ │ +0001ef70: 2e20 2e20 2e20 2e20 2e20 2031 2020 3920 . . . . . 1 9 │ │ │ │ +0001ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efb0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001efa0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001eff0: 2020 2020 2020 2020 7c0a 7c6f 3720 3a20 |.|o7 : │ │ │ │ -0001f000: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +0001efe0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001eff0: 6f37 203a 2042 6574 7469 5461 6c6c 7920 o7 : BettiTally │ │ │ │ +0001f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001f030: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f080: 2d2d 2d2d 2b0a 7c69 3820 3a20 4d53 203d ----+.|i8 : MS = │ │ │ │ -0001f090: 2070 7275 6e65 2070 7573 6846 6f72 7761 prune pushForwa │ │ │ │ -0001f0a0: 7264 2870 2c20 636f 6b65 7220 4646 2e64 rd(p, coker FF.d │ │ │ │ -0001f0b0: 645f 3629 3b20 2020 2020 2020 2020 2020 d_6); │ │ │ │ -0001f0c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0001f070: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ +0001f080: 204d 5320 3d20 7072 756e 6520 7075 7368 MS = prune push │ │ │ │ +0001f090: 466f 7277 6172 6428 702c 2063 6f6b 6572 Forward(p, coker │ │ │ │ +0001f0a0: 2046 462e 6464 5f36 293b 2020 2020 2020 FF.dd_6); │ │ │ │ +0001f0b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f0c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001f0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f110: 2b0a 7c69 3920 3a20 5420 3d20 6578 7465 +.|i9 : T = exte │ │ │ │ -0001f120: 7269 6f72 546f 724d 6f64 756c 6528 662c riorTorModule(f, │ │ │ │ -0001f130: 4d53 293b 2020 2020 2020 2020 2020 2020 MS); │ │ │ │ -0001f140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f150: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001f100: 2d2d 2d2d 2d2b 0a7c 6939 203a 2054 203d -----+.|i9 : T = │ │ │ │ +0001f110: 2065 7874 6572 696f 7254 6f72 4d6f 6475 exteriorTorModu │ │ │ │ +0001f120: 6c65 2866 2c4d 5329 3b20 2020 2020 2020 le(f,MS); │ │ │ │ +0001f130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f140: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001f150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001f1a0: 3130 203a 2062 6574 7469 2054 2020 2020 10 : betti T │ │ │ │ +0001f190: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920 -+.|i10 : betti │ │ │ │ +0001f1a0: 5420 2020 2020 2020 2020 2020 2020 2020 T │ │ │ │ 0001f1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f1e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001f1d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f220: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f230: 2020 2020 2020 2020 2030 2020 2031 2020 0 1 │ │ │ │ +0001f210: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f220: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001f230: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001f240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f270: 7c6f 3130 203d 2074 6f74 616c 3a20 3834 |o10 = total: 84 │ │ │ │ -0001f280: 2032 3532 2020 2020 2020 2020 2020 2020 252 │ │ │ │ +0001f260: 2020 207c 0a7c 6f31 3020 3d20 746f 7461 |.|o10 = tota │ │ │ │ +0001f270: 6c3a 2038 3420 3235 3220 2020 2020 2020 l: 84 252 │ │ │ │ +0001f280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001f2c0: 2030 3a20 3133 2020 3339 2020 2020 2020 0: 13 39 │ │ │ │ +0001f2a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001f2b0: 2020 2020 2020 303a 2031 3320 2033 3920 0: 13 39 │ │ │ │ +0001f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f2f0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001f300: 2020 2020 2020 2031 3a20 3333 2020 3939 1: 33 99 │ │ │ │ +0001f2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f2f0: 0a7c 2020 2020 2020 2020 2020 313a 2033 .| 1: 3 │ │ │ │ +0001f300: 3320 2039 3920 2020 2020 2020 2020 2020 3 99 │ │ │ │ 0001f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ -0001f350: 3239 2020 3837 2020 2020 2020 2020 2020 29 87 │ │ │ │ +0001f330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0001f340: 2020 323a 2032 3920 2038 3720 2020 2020 2: 29 87 │ │ │ │ +0001f350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f380: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001f390: 2020 2033 3a20 2039 2020 3237 2020 2020 3: 9 27 │ │ │ │ +0001f370: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f380: 2020 2020 2020 2020 333a 2020 3920 2032 3: 9 2 │ │ │ │ +0001f390: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ 0001f3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f3c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f3c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001f3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f410: 2020 7c0a 7c6f 3130 203a 2042 6574 7469 |.|o10 : Betti │ │ │ │ -0001f420: 5461 6c6c 7920 2020 2020 2020 2020 2020 Tally │ │ │ │ +0001f400: 2020 2020 2020 207c 0a7c 6f31 3020 3a20 |.|o10 : │ │ │ │ +0001f410: 4265 7474 6954 616c 6c79 2020 2020 2020 BettiTally │ │ │ │ +0001f420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0001f440: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0001f450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0001f4a0: 7c69 3131 203a 2062 6574 7469 2066 7265 |i11 : betti fre │ │ │ │ -0001f4b0: 6552 6573 6f6c 7574 696f 6e20 2850 5420 eResolution (PT │ │ │ │ -0001f4c0: 3d20 7072 756e 6520 542c 204c 656e 6774 = prune T, Lengt │ │ │ │ -0001f4d0: 684c 696d 6974 203d 3e20 3429 2020 2020 hLimit => 4) │ │ │ │ -0001f4e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0001f490: 2d2d 2d2b 0a7c 6931 3120 3a20 6265 7474 ---+.|i11 : bett │ │ │ │ +0001f4a0: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ +0001f4b0: 2028 5054 203d 2070 7275 6e65 2054 2c20 (PT = prune T, │ │ │ │ +0001f4c0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034 LengthLimit => 4 │ │ │ │ +0001f4d0: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +0001f4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f520: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0001f530: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ -0001f540: 2032 2020 2033 2020 2034 2020 2020 2020 2 3 4 │ │ │ │ +0001f510: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f520: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0001f530: 3020 2031 2020 3220 2020 3320 2020 3420 0 1 2 3 4 │ │ │ │ +0001f540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f570: 7c0a 7c6f 3131 203d 2074 6f74 616c 3a20 |.|o11 = total: │ │ │ │ -0001f580: 3331 2035 3520 3837 2031 3237 2031 3735 31 55 87 127 175 │ │ │ │ +0001f560: 2020 2020 207c 0a7c 6f31 3120 3d20 746f |.|o11 = to │ │ │ │ +0001f570: 7461 6c3a 2033 3120 3535 2038 3720 3132 tal: 31 55 87 12 │ │ │ │ +0001f580: 3720 3137 3520 2020 2020 2020 2020 2020 7 175 │ │ │ │ 0001f590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001f5c0: 2020 2030 3a20 3133 2032 3420 3339 2020 0: 13 24 39 │ │ │ │ -0001f5d0: 3538 2020 3831 2020 2020 2020 2020 2020 58 81 │ │ │ │ +0001f5a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f5b0: 2020 2020 2020 2020 303a 2031 3320 3234 0: 13 24 │ │ │ │ +0001f5c0: 2033 3920 2035 3820 2038 3120 2020 2020 39 58 81 │ │ │ │ +0001f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f5f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0001f600: 2020 2020 2020 2020 2031 3a20 3138 2033 1: 18 3 │ │ │ │ -0001f610: 3120 3438 2020 3639 2020 3934 2020 2020 1 48 69 94 │ │ │ │ +0001f5f0: 207c 0a7c 2020 2020 2020 2020 2020 313a |.| 1: │ │ │ │ +0001f600: 2031 3820 3331 2034 3820 2036 3920 2039 18 31 48 69 9 │ │ │ │ +0001f610: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0001f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f640: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001f630: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001f640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f680: 2020 2020 2020 2020 7c0a 7c6f 3131 203a |.|o11 : │ │ │ │ -0001f690: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0001f670: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f680: 6f31 3120 3a20 4265 7474 6954 616c 6c79 o11 : BettiTally │ │ │ │ +0001f690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f6c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f6d0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001f6c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001f6d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f6e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f710: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 6e6e ----+.|i12 : ann │ │ │ │ -0001f720: 2050 5420 2020 2020 2020 2020 2020 2020 PT │ │ │ │ +0001f700: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ +0001f710: 3a20 616e 6e20 5054 2020 2020 2020 2020 : ann PT │ │ │ │ +0001f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f750: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001f740: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001f750: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7a0: 7c0a 7c6f 3132 203d 2069 6465 616c 2865 |.|o12 = ideal(e │ │ │ │ -0001f7b0: 2065 2065 2029 2020 2020 2020 2020 2020 e e ) │ │ │ │ +0001f790: 2020 2020 207c 0a7c 6f31 3220 3d20 6964 |.|o12 = id │ │ │ │ +0001f7a0: 6561 6c28 6520 6520 6520 2920 2020 2020 eal(e e e ) │ │ │ │ +0001f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0001f7f0: 2020 2020 2020 3020 3120 3220 2020 2020 0 1 2 │ │ │ │ +0001f7d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0001f7e0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +0001f7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f820: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0001f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f870: 2020 7c0a 7c6f 3132 203a 2049 6465 616c |.|o12 : Ideal │ │ │ │ -0001f880: 206f 6620 6b6b 5b65 202e 2e65 205d 2020 of kk[e ..e ] │ │ │ │ +0001f860: 2020 2020 2020 207c 0a7c 6f31 3220 3a20 |.|o12 : │ │ │ │ +0001f870: 4964 6561 6c20 6f66 206b 6b5b 6520 2e2e Ideal of kk[e .. │ │ │ │ +0001f880: 6520 5d20 2020 2020 2020 2020 2020 2020 e ] │ │ │ │ 0001f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001f8c0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ -0001f8d0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001f8a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001f8c0: 2020 2030 2020 2032 2020 2020 2020 2020 0 2 │ │ │ │ +0001f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f8f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001f900: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0001f8f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0001f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f940: 2d2d 2d2d 2b0a 7c69 3133 203a 2050 5430 ----+.|i13 : PT0 │ │ │ │ -0001f950: 203d 2069 6d61 6765 2028 696e 6475 6365 = image (induce │ │ │ │ -0001f960: 644d 6170 2850 542c 636f 7665 7220 5054 dMap(PT,cover PT │ │ │ │ -0001f970: 292a 2028 2863 6f76 6572 2050 5429 5f7b )* ((cover PT)_{ │ │ │ │ -0001f980: 302e 2e31 327d 2929 3b20 7c0a 2b2d 2d2d 0..12})); |.+--- │ │ │ │ +0001f930: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ +0001f940: 3a20 5054 3020 3d20 696d 6167 6520 2869 : PT0 = image (i │ │ │ │ +0001f950: 6e64 7563 6564 4d61 7028 5054 2c63 6f76 nducedMap(PT,cov │ │ │ │ +0001f960: 6572 2050 5429 2a20 2828 636f 7665 7220 er PT)* ((cover │ │ │ │ +0001f970: 5054 295f 7b30 2e2e 3132 7d29 293b 207c PT)_{0..12})); | │ │ │ │ +0001f980: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0001f990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f9a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f9d0: 2b0a 7c69 3134 203a 2050 5431 203d 2069 +.|i14 : PT1 = i │ │ │ │ -0001f9e0: 6d61 6765 2028 696e 6475 6365 644d 6170 mage (inducedMap │ │ │ │ -0001f9f0: 2850 542c 636f 7665 7220 5054 292a 2028 (PT,cover PT)* ( │ │ │ │ -0001fa00: 2863 6f76 6572 2050 5429 5f7b 3133 2e2e (cover PT)_{13.. │ │ │ │ -0001fa10: 3330 7d29 293b 7c0a 2b2d 2d2d 2d2d 2d2d 30}));|.+------- │ │ │ │ +0001f9c0: 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 5054 -----+.|i14 : PT │ │ │ │ +0001f9d0: 3120 3d20 696d 6167 6520 2869 6e64 7563 1 = image (induc │ │ │ │ +0001f9e0: 6564 4d61 7028 5054 2c63 6f76 6572 2050 edMap(PT,cover P │ │ │ │ +0001f9f0: 5429 2a20 2828 636f 7665 7220 5054 295f T)* ((cover PT)_ │ │ │ │ +0001fa00: 7b31 332e 2e33 307d 2929 3b7c 0a2b 2d2d {13..30}));|.+-- │ │ │ │ +0001fa10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001fa60: 3135 203a 2062 6574 7469 2066 7265 6552 15 : betti freeR │ │ │ │ -0001fa70: 6573 6f6c 7574 696f 6e28 7072 756e 6520 esolution(prune │ │ │ │ -0001fa80: 5054 302c 204c 656e 6774 684c 696d 6974 PT0, LengthLimit │ │ │ │ -0001fa90: 203d 3e20 3429 2020 2020 2020 2020 2020 => 4) │ │ │ │ -0001faa0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001fa50: 2d2b 0a7c 6931 3520 3a20 6265 7474 6920 -+.|i15 : betti │ │ │ │ +0001fa60: 6672 6565 5265 736f 6c75 7469 6f6e 2870 freeResolution(p │ │ │ │ +0001fa70: 7275 6e65 2050 5430 2c20 4c65 6e67 7468 rune PT0, Length │ │ │ │ +0001fa80: 4c69 6d69 7420 3d3e 2034 2920 2020 2020 Limit => 4) │ │ │ │ +0001fa90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001faa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fae0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001faf0: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0001fb00: 2020 3320 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0001fad0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001fae0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001faf0: 2031 2020 3220 2033 2020 3420 2020 2020 1 2 3 4 │ │ │ │ +0001fb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fb20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001fb30: 7c6f 3135 203d 2074 6f74 616c 3a20 3133 |o15 = total: 13 │ │ │ │ -0001fb40: 2032 3420 3339 2035 3820 3831 2020 2020 24 39 58 81 │ │ │ │ +0001fb20: 2020 207c 0a7c 6f31 3520 3d20 746f 7461 |.|o15 = tota │ │ │ │ +0001fb30: 6c3a 2031 3320 3234 2033 3920 3538 2038 l: 13 24 39 58 8 │ │ │ │ +0001fb40: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0001fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fb70: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001fb80: 2030 3a20 3133 2032 3420 3339 2035 3820 0: 13 24 39 58 │ │ │ │ -0001fb90: 3831 2020 2020 2020 2020 2020 2020 2020 81 │ │ │ │ -0001fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fbb0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001fb60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001fb70: 2020 2020 2020 303a 2031 3320 3234 2033 0: 13 24 3 │ │ │ │ +0001fb80: 3920 3538 2038 3120 2020 2020 2020 2020 9 58 81 │ │ │ │ +0001fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fba0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001fbb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc00: 7c0a 7c6f 3135 203a 2042 6574 7469 5461 |.|o15 : BettiTa │ │ │ │ -0001fc10: 6c6c 7920 2020 2020 2020 2020 2020 2020 lly │ │ │ │ +0001fbf0: 2020 2020 207c 0a7c 6f31 3520 3a20 4265 |.|o15 : Be │ │ │ │ +0001fc00: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +0001fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fc40: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001fc30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001fc40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fc50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fc60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001fc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001fc90: 3136 203a 2062 6574 7469 2066 7265 6552 16 : betti freeR │ │ │ │ -0001fca0: 6573 6f6c 7574 696f 6e28 7072 756e 6520 esolution(prune │ │ │ │ -0001fcb0: 5054 312c 204c 656e 6774 684c 696d 6974 PT1, LengthLimit │ │ │ │ -0001fcc0: 203d 3e20 3429 2020 2020 2020 2020 2020 => 4) │ │ │ │ -0001fcd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001fc80: 2d2b 0a7c 6931 3620 3a20 6265 7474 6920 -+.|i16 : betti │ │ │ │ +0001fc90: 6672 6565 5265 736f 6c75 7469 6f6e 2870 freeResolution(p │ │ │ │ +0001fca0: 7275 6e65 2050 5431 2c20 4c65 6e67 7468 rune PT1, Length │ │ │ │ +0001fcb0: 4c69 6d69 7420 3d3e 2034 2920 2020 2020 Limit => 4) │ │ │ │ +0001fcc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001fcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001fd20: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0001fd30: 2020 3320 2034 2020 2020 2020 2020 2020 3 4 │ │ │ │ +0001fd00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001fd10: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001fd20: 2031 2020 3220 2033 2020 3420 2020 2020 1 2 3 4 │ │ │ │ +0001fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001fd60: 7c6f 3136 203d 2074 6f74 616c 3a20 3138 |o16 = total: 18 │ │ │ │ -0001fd70: 2032 3820 3339 2035 3120 3634 2020 2020 28 39 51 64 │ │ │ │ +0001fd50: 2020 207c 0a7c 6f31 3620 3d20 746f 7461 |.|o16 = tota │ │ │ │ +0001fd60: 6c3a 2031 3820 3238 2033 3920 3531 2036 l: 18 28 39 51 6 │ │ │ │ +0001fd70: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 0001fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fda0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001fdb0: 2031 3a20 3138 2032 3820 3339 2035 3120 1: 18 28 39 51 │ │ │ │ -0001fdc0: 3634 2020 2020 2020 2020 2020 2020 2020 64 │ │ │ │ -0001fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fde0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0001fd90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001fda0: 2020 2020 2020 313a 2031 3820 3238 2033 1: 18 28 3 │ │ │ │ +0001fdb0: 3920 3531 2036 3420 2020 2020 2020 2020 9 51 64 │ │ │ │ +0001fdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001fdd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0001fde0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0001fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe30: 7c0a 7c6f 3136 203a 2042 6574 7469 5461 |.|o16 : BettiTa │ │ │ │ -0001fe40: 6c6c 7920 2020 2020 2020 2020 2020 2020 lly │ │ │ │ +0001fe20: 2020 2020 207c 0a7c 6f31 3620 3a20 4265 |.|o16 : Be │ │ │ │ +0001fe30: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +0001fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001fe50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001fe70: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001fe60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0001fe70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fe90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001fea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001feb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0001fec0: 3137 203a 2062 6574 7469 2066 7265 6552 17 : betti freeR │ │ │ │ -0001fed0: 6573 6f6c 7574 696f 6e28 7072 756e 6520 esolution(prune │ │ │ │ -0001fee0: 5054 2c20 4c65 6e67 7468 4c69 6d69 7420 PT, LengthLimit │ │ │ │ -0001fef0: 3d3e 2034 2920 2020 2020 2020 2020 2020 => 4) │ │ │ │ -0001ff00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0001feb0: 2d2b 0a7c 6931 3720 3a20 6265 7474 6920 -+.|i17 : betti │ │ │ │ +0001fec0: 6672 6565 5265 736f 6c75 7469 6f6e 2870 freeResolution(p │ │ │ │ +0001fed0: 7275 6e65 2050 542c 204c 656e 6774 684c rune PT, LengthL │ │ │ │ +0001fee0: 696d 6974 203d 3e20 3429 2020 2020 2020 imit => 4) │ │ │ │ +0001fef0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0001ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0001ff50: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0001ff60: 2020 2033 2020 2034 2020 2020 2020 2020 3 4 │ │ │ │ +0001ff30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0001ff40: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +0001ff50: 2031 2020 3220 2020 3320 2020 3420 2020 1 2 3 4 │ │ │ │ +0001ff60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001ff70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ff80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0001ff90: 7c6f 3137 203d 2074 6f74 616c 3a20 3331 |o17 = total: 31 │ │ │ │ -0001ffa0: 2035 3520 3837 2031 3237 2031 3735 2020 55 87 127 175 │ │ │ │ +0001ff80: 2020 207c 0a7c 6f31 3720 3d20 746f 7461 |.|o17 = tota │ │ │ │ +0001ff90: 6c3a 2033 3120 3535 2038 3720 3132 3720 l: 31 55 87 127 │ │ │ │ +0001ffa0: 3137 3520 2020 2020 2020 2020 2020 2020 175 │ │ │ │ 0001ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001ffd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0001ffe0: 2030 3a20 3133 2032 3420 3339 2020 3538 0: 13 24 39 58 │ │ │ │ -0001fff0: 2020 3831 2020 2020 2020 2020 2020 2020 81 │ │ │ │ -00020000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00020020: 2020 2020 2020 2031 3a20 3138 2033 3120 1: 18 31 │ │ │ │ -00020030: 3438 2020 3639 2020 3934 2020 2020 2020 48 69 94 │ │ │ │ +0001ffc0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0001ffd0: 2020 2020 2020 303a 2031 3320 3234 2033 0: 13 24 3 │ │ │ │ +0001ffe0: 3920 2035 3820 2038 3120 2020 2020 2020 9 58 81 │ │ │ │ +0001fff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020000: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00020010: 0a7c 2020 2020 2020 2020 2020 313a 2031 .| 1: 1 │ │ │ │ +00020020: 3820 3331 2034 3820 2036 3920 2039 3420 8 31 48 69 94 │ │ │ │ +00020030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020060: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020050: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00020060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000200a0: 2020 2020 2020 7c0a 7c6f 3137 203a 2042 |.|o17 : B │ │ │ │ -000200b0: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +00020090: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000200a0: 3720 3a20 4265 7474 6954 616c 6c79 2020 7 : BettiTally │ │ │ │ +000200b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000200d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000200e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000200e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000200f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020130: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ -00020140: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00020150: 6520 6d61 6b65 4d6f 6475 6c65 3a20 6d61 e makeModule: ma │ │ │ │ -00020160: 6b65 4d6f 6475 6c65 2c20 2d2d 206d 616b keModule, -- mak │ │ │ │ -00020170: 6573 2061 204d 6f64 756c 6520 6f75 7420 es a Module out │ │ │ │ -00020180: 6f66 2061 2063 6f6c 6c65 6374 696f 6e20 of a collection │ │ │ │ -00020190: 6f66 0a20 2020 206d 6f64 756c 6573 2061 of. modules a │ │ │ │ -000201a0: 6e64 206d 6170 730a 0a57 6179 7320 746f nd maps..Ways to │ │ │ │ -000201b0: 2075 7365 2065 7874 6572 696f 7254 6f72 use exteriorTor │ │ │ │ -000201c0: 4d6f 6475 6c65 3a0a 3d3d 3d3d 3d3d 3d3d Module:.======== │ │ │ │ -000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000201e0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 7874 ======.. * "ext │ │ │ │ -000201f0: 6572 696f 7254 6f72 4d6f 6475 6c65 284d eriorTorModule(M │ │ │ │ -00020200: 6174 7269 782c 4d6f 6475 6c65 2922 0a20 atrix,Module)". │ │ │ │ -00020210: 202a 2022 6578 7465 7269 6f72 546f 724d * "exteriorTorM │ │ │ │ -00020220: 6f64 756c 6528 4d61 7472 6978 2c4d 6f64 odule(Matrix,Mod │ │ │ │ -00020230: 756c 652c 4d6f 6475 6c65 2922 0a0a 466f ule,Module)"..Fo │ │ │ │ -00020240: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00020250: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00020260: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00020270: 2a6e 6f74 6520 6578 7465 7269 6f72 546f *note exteriorTo │ │ │ │ -00020280: 724d 6f64 756c 653a 2065 7874 6572 696f rModule: exterio │ │ │ │ -00020290: 7254 6f72 4d6f 6475 6c65 2c20 6973 2061 rTorModule, is a │ │ │ │ -000202a0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -000202b0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -000202c0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -000202d0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +00020120: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +00020130: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +00020140: 202a 6e6f 7465 206d 616b 654d 6f64 756c *note makeModul │ │ │ │ +00020150: 653a 206d 616b 654d 6f64 756c 652c 202d e: makeModule, - │ │ │ │ +00020160: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65 - makes a Module │ │ │ │ +00020170: 206f 7574 206f 6620 6120 636f 6c6c 6563 out of a collec │ │ │ │ +00020180: 7469 6f6e 206f 660a 2020 2020 6d6f 6475 tion of. modu │ │ │ │ +00020190: 6c65 7320 616e 6420 6d61 7073 0a0a 5761 les and maps..Wa │ │ │ │ +000201a0: 7973 2074 6f20 7573 6520 6578 7465 7269 ys to use exteri │ │ │ │ +000201b0: 6f72 546f 724d 6f64 756c 653a 0a3d 3d3d orTorModule:.=== │ │ │ │ +000201c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000201d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000201e0: 2022 6578 7465 7269 6f72 546f 724d 6f64 "exteriorTorMod │ │ │ │ +000201f0: 756c 6528 4d61 7472 6978 2c4d 6f64 756c ule(Matrix,Modul │ │ │ │ +00020200: 6529 220a 2020 2a20 2265 7874 6572 696f e)". * "exterio │ │ │ │ +00020210: 7254 6f72 4d6f 6475 6c65 284d 6174 7269 rTorModule(Matri │ │ │ │ +00020220: 782c 4d6f 6475 6c65 2c4d 6f64 756c 6529 x,Module,Module) │ │ │ │ +00020230: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +00020240: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00020250: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00020260: 6a65 6374 202a 6e6f 7465 2065 7874 6572 ject *note exter │ │ │ │ +00020270: 696f 7254 6f72 4d6f 6475 6c65 3a20 6578 iorTorModule: ex │ │ │ │ +00020280: 7465 7269 6f72 546f 724d 6f64 756c 652c teriorTorModule, │ │ │ │ +00020290: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +000202a0: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61 od.function: (Ma │ │ │ │ +000202b0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +000202c0: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +000202d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000202e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000202f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020320: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00020330: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00020340: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00020350: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00020360: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00020370: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -00020380: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00020390: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -000203a0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -000203b0: 6d32 3a34 3138 323a 302e 0a1f 0a46 696c m2:4182:0....Fil │ │ │ │ -000203c0: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -000203d0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -000203e0: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2065 ns.info, Node: e │ │ │ │ -000203f0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961 xtIsOnePolynomia │ │ │ │ -00020400: 6c2c 204e 6578 743a 2045 7874 4d6f 6475 l, Next: ExtModu │ │ │ │ -00020410: 6c65 2c20 5072 6576 3a20 6578 7465 7269 le, Prev: exteri │ │ │ │ -00020420: 6f72 546f 724d 6f64 756c 652c 2055 703a orTorModule, Up: │ │ │ │ -00020430: 2054 6f70 0a0a 6578 7449 734f 6e65 506f Top..extIsOnePo │ │ │ │ -00020440: 6c79 6e6f 6d69 616c 202d 2d20 6368 6563 lynomial -- chec │ │ │ │ -00020450: 6b20 7768 6574 6865 7220 7468 6520 4869 k whether the Hi │ │ │ │ -00020460: 6c62 6572 7420 6675 6e63 7469 6f6e 206f lbert function o │ │ │ │ -00020470: 6620 4578 7428 4d2c 6b29 2069 7320 6f6e f Ext(M,k) is on │ │ │ │ -00020480: 6520 706f 6c79 6e6f 6d69 616c 0a2a 2a2a e polynomial.*** │ │ │ │ +00020310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00020320: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00020330: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00020340: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00020350: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00020360: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00020370: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00020380: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00020390: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +000203a0: 696f 6e73 2e6d 323a 3431 3832 3a30 2e0a ions.m2:4182:0.. │ │ │ │ +000203b0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +000203c0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +000203d0: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +000203e0: 6465 3a20 6578 7449 734f 6e65 506f 6c79 de: extIsOnePoly │ │ │ │ +000203f0: 6e6f 6d69 616c 2c20 4e65 7874 3a20 4578 nomial, Next: Ex │ │ │ │ +00020400: 744d 6f64 756c 652c 2050 7265 763a 2065 tModule, Prev: e │ │ │ │ +00020410: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ +00020420: 2c20 5570 3a20 546f 700a 0a65 7874 4973 , Up: Top..extIs │ │ │ │ +00020430: 4f6e 6550 6f6c 796e 6f6d 6961 6c20 2d2d OnePolynomial -- │ │ │ │ +00020440: 2063 6865 636b 2077 6865 7468 6572 2074 check whether t │ │ │ │ +00020450: 6865 2048 696c 6265 7274 2066 756e 6374 he Hilbert funct │ │ │ │ +00020460: 696f 6e20 6f66 2045 7874 284d 2c6b 2920 ion of Ext(M,k) │ │ │ │ +00020470: 6973 206f 6e65 2070 6f6c 796e 6f6d 6961 is one polynomia │ │ │ │ +00020480: 6c0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a l.************** │ │ │ │ 00020490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000204a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000204b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000204c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000204d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000204e0: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -000204f0: 0a20 2020 2020 2020 2028 702c 7429 203d . (p,t) = │ │ │ │ -00020500: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -00020510: 6961 6c20 4d0a 2020 2a20 496e 7075 7473 ial M. * Inputs │ │ │ │ -00020520: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ -00020530: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00020540: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00020550: 652c 2c20 6d6f 6475 6c65 206f 7665 7220 e,, module over │ │ │ │ -00020560: 6120 636f 6d70 6c65 7465 0a20 2020 2020 a complete. │ │ │ │ -00020570: 2020 2069 6e74 6572 7365 6374 696f 6e0a intersection. │ │ │ │ -00020580: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00020590: 2020 202a 2070 2c20 6120 2a6e 6f74 6520 * p, a *note │ │ │ │ -000205a0: 7269 6e67 2065 6c65 6d65 6e74 3a20 284d ring element: (M │ │ │ │ -000205b0: 6163 6175 6c61 7932 446f 6329 5269 6e67 acaulay2Doc)Ring │ │ │ │ -000205c0: 456c 656d 656e 742c 2c20 7028 7a29 3d70 Element,, p(z)=p │ │ │ │ -000205d0: 6528 7a2f 3229 2c0a 2020 2020 2020 2020 e(z/2),. │ │ │ │ -000205e0: 7768 6572 6520 7065 2069 7320 7468 6520 where pe is the │ │ │ │ -000205f0: 4869 6c62 6572 7420 706f 6c79 206f 6620 Hilbert poly of │ │ │ │ -00020600: 4578 745e 7b65 7665 6e7d 284d 2c6b 290a Ext^{even}(M,k). │ │ │ │ -00020610: 2020 2020 2020 2a20 742c 2061 202a 6e6f * t, a *no │ │ │ │ -00020620: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ -00020630: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00020640: 426f 6f6c 6561 6e2c 2c20 7472 7565 2069 Boolean,, true i │ │ │ │ -00020650: 6620 7468 6520 6576 656e 2061 6e64 0a20 f the even and. │ │ │ │ -00020660: 2020 2020 2020 206f 6464 2070 6f6c 796e odd polyn │ │ │ │ -00020670: 6f6d 6961 6c73 206d 6174 6368 2074 6f20 omials match to │ │ │ │ -00020680: 666f 726d 206f 6e65 2070 6f6c 796e 6f6d form one polynom │ │ │ │ -00020690: 6961 6c0a 0a44 6573 6372 6970 7469 6f6e ial..Description │ │ │ │ -000206a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 436f .===========..Co │ │ │ │ -000206b0: 6d70 7574 6573 2074 6865 2048 696c 6265 mputes the Hilbe │ │ │ │ -000206c0: 7274 2070 6f6c 796e 6f6d 6961 6c73 2070 rt polynomials p │ │ │ │ -000206d0: 6528 7a29 2c20 706f 287a 2920 6f66 2065 e(z), po(z) of e │ │ │ │ -000206e0: 7665 6e45 7874 4d6f 6475 6c65 2061 6e64 venExtModule and │ │ │ │ -000206f0: 0a6f 6464 4578 744d 6f64 756c 652e 2049 .oddExtModule. I │ │ │ │ -00020700: 7420 7265 7475 726e 7320 7065 287a 2f32 t returns pe(z/2 │ │ │ │ -00020710: 292c 2061 6e64 2063 6f6d 7061 7265 7320 ), and compares │ │ │ │ -00020720: 746f 2073 6565 2077 6865 7468 6572 2074 to see whether t │ │ │ │ -00020730: 6869 7320 6973 2065 7175 616c 2074 6f0a his is equal to. │ │ │ │ -00020740: 706f 287a 2f32 2d31 2f32 292e 2041 7672 po(z/2-1/2). Avr │ │ │ │ -00020750: 616d 6f76 2c20 5365 6365 6c65 616e 7520 amov, Seceleanu │ │ │ │ -00020760: 616e 6420 5a68 656e 6720 6861 7665 2070 and Zheng have p │ │ │ │ -00020770: 726f 7665 6e20 7468 6174 2069 6620 7468 roven that if th │ │ │ │ -00020780: 6520 6964 6561 6c20 6f66 0a71 7561 6472 e ideal of.quadr │ │ │ │ -00020790: 6174 6963 206c 6561 6469 6e67 2066 6f72 atic leading for │ │ │ │ -000207a0: 6d73 206f 6620 6120 636f 6d70 6c65 7465 ms of a complete │ │ │ │ -000207b0: 2069 6e74 6572 7365 6374 696f 6e20 6f66 intersection of │ │ │ │ -000207c0: 2063 6f64 696d 656e 7369 6f6e 2063 2067 codimension c g │ │ │ │ -000207d0: 656e 6572 6174 6520 616e 0a69 6465 616c enerate an.ideal │ │ │ │ -000207e0: 206f 6620 636f 6469 6d65 6e73 696f 6e20 of codimension │ │ │ │ -000207f0: 6174 206c 6561 7374 2063 2d31 2c20 7468 at least c-1, th │ │ │ │ -00020800: 656e 2074 6865 2042 6574 7469 206e 756d en the Betti num │ │ │ │ -00020810: 6265 7273 206f 6620 616e 7920 6d6f 6475 bers of any modu │ │ │ │ -00020820: 6c65 2067 726f 772c 0a65 7665 6e74 7561 le grow,.eventua │ │ │ │ -00020830: 6c6c 792c 2061 7320 6120 7369 6e67 6c65 lly, as a single │ │ │ │ -00020840: 2070 6f6c 796e 6f6d 6961 6c20 2869 6e73 polynomial (ins │ │ │ │ -00020850: 7465 6164 206f 6620 7265 7175 6972 696e tead of requirin │ │ │ │ -00020860: 6720 7365 7061 7261 7465 2070 6f6c 796e g separate polyn │ │ │ │ -00020870: 6f6d 6961 6c73 0a66 6f72 2065 7665 6e20 omials.for even │ │ │ │ -00020880: 616e 6420 6f64 6420 7465 726d 732e 2920 and odd terms.) │ │ │ │ -00020890: 5468 6973 2073 6372 6970 7420 6368 6563 This script chec │ │ │ │ -000208a0: 6b73 2074 6865 2072 6573 756c 7420 696e ks the result in │ │ │ │ -000208b0: 2074 6865 2068 6f6d 6f67 656e 656f 7573 the homogeneous │ │ │ │ -000208c0: 2063 6173 650a 2869 6e20 7768 6963 6820 case.(in which │ │ │ │ -000208d0: 6361 7365 2074 6865 2063 6f6e 6469 7469 case the conditi │ │ │ │ -000208e0: 6f6e 2069 7320 6e65 6365 7373 6172 7920 on is necessary │ │ │ │ -000208f0: 616e 6420 7375 6666 6963 6965 6e74 2e29 and sufficient.) │ │ │ │ -00020900: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +000204d0: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +000204e0: 6167 653a 200a 2020 2020 2020 2020 2870 age: . (p │ │ │ │ +000204f0: 2c74 2920 3d20 6578 7449 734f 6e65 506f ,t) = extIsOnePo │ │ │ │ +00020500: 6c79 6e6f 6d69 616c 204d 0a20 202a 2049 lynomial M. * I │ │ │ │ +00020510: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +00020520: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00020530: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00020540: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520 Module,, module │ │ │ │ +00020550: 6f76 6572 2061 2063 6f6d 706c 6574 650a over a complete. │ │ │ │ +00020560: 2020 2020 2020 2020 696e 7465 7273 6563 intersec │ │ │ │ +00020570: 7469 6f6e 0a20 202a 204f 7574 7075 7473 tion. * Outputs │ │ │ │ +00020580: 3a0a 2020 2020 2020 2a20 702c 2061 202a :. * p, a * │ │ │ │ +00020590: 6e6f 7465 2072 696e 6720 656c 656d 656e note ring elemen │ │ │ │ +000205a0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +000205b0: 2952 696e 6745 6c65 6d65 6e74 2c2c 2070 )RingElement,, p │ │ │ │ +000205c0: 287a 293d 7065 287a 2f32 292c 0a20 2020 (z)=pe(z/2),. │ │ │ │ +000205d0: 2020 2020 2077 6865 7265 2070 6520 6973 where pe is │ │ │ │ +000205e0: 2074 6865 2048 696c 6265 7274 2070 6f6c the Hilbert pol │ │ │ │ +000205f0: 7920 6f66 2045 7874 5e7b 6576 656e 7d28 y of Ext^{even}( │ │ │ │ +00020600: 4d2c 6b29 0a20 2020 2020 202a 2074 2c20 M,k). * t, │ │ │ │ +00020610: 6120 2a6e 6f74 6520 426f 6f6c 6561 6e20 a *note Boolean │ │ │ │ +00020620: 7661 6c75 653a 2028 4d61 6361 756c 6179 value: (Macaulay │ │ │ │ +00020630: 3244 6f63 2942 6f6f 6c65 616e 2c2c 2074 2Doc)Boolean,, t │ │ │ │ +00020640: 7275 6520 6966 2074 6865 2065 7665 6e20 rue if the even │ │ │ │ +00020650: 616e 640a 2020 2020 2020 2020 6f64 6420 and. odd │ │ │ │ +00020660: 706f 6c79 6e6f 6d69 616c 7320 6d61 7463 polynomials matc │ │ │ │ +00020670: 6820 746f 2066 6f72 6d20 6f6e 6520 706f h to form one po │ │ │ │ +00020680: 6c79 6e6f 6d69 616c 0a0a 4465 7363 7269 lynomial..Descri │ │ │ │ +00020690: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +000206a0: 3d0a 0a43 6f6d 7075 7465 7320 7468 6520 =..Computes the │ │ │ │ +000206b0: 4869 6c62 6572 7420 706f 6c79 6e6f 6d69 Hilbert polynomi │ │ │ │ +000206c0: 616c 7320 7065 287a 292c 2070 6f28 7a29 als pe(z), po(z) │ │ │ │ +000206d0: 206f 6620 6576 656e 4578 744d 6f64 756c of evenExtModul │ │ │ │ +000206e0: 6520 616e 640a 6f64 6445 7874 4d6f 6475 e and.oddExtModu │ │ │ │ +000206f0: 6c65 2e20 4974 2072 6574 7572 6e73 2070 le. It returns p │ │ │ │ +00020700: 6528 7a2f 3229 2c20 616e 6420 636f 6d70 e(z/2), and comp │ │ │ │ +00020710: 6172 6573 2074 6f20 7365 6520 7768 6574 ares to see whet │ │ │ │ +00020720: 6865 7220 7468 6973 2069 7320 6571 7561 her this is equa │ │ │ │ +00020730: 6c20 746f 0a70 6f28 7a2f 322d 312f 3229 l to.po(z/2-1/2) │ │ │ │ +00020740: 2e20 4176 7261 6d6f 762c 2053 6563 656c . Avramov, Secel │ │ │ │ +00020750: 6561 6e75 2061 6e64 205a 6865 6e67 2068 eanu and Zheng h │ │ │ │ +00020760: 6176 6520 7072 6f76 656e 2074 6861 7420 ave proven that │ │ │ │ +00020770: 6966 2074 6865 2069 6465 616c 206f 660a if the ideal of. │ │ │ │ +00020780: 7175 6164 7261 7469 6320 6c65 6164 696e quadratic leadin │ │ │ │ +00020790: 6720 666f 726d 7320 6f66 2061 2063 6f6d g forms of a com │ │ │ │ +000207a0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +000207b0: 6f6e 206f 6620 636f 6469 6d65 6e73 696f on of codimensio │ │ │ │ +000207c0: 6e20 6320 6765 6e65 7261 7465 2061 6e0a n c generate an. │ │ │ │ +000207d0: 6964 6561 6c20 6f66 2063 6f64 696d 656e ideal of codimen │ │ │ │ +000207e0: 7369 6f6e 2061 7420 6c65 6173 7420 632d sion at least c- │ │ │ │ +000207f0: 312c 2074 6865 6e20 7468 6520 4265 7474 1, then the Bett │ │ │ │ +00020800: 6920 6e75 6d62 6572 7320 6f66 2061 6e79 i numbers of any │ │ │ │ +00020810: 206d 6f64 756c 6520 6772 6f77 2c0a 6576 module grow,.ev │ │ │ │ +00020820: 656e 7475 616c 6c79 2c20 6173 2061 2073 entually, as a s │ │ │ │ +00020830: 696e 676c 6520 706f 6c79 6e6f 6d69 616c ingle polynomial │ │ │ │ +00020840: 2028 696e 7374 6561 6420 6f66 2072 6571 (instead of req │ │ │ │ +00020850: 7569 7269 6e67 2073 6570 6172 6174 6520 uiring separate │ │ │ │ +00020860: 706f 6c79 6e6f 6d69 616c 730a 666f 7220 polynomials.for │ │ │ │ +00020870: 6576 656e 2061 6e64 206f 6464 2074 6572 even and odd ter │ │ │ │ +00020880: 6d73 2e29 2054 6869 7320 7363 7269 7074 ms.) This script │ │ │ │ +00020890: 2063 6865 636b 7320 7468 6520 7265 7375 checks the resu │ │ │ │ +000208a0: 6c74 2069 6e20 7468 6520 686f 6d6f 6765 lt in the homoge │ │ │ │ +000208b0: 6e65 6f75 7320 6361 7365 0a28 696e 2077 neous case.(in w │ │ │ │ +000208c0: 6869 6368 2063 6173 6520 7468 6520 636f hich case the co │ │ │ │ +000208d0: 6e64 6974 696f 6e20 6973 206e 6563 6573 ndition is neces │ │ │ │ +000208e0: 7361 7279 2061 6e64 2073 7566 6669 6369 sary and suffici │ │ │ │ +000208f0: 656e 742e 290a 0a2b 2d2d 2d2d 2d2d 2d2d ent.)..+-------- │ │ │ │ +00020900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00020940: 203a 2052 313d 5a5a 2f31 3031 5b61 2c62 : R1=ZZ/101[a,b │ │ │ │ -00020950: 2c63 5d2f 6964 6561 6c28 615e 322c 625e ,c]/ideal(a^2,b^ │ │ │ │ -00020960: 322c 635e 3529 2020 2020 2020 2020 2020 2,c^5) │ │ │ │ -00020970: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020930: 2b0a 7c69 3120 3a20 5231 3d5a 5a2f 3130 +.|i1 : R1=ZZ/10 │ │ │ │ +00020940: 315b 612c 622c 635d 2f69 6465 616c 2861 1[a,b,c]/ideal(a │ │ │ │ +00020950: 5e32 2c62 5e32 2c63 5e35 2920 2020 2020 ^2,b^2,c^5) │ │ │ │ +00020960: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00020970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000209a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000209b0: 207c 0a7c 6f31 203d 2052 3120 2020 2020 |.|o1 = R1 │ │ │ │ +000209a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 5231 |.|o1 = R1 │ │ │ │ +000209b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000209d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000209e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000209e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000209f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a20: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -00020a30: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00020a10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00020a20: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +00020a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020a60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00020a50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00020a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00020aa0: 6932 203a 2052 323d 5a5a 2f31 3031 5b61 i2 : R2=ZZ/101[a │ │ │ │ -00020ab0: 2c62 2c63 5d2f 6964 6561 6c28 615e 332c ,b,c]/ideal(a^3, │ │ │ │ -00020ac0: 625e 3329 2020 2020 2020 2020 2020 2020 b^3) │ │ │ │ -00020ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00020a90: 2d2d 2b0a 7c69 3220 3a20 5232 3d5a 5a2f --+.|i2 : R2=ZZ/ │ │ │ │ +00020aa0: 3130 315b 612c 622c 635d 2f69 6465 616c 101[a,b,c]/ideal │ │ │ │ +00020ab0: 2861 5e33 2c62 5e33 2920 2020 2020 2020 (a^3,b^3) │ │ │ │ +00020ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b10: 2020 207c 0a7c 6f32 203d 2052 3220 2020 |.|o2 = R2 │ │ │ │ +00020b00: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +00020b10: 5232 2020 2020 2020 2020 2020 2020 2020 R2 │ │ │ │ 00020b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020b50: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00020b40: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00020b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020b80: 2020 2020 2020 2020 207c 0a7c 6f32 203a |.|o2 : │ │ │ │ -00020b90: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00020b70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00020b80: 7c6f 3220 3a20 5175 6f74 6965 6e74 5269 |o2 : QuotientRi │ │ │ │ +00020b90: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00020ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020bc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00020bb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00020bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00020c00: 0a7c 6933 203a 2065 7874 4973 4f6e 6550 .|i3 : extIsOneP │ │ │ │ -00020c10: 6f6c 796e 6f6d 6961 6c20 636f 6b65 7220 olynomial coker │ │ │ │ -00020c20: 7261 6e64 6f6d 2852 315e 7b30 2c31 7d2c random(R1^{0,1}, │ │ │ │ -00020c30: 5231 5e7b 333a 2d31 7d29 7c0a 7c20 2020 R1^{3:-1})|.| │ │ │ │ +00020bf0: 2d2d 2d2d 2b0a 7c69 3320 3a20 6578 7449 ----+.|i3 : extI │ │ │ │ +00020c00: 734f 6e65 506f 6c79 6e6f 6d69 616c 2063 sOnePolynomial c │ │ │ │ +00020c10: 6f6b 6572 2072 616e 646f 6d28 5231 5e7b oker random(R1^{ │ │ │ │ +00020c20: 302c 317d 2c52 315e 7b33 3a2d 317d 297c 0,1},R1^{3:-1})| │ │ │ │ +00020c30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00020c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020c70: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ -00020c80: 3220 2020 3120 2020 2020 2020 2020 2020 2 1 │ │ │ │ +00020c60: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00020c70: 2020 2031 2032 2020 2031 2020 2020 2020 1 2 1 │ │ │ │ +00020c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020cb0: 7c0a 7c6f 3320 3d20 282d 7a20 202d 202d |.|o3 = (-z - - │ │ │ │ -00020cc0: 7a20 2b20 332c 2074 7275 6529 2020 2020 z + 3, true) │ │ │ │ +00020ca0: 2020 2020 207c 0a7c 6f33 203d 2028 2d7a |.|o3 = (-z │ │ │ │ +00020cb0: 2020 2d20 2d7a 202b 2033 2c20 7472 7565 - -z + 3, true │ │ │ │ +00020cc0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00020cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00020cf0: 2020 2020 3220 2020 2020 3220 2020 2020 2 2 │ │ │ │ +00020ce0: 7c0a 7c20 2020 2020 2032 2020 2020 2032 |.| 2 2 │ │ │ │ +00020cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d20: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00020d10: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00020d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d60: 207c 0a7c 6f33 203a 2053 6571 7565 6e63 |.|o3 : Sequenc │ │ │ │ -00020d70: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ +00020d50: 2020 2020 2020 7c0a 7c6f 3320 3a20 5365 |.|o3 : Se │ │ │ │ +00020d60: 7175 656e 6365 2020 2020 2020 2020 2020 quence │ │ │ │ +00020d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020d90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00020d90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00020da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020dd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2065 -------+.|i4 : e │ │ │ │ -00020de0: 7874 4973 4f6e 6550 6f6c 796e 6f6d 6961 xtIsOnePolynomia │ │ │ │ -00020df0: 6c20 636f 6b65 7220 7261 6e64 6f6d 2852 l coker random(R │ │ │ │ -00020e00: 325e 7b30 2c31 7d2c 5232 5e7b 333a 2d31 2^{0,1},R2^{3:-1 │ │ │ │ -00020e10: 7d29 7c0a 7c20 2020 2020 2020 2020 2020 })|.| │ │ │ │ +00020dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00020dd0: 3420 3a20 6578 7449 734f 6e65 506f 6c79 4 : extIsOnePoly │ │ │ │ +00020de0: 6e6f 6d69 616c 2063 6f6b 6572 2072 616e nomial coker ran │ │ │ │ +00020df0: 646f 6d28 5232 5e7b 302c 317d 2c52 325e dom(R2^{0,1},R2^ │ │ │ │ +00020e00: 7b33 3a2d 317d 297c 0a7c 2020 2020 2020 {3:-1})|.| │ │ │ │ +00020e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00020e50: 6f34 203d 2028 337a 202d 2032 2c20 6661 o4 = (3z - 2, fa │ │ │ │ -00020e60: 6c73 6529 2020 2020 2020 2020 2020 2020 lse) │ │ │ │ -00020e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020e80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00020e40: 2020 7c0a 7c6f 3420 3d20 2833 7a20 2d20 |.|o4 = (3z - │ │ │ │ +00020e50: 322c 2066 616c 7365 2920 2020 2020 2020 2, false) │ │ │ │ +00020e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00020e70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00020e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ec0: 2020 207c 0a7c 6f34 203a 2053 6571 7565 |.|o4 : Seque │ │ │ │ -00020ed0: 6e63 6520 2020 2020 2020 2020 2020 2020 nce │ │ │ │ +00020eb0: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +00020ec0: 5365 7175 656e 6365 2020 2020 2020 2020 Sequence │ │ │ │ +00020ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00020ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00020ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00020f00: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00020ef0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00020f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00020f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00020f30: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ -00020f40: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ -00020f50: 202a 202a 6e6f 7465 2065 7665 6e45 7874 * *note evenExt │ │ │ │ -00020f60: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -00020f70: 6f64 756c 652c 202d 2d20 6576 656e 2070 odule, -- even p │ │ │ │ -00020f80: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ -00020f90: 2920 6f76 6572 2061 0a20 2020 2063 6f6d ) over a. com │ │ │ │ -00020fa0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -00020fb0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -00020fc0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -00020fd0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464 ng. * *note odd │ │ │ │ -00020fe0: 4578 744d 6f64 756c 653a 206f 6464 4578 ExtModule: oddEx │ │ │ │ -00020ff0: 744d 6f64 756c 652c 202d 2d20 6f64 6420 tModule, -- odd │ │ │ │ -00021000: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ -00021010: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ -00021020: 7465 0a20 2020 2069 6e74 6572 7365 6374 te. intersect │ │ │ │ -00021030: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -00021040: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -00021050: 696e 670a 0a57 6179 7320 746f 2075 7365 ing..Ways to use │ │ │ │ -00021060: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -00021070: 6961 6c3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ial:.=========== │ │ │ │ -00021080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00021090: 3d3d 3d3d 0a0a 2020 2a20 2265 7874 4973 ====.. * "extIs │ │ │ │ -000210a0: 4f6e 6550 6f6c 796e 6f6d 6961 6c28 4d6f OnePolynomial(Mo │ │ │ │ -000210b0: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ -000210c0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -000210d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -000210e0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -000210f0: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69 extIsOnePolynomi │ │ │ │ -00021100: 616c 3a20 6578 7449 734f 6e65 506f 6c79 al: extIsOnePoly │ │ │ │ -00021110: 6e6f 6d69 616c 2c20 6973 2061 202a 6e6f nomial, is a *no │ │ │ │ -00021120: 7465 206d 6574 686f 640a 6675 6e63 7469 te method.functi │ │ │ │ -00021130: 6f6e 3a20 284d 6163 6175 6c61 7932 446f on: (Macaulay2Do │ │ │ │ -00021140: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -00021150: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +00020f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00020f30: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ +00020f40: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6576 ==.. * *note ev │ │ │ │ +00020f50: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +00020f60: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065 nExtModule, -- e │ │ │ │ +00020f70: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ +00020f80: 2a28 4d2c 6b29 206f 7665 7220 610a 2020 *(M,k) over a. │ │ │ │ +00020f90: 2020 636f 6d70 6c65 7465 2069 6e74 6572 complete inter │ │ │ │ +00020fa0: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +00020fb0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +00020fc0: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74 or ring. * *not │ │ │ │ +00020fd0: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20 e oddExtModule: │ │ │ │ +00020fe0: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d oddExtModule, -- │ │ │ │ +00020ff0: 206f 6464 2070 6172 7420 6f66 2045 7874 odd part of Ext │ │ │ │ +00021000: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063 ^*(M,k) over a c │ │ │ │ +00021010: 6f6d 706c 6574 650a 2020 2020 696e 7465 omplete. inte │ │ │ │ +00021020: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +00021030: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +00021040: 746f 7220 7269 6e67 0a0a 5761 7973 2074 tor ring..Ways t │ │ │ │ +00021050: 6f20 7573 6520 6578 7449 734f 6e65 506f o use extIsOnePo │ │ │ │ +00021060: 6c79 6e6f 6d69 616c 3a0a 3d3d 3d3d 3d3d lynomial:.====== │ │ │ │ +00021070: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00021080: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00021090: 6578 7449 734f 6e65 506f 6c79 6e6f 6d69 extIsOnePolynomi │ │ │ │ +000210a0: 616c 284d 6f64 756c 6529 220a 0a46 6f72 al(Module)"..For │ │ │ │ +000210b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +000210c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000210d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +000210e0: 6e6f 7465 2065 7874 4973 4f6e 6550 6f6c note extIsOnePol │ │ │ │ +000210f0: 796e 6f6d 6961 6c3a 2065 7874 4973 4f6e ynomial: extIsOn │ │ │ │ +00021100: 6550 6f6c 796e 6f6d 6961 6c2c 2069 7320 ePolynomial, is │ │ │ │ +00021110: 6120 2a6e 6f74 6520 6d65 7468 6f64 0a66 a *note method.f │ │ │ │ +00021120: 756e 6374 696f 6e3a 2028 4d61 6361 756c unction: (Macaul │ │ │ │ +00021130: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +00021140: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00021150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000211a0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -000211b0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -000211c0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -000211d0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -000211e0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -000211f0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -00021200: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -00021210: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -00021220: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ -00021230: 3933 313a 302e 0a1f 0a46 696c 653a 2043 931:0....File: C │ │ │ │ -00021240: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00021250: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -00021260: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f nfo, Node: ExtMo │ │ │ │ -00021270: 6475 6c65 2c20 4e65 7874 3a20 4578 744d dule, Next: ExtM │ │ │ │ -00021280: 6f64 756c 6544 6174 612c 2050 7265 763a oduleData, Prev: │ │ │ │ -00021290: 2065 7874 4973 4f6e 6550 6f6c 796e 6f6d extIsOnePolynom │ │ │ │ -000212a0: 6961 6c2c 2055 703a 2054 6f70 0a0a 4578 ial, Up: Top..Ex │ │ │ │ -000212b0: 744d 6f64 756c 6520 2d2d 2045 7874 5e2a tModule -- Ext^* │ │ │ │ -000212c0: 284d 2c6b 2920 6f76 6572 2061 2063 6f6d (M,k) over a com │ │ │ │ -000212d0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -000212e0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -000212f0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -00021300: 6e67 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ng.************* │ │ │ │ +00021190: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +000211a0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +000211b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +000211c0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +000211d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +000211e0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +000211f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +00021200: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +00021210: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +00021220: 2e6d 323a 3439 3331 3a30 2e0a 1f0a 4669 .m2:4931:0....Fi │ │ │ │ +00021230: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +00021240: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00021250: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +00021260: 4578 744d 6f64 756c 652c 204e 6578 743a ExtModule, Next: │ │ │ │ +00021270: 2045 7874 4d6f 6475 6c65 4461 7461 2c20 ExtModuleData, │ │ │ │ +00021280: 5072 6576 3a20 6578 7449 734f 6e65 506f Prev: extIsOnePo │ │ │ │ +00021290: 6c79 6e6f 6d69 616c 2c20 5570 3a20 546f lynomial, Up: To │ │ │ │ +000212a0: 700a 0a45 7874 4d6f 6475 6c65 202d 2d20 p..ExtModule -- │ │ │ │ +000212b0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ +000212c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ +000212d0: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +000212e0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +000212f0: 6f72 2072 696e 670a 2a2a 2a2a 2a2a 2a2a or ring.******** │ │ │ │ +00021300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00021310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00021320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00021330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00021350: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -00021360: 6765 3a20 0a20 2020 2020 2020 2045 203d ge: . E = │ │ │ │ -00021370: 2045 7874 4d6f 6475 6c65 204d 0a20 202a ExtModule M. * │ │ │ │ -00021380: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00021390: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -000213a0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -000213b0: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220 c)Module,, over │ │ │ │ -000213c0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ -000213d0: 7365 6374 696f 6e0a 2020 2020 2020 2020 section. │ │ │ │ -000213e0: 7269 6e67 0a20 202a 204f 7574 7075 7473 ring. * Outputs │ │ │ │ -000213f0: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -00021400: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00021410: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00021420: 652c 2c20 6f76 6572 2061 2070 6f6c 796e e,, over a polyn │ │ │ │ -00021430: 6f6d 6961 6c20 7269 6e67 2077 6974 680a omial ring with. │ │ │ │ -00021440: 2020 2020 2020 2020 6765 6e73 2069 6e20 gens in │ │ │ │ -00021450: 6576 656e 2064 6567 7265 650a 0a44 6573 even degree..Des │ │ │ │ -00021460: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00021470: 3d3d 3d3d 0a0a 5573 6573 2063 6f64 6520 ====..Uses code │ │ │ │ -00021480: 6f66 2041 7672 616d 6f76 2d47 7261 7973 of Avramov-Grays │ │ │ │ -00021490: 6f6e 2064 6573 6372 6962 6564 2069 6e20 on described in │ │ │ │ -000214a0: 4d61 6361 756c 6179 3220 626f 6f6b 0a0a Macaulay2 book.. │ │ │ │ -000214b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00021340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +00021350: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00021360: 2020 4520 3d20 4578 744d 6f64 756c 6520 E = ExtModule │ │ │ │ +00021370: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ +00021380: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00021390: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +000213a0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +000213b0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ +000213c0: 696e 7465 7273 6563 7469 6f6e 0a20 2020 intersection. │ │ │ │ +000213d0: 2020 2020 2072 696e 670a 2020 2a20 4f75 ring. * Ou │ │ │ │ +000213e0: 7470 7574 733a 0a20 2020 2020 202a 2045 tputs:. * E │ │ │ │ +000213f0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00021400: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00021410: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120 Module,, over a │ │ │ │ +00021420: 706f 6c79 6e6f 6d69 616c 2072 696e 6720 polynomial ring │ │ │ │ +00021430: 7769 7468 0a20 2020 2020 2020 2067 656e with. gen │ │ │ │ +00021440: 7320 696e 2065 7665 6e20 6465 6772 6565 s in even degree │ │ │ │ +00021450: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00021460: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 7320 =========..Uses │ │ │ │ +00021470: 636f 6465 206f 6620 4176 7261 6d6f 762d code of Avramov- │ │ │ │ +00021480: 4772 6179 736f 6e20 6465 7363 7269 6265 Grayson describe │ │ │ │ +00021490: 6420 696e 204d 6163 6175 6c61 7932 2062 d in Macaulay2 b │ │ │ │ +000214a0: 6f6f 6b0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ook..+---------- │ │ │ │ +000214b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000214c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000214d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000214e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000214f0: 203a 206b 6b3d 205a 5a2f 3130 3120 2020 : kk= ZZ/101 │ │ │ │ +000214e0: 2b0a 7c69 3120 3a20 6b6b 3d20 5a5a 2f31 +.|i1 : kk= ZZ/1 │ │ │ │ +000214f0: 3031 2020 2020 2020 2020 2020 2020 2020 01 │ │ │ │ 00021500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021520: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00021510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00021520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021560: 2020 2020 207c 0a7c 6f31 203d 206b 6b20 |.|o1 = kk │ │ │ │ +00021550: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ +00021560: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 00021570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00021590: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000215a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000215c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000215d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000215e0: 0a7c 6f31 203a 2051 756f 7469 656e 7452 .|o1 : QuotientR │ │ │ │ -000215f0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ +000215d0: 2020 2020 7c0a 7c6f 3120 3a20 5175 6f74 |.|o1 : Quot │ │ │ │ +000215e0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +000215f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021610: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00021610: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00021620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021650: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -00021660: 2053 203d 206b 6b5b 782c 792c 7a5d 2020 S = kk[x,y,z] │ │ │ │ +00021640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00021650: 7c69 3220 3a20 5320 3d20 6b6b 5b78 2c79 |i2 : S = kk[x,y │ │ │ │ +00021660: 2c7a 5d20 2020 2020 2020 2020 2020 2020 ,z] │ │ │ │ 00021670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021690: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00021680: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00021690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000216d0: 2020 207c 0a7c 6f32 203d 2053 2020 2020 |.|o2 = S │ │ │ │ +000216c0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +000216d0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 000216e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000216f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021710: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00021700: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00021710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021740: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00021750: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52 o2 : PolynomialR │ │ │ │ -00021760: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -00021770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021780: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00021740: 2020 7c0a 7c6f 3220 3a20 506f 6c79 6e6f |.|o2 : Polyno │ │ │ │ +00021750: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00021760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00021770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00021780: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00021790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000217a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000217c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2049 -------+.|i3 : I │ │ │ │ -000217d0: 3120 3d20 6964 6561 6c20 2278 3379 2220 1 = ideal "x3y" │ │ │ │ +000217b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000217c0: 3320 3a20 4931 203d 2069 6465 616c 2022 3 : I1 = ideal " │ │ │ │ +000217d0: 7833 7922 2020 2020 2020 2020 2020 2020 x3y" │ │ │ │ 000217e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000217f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021800: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000217f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00021800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021840: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00021850: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +00021830: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00021840: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +00021850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021870: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00021880: 7c6f 3320 3d20 6964 6561 6c28 7820 7929 |o3 = ideal(x y) │ │ │ │ +00021870: 2020 207c 0a7c 6f33 203d 2069 6465 616c |.|o3 = ideal │ │ │ │ +00021880: 2878 2079 2920 2020 2020 2020 2020 2020 (x y) │ │ │ │ 00021890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000218b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000218c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000218d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000218f0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00021900: 4964 6561 6c20 6f66 2053 2020 2020 2020 Ideal of S │ │ │ │ +000218e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000218f0: 6f33 203a 2049 6465 616c 206f 6620 5320 o3 : Ideal of S │ │ │ │ +00021900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021930: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00021920: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00021930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021970: 2d2d 2b0a 7c69 3420 3a20 5231 203d 2053 --+.|i4 : R1 = S │ │ │ │ -00021980: 2f49 3120 2020 2020 2020 2020 2020 2020 /I1 │ │ │ │ +00021960: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ +00021970: 3120 3d20 532f 4931 2020 2020 2020 2020 1 = S/I1 │ │ │ │ +00021980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000219b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000219a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000219b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000219c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000219d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000219e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -000219f0: 3420 3d20 5231 2020 2020 2020 2020 2020 4 = R1 │ │ │ │ +000219e0: 207c 0a7c 6f34 203d 2052 3120 2020 2020 |.|o4 = R1 │ │ │ │ +000219f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a20: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00021a10: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00021a20: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00021a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a60: 2020 2020 2020 7c0a 7c6f 3420 3a20 5175 |.|o4 : Qu │ │ │ │ -00021a70: 6f74 6965 6e74 5269 6e67 2020 2020 2020 otientRing │ │ │ │ +00021a50: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +00021a60: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ +00021a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00021a90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00021aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00021ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00021ae0: 2b0a 7c69 3520 3a20 4d31 203d 2052 315e +.|i5 : M1 = R1^ │ │ │ │ -00021af0: 312f 6964 6561 6c28 785e 3229 2020 2020 1/ideal(x^2) │ │ │ │ +00021ad0: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 3120 -----+.|i5 : M1 │ │ │ │ +00021ae0: 3d20 5231 5e31 2f69 6465 616c 2878 5e32 = R1^1/ideal(x^2 │ │ │ │ +00021af0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00021b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00021b10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00021b10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00021b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00021b30: 2020 2020 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312c 2031 2c20 {1, 1, 1, 1, 1, │ │ │ │ +00022250: 312c 2031 2c20 312c 2031 2c20 312c 2031 1, 1, 1, 1, 1, 1 │ │ │ │ +00022260: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00022270: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00022280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000222c0: 6f38 203a 204c 6973 7420 2020 2020 2020 o8 : List │ │ │ │ +000222b0: 2020 7c0a 7c6f 3820 3a20 4c69 7374 2020 |.|o8 : List │ │ │ │ +000222c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000222d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000222f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000222e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000222f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 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2020 2020 2020 2020 2020 │ │ │ │ 00022390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000223c0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000223a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000223b0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +000223c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000223d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000223e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000223f0: 7c6f 3920 3d20 286b 6b5b 5820 5d29 2020 |o9 = (kk[X ]) │ │ │ │ +000223e0: 2020 207c 0a7c 6f39 203d 2028 6b6b 5b58 |.|o9 = (kk[X │ │ │ │ +000223f0: 205d 2920 2020 2020 2020 2020 2020 2020 ]) │ │ │ │ 00022400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00022430: 2020 2020 2020 2020 3020 2020 2020 2020 0 │ │ │ │ +00022420: 7c0a 7c20 2020 2020 2020 2020 2030 2020 |.| 0 │ │ │ │ +00022430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022460: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224a0: 2020 2020 207c 0a7c 6f39 203a 206b 6b5b |.|o9 : kk[ │ │ │ │ -000224b0: 5820 5d2d 6d6f 6475 6c65 2c20 6672 6565 X ]-module, free │ │ │ │ +00022490: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +000224a0: 3a20 6b6b 5b58 205d 2d6d 6f64 756c 652c : kk[X ]-module, │ │ │ │ +000224b0: 2066 7265 6520 2020 2020 2020 2020 2020 free │ │ │ │ 000224c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000224e0: 2020 7c0a 7c20 2020 2020 2020 2020 3020 |.| 0 │ │ │ │ +000224d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000224e0: 2020 2030 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 000224f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022510: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00022520: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00022510: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00022520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00022560: 3130 203a 2061 7070 6c79 2874 6f4c 6973 10 : apply(toLis │ │ │ │ -00022570: 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2020 2020 │ │ │ │ -00022610: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000225c0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +000225d0: 3020 3d20 7b31 2c20 312c 2031 2c20 312c 0 = {1, 1, 1, 1, │ │ │ │ +000225e0: 2031 2c20 317d 2020 2020 2020 2020 2020 1, 1} │ │ │ │ +000225f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022600: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022650: 7c0a 7c6f 3130 203a 204c 6973 7420 2020 |.|o10 : List │ │ │ │ +00022640: 2020 2020 207c 0a7c 6f31 3020 3a20 4c69 |.|o10 : Li │ │ │ │ +00022650: 7374 2020 2020 2020 2020 2020 2020 2020 st │ │ │ │ 00022660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022680: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00022680: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00022690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000226a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000226c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 ----------+.|i11 │ │ │ │ -000226d0: 203a 2045 6f64 6420 3d20 6f64 6445 7874 : Eodd = oddExt │ │ │ │ -000226e0: 4d6f 6475 6c65 284d 3129 2020 2020 2020 Module(M1) │ │ │ │ -000226f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022700: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000226b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000226c0: 0a7c 6931 3120 3a20 456f 6464 203d 206f .|i11 : Eodd = o │ │ │ │ +000226d0: 6464 4578 744d 6f64 756c 6528 4d31 2920 ddExtModule(M1) │ │ │ │ +000226e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000226f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00022700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022740: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00022750: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00022730: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022740: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00022750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022780: 207c 0a7c 6f31 3120 3d20 286b 6b5b 5820 |.|o11 = (kk[X │ │ │ │ -00022790: 5d29 2020 2020 2020 2020 2020 2020 2020 ]) │ │ │ │ +00022770: 2020 2020 2020 7c0a 7c6f 3131 203d 2028 |.|o11 = ( │ │ │ │ +00022780: 6b6b 5b58 205d 2920 2020 2020 2020 2020 kk[X ]) │ │ │ │ +00022790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000227b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000227c0: 7c20 2020 2020 2020 2020 2020 3020 2020 | 0 │ │ │ │ +000227b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000227c0: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 000227d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000227e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000227f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000227f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022830: 2020 2020 2020 2020 7c0a 7c6f 3131 203a |.|o11 : │ │ │ │ -00022840: 206b 6b5b 5820 5d2d 6d6f 6475 6c65 2c20 kk[X ]-module, │ │ │ │ -00022850: 6672 6565 2020 2020 2020 2020 2020 2020 free │ │ │ │ -00022860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022870: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022880: 2020 3020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00022820: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022830: 6f31 3120 3a20 6b6b 5b58 205d 2d6d 6f64 o11 : kk[X ]-mod │ │ │ │ +00022840: 756c 652c 2066 7265 6520 2020 2020 2020 ule, free │ │ │ │ +00022850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022860: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022870: 2020 2020 2020 2030 2020 2020 2020 2020 0 │ │ │ │ +00022880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000228a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000228b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000228a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000228b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000228d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000228e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000228f0: 0a7c 6931 3220 3a20 6170 706c 7928 746f .|i12 : apply(to │ │ │ │ -00022900: 4c69 7374 2830 2e2e 3529 2c20 692d 3e68 List(0..5), i->h │ │ │ │ -00022910: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ -00022920: 2c20 456f 6464 2929 2020 2020 7c0a 7c20 , Eodd)) |.| │ │ │ │ +000228e0: 2d2d 2d2d 2b0a 7c69 3132 203a 2061 7070 ----+.|i12 : app │ │ │ │ +000228f0: 6c79 2874 6f4c 6973 7428 302e 2e35 292c ly(toList(0..5), │ │ │ │ +00022900: 2069 2d3e 6869 6c62 6572 7446 756e 6374 i->hilbertFunct │ │ │ │ +00022910: 696f 6e28 692c 2045 6f64 6429 2920 2020 ion(i, Eodd)) │ │ │ │ +00022920: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022960: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00022970: 3d20 7b31 2c20 312c 2031 2c20 312c 2031 = {1, 1, 1, 1, 1 │ │ │ │ -00022980: 2c20 317d 2020 2020 2020 2020 2020 2020 , 1} │ │ │ │ -00022990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00022950: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022960: 7c6f 3132 203d 207b 312c 2031 2c20 312c |o12 = {1, 1, 1, │ │ │ │ +00022970: 2031 2c20 312c 2031 7d20 2020 2020 2020 1, 1, 1} │ │ │ │ +00022980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022990: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000229a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000229c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000229e0: 2020 207c 0a7c 6f31 3220 3a20 4c69 7374 |.|o12 : List │ │ │ │ +000229d0: 2020 2020 2020 2020 7c0a 7c6f 3132 203a |.|o12 : │ │ │ │ +000229e0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 000229f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a20: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00022a10: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00022a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00022a60: 6931 3320 3a20 7573 6520 5320 2020 2020 i13 : use S │ │ │ │ +00022a50: 2d2d 2b0a 7c69 3133 203a 2075 7365 2053 --+.|i13 : use S │ │ │ │ +00022a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022a90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022a80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00022a90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00022aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ad0: 2020 2020 2020 207c 0a7c 6f31 3320 3d20 |.|o13 = │ │ │ │ -00022ae0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00022ac0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00022ad0: 3133 203d 2053 2020 2020 2020 2020 2020 13 = S │ │ │ │ +00022ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022b00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00022b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b50: 207c 0a7c 6f31 3320 3a20 506f 6c79 6e6f |.|o13 : Polyno │ │ │ │ -00022b60: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +00022b40: 2020 2020 2020 7c0a 7c6f 3133 203a 2050 |.|o13 : P │ │ │ │ +00022b50: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00022b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022b80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00022b90: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00022b80: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00022b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00022bd0: 3420 3a20 4932 203d 2069 6465 616c 2278 4 : I2 = ideal"x │ │ │ │ -00022be0: 332c 797a 2220 2020 2020 2020 2020 2020 3,yz" │ │ │ │ -00022bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022bc0: 2b0a 7c69 3134 203a 2049 3220 3d20 6964 +.|i14 : I2 = id │ │ │ │ +00022bd0: 6561 6c22 7833 2c79 7a22 2020 2020 2020 eal"x3,yz" │ │ │ │ +00022be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00022c50: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +00022c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00022c40: 2020 2020 2020 2020 2020 2033 2020 2020 3 │ │ │ │ +00022c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022c80: 2020 7c0a 7c6f 3134 203d 2069 6465 616c |.|o14 = ideal │ │ │ │ -00022c90: 2028 7820 2c20 792a 7a29 2020 2020 2020 (x , y*z) │ │ │ │ +00022c70: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ +00022c80: 6964 6561 6c20 2878 202c 2079 2a7a 2920 ideal (x , y*z) │ │ │ │ +00022c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00022cc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00022cb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00022cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022cf0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00022d00: 3134 203a 2049 6465 616c 206f 6620 5320 14 : Ideal of S │ │ │ │ +00022cf0: 207c 0a7c 6f31 3420 3a20 4964 6561 6c20 |.|o14 : Ideal │ │ │ │ +00022d00: 6f66 2053 2020 2020 2020 2020 2020 2020 of S │ │ │ │ 00022d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022d30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00022d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00022d30: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00022d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00022d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00022d70: 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a 2052 ------+.|i15 : R │ │ │ │ -00022d80: 3220 3d20 532f 4932 2020 2020 2020 2020 2 = S/I2 │ │ │ │ +00022d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00022d70: 3520 3a20 5232 203d 2053 2f49 3220 2020 5 : R2 = S/I2 │ │ │ │ +00022d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022db0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00022da0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00022db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022df0: 7c0a 7c6f 3135 203d 2052 3220 2020 2020 |.|o15 = R2 │ │ │ │ +00022de0: 2020 2020 207c 0a7c 6f31 3520 3d20 5232 |.|o15 = R2 │ │ │ │ +00022df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022e20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e60: 2020 2020 2020 2020 2020 7c0a 7c6f 3135 |.|o15 │ │ │ │ -00022e70: 203a 2051 756f 7469 656e 7452 696e 6720 : QuotientRing │ │ │ │ +00022e50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00022e60: 0a7c 6f31 3520 3a20 5175 6f74 6965 6e74 .|o15 : Quotient │ │ │ │ +00022e70: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00022e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ea0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00022e90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00022ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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2020 2020 2020 │ │ │ │ 00022f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00022f60: 7c6f 3136 203d 2063 6f6b 6572 6e65 6c20 |o16 = cokernel │ │ │ │ -00022f70: 7c20 7832 2079 207a 207c 2020 2020 2020 | x2 y z | │ │ │ │ +00022f50: 2020 207c 0a7c 6f31 3620 3d20 636f 6b65 |.|o16 = coke │ │ │ │ +00022f60: 726e 656c 207c 2078 3220 7920 7a20 7c20 rnel | x2 y z | │ │ │ │ +00022f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022f90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00022f90: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00022fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00022fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022fd0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00022fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00022ff0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ -00023000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023010: 2020 2020 207c 0a7c 6f31 3620 3a20 5232 |.|o16 : R2 │ │ │ │ -00023020: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00023030: 7420 6f66 2052 3220 2020 2020 2020 2020 t of R2 │ │ │ │ -00023040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023050: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00022fc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00022fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00022fe0: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ +00022ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023000: 2020 2020 2020 2020 2020 7c0a 7c6f 3136 |.|o16 │ │ │ │ +00023010: 203a 2052 322d 6d6f 6475 6c65 2c20 7175 : R2-module, qu │ │ │ │ +00023020: 6f74 6965 6e74 206f 6620 5232 2020 2020 otient of R2 │ │ │ │ +00023030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023040: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00023050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00023090: 0a7c 6931 3720 3a20 6265 7474 6920 6672 .|i17 : betti fr │ │ │ │ -000230a0: 6565 5265 736f 6c75 7469 6f6e 2028 4d32 eeResolution (M2 │ │ │ │ -000230b0: 2c20 4c65 6e67 7468 4c69 6d69 7420 3d3e , LengthLimit => │ │ │ │ -000230c0: 3130 2920 2020 2020 2020 2020 7c0a 7c20 10) |.| │ │ │ │ +00023080: 2d2d 2d2d 2b0a 7c69 3137 203a 2062 6574 ----+.|i17 : bet │ │ │ │ +00023090: 7469 2066 7265 6552 6573 6f6c 7574 696f ti freeResolutio │ │ │ │ +000230a0: 6e20 284d 322c 204c 656e 6774 684c 696d n (M2, LengthLim │ │ │ │ +000230b0: 6974 203d 3e31 3029 2020 2020 2020 2020 it =>10) │ │ │ │ +000230c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000230d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000230e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000230f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023100: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00023110: 2020 2020 2020 2020 2030 2031 2032 2033 0 1 2 3 │ │ │ │ -00023120: 2034 2020 3520 2036 2020 3720 2038 2020 4 5 6 7 8 │ │ │ │ -00023130: 3920 3130 2020 2020 2020 2020 2020 2020 9 10 │ │ │ │ -00023140: 2020 2020 2020 7c0a 7c6f 3137 203d 2074 |.|o17 = t │ │ │ │ -00023150: 6f74 616c 3a20 3120 3320 3520 3720 3920 otal: 1 3 5 7 9 │ │ │ │ -00023160: 3131 2031 3320 3135 2031 3720 3139 2032 11 13 15 17 19 2 │ │ │ │ -00023170: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00023180: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023190: 303a 2031 2032 2032 2032 2032 2020 3220 0: 1 2 2 2 2 2 │ │ │ │ -000231a0: 2032 2020 3220 2032 2020 3220 2032 2020 2 2 2 2 2 │ │ │ │ -000231b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000231c0: 7c0a 7c20 2020 2020 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3220 2032 2020 3220 2032 2 2 2 2 2 2 │ │ │ │ +000231a0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +000231b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000231c0: 2020 313a 202e 2031 2033 2034 2034 2020 1: . 1 3 4 4 │ │ │ │ +000231d0: 3420 2034 2020 3420 2034 2020 3420 2034 4 4 4 4 4 4 │ │ │ │ +000231e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000231f0: 2020 7c0a 7c20 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ +00023200: 3a20 2e20 2e20 2e20 3120 3320 2034 2020 : . . . 1 3 4 │ │ │ │ +00023210: 3420 2034 2020 3420 2034 2020 3420 2020 4 4 4 4 4 │ │ │ │ +00023220: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00023230: 0a7c 2020 2020 2020 2020 2020 333a 202e .| 3: . │ │ │ │ +00023240: 202e 202e 202e 202e 2020 3120 2033 2020 . . . . 1 3 │ │ │ │ +00023250: 3420 2034 2020 3420 2034 2020 2020 2020 4 4 4 4 │ │ │ │ +00023260: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00023270: 2020 2020 2020 2020 2034 3a20 2e20 2e20 4: . . │ │ │ │ +00023280: 2e20 2e20 2e20 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2020 │ │ │ │ 00023350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023360: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00023360: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00023370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000233a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ -000233b0: 2045 203d 2045 7874 4d6f 6475 6c65 204d E = ExtModule M │ │ │ │ -000233c0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000233d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000233e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00023390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +000233a0: 6931 3820 3a20 4520 3d20 4578 744d 6f64 i18 : E = ExtMod │ │ │ │ +000233b0: 756c 6520 4d32 2020 2020 2020 2020 2020 ule M2 │ │ │ │ +000233c0: 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+000237c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000237d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000237e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000237f0: 2020 7c0a 7c6f 3230 203d 2028 6b6b 5b58 |.|o20 = (kk[X │ │ │ │ -00023800: 202e 2e58 205d 2920 2020 2020 2020 2020 ..X ]) │ │ │ │ +000237e0: 2020 2020 2020 207c 0a7c 6f32 3020 3d20 |.|o20 = │ │ │ │ +000237f0: 286b 6b5b 5820 2e2e 5820 5d29 2020 2020 (kk[X ..X ]) │ │ │ │ +00023800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023820: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00023830: 0a7c 2020 2020 2020 2020 2020 2030 2020 .| 0 │ │ │ │ -00023840: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00023820: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00023830: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +00023840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023860: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00023860: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000238a0: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ -000238b0: 3a20 6b6b 5b58 202e 2e58 205d 2d6d 6f64 : kk[X ..X ]-mod │ │ │ │ -000238c0: 756c 652c 2066 7265 652c 2064 6567 7265 ule, free, degre │ │ │ │ -000238d0: 6573 207b 302e 2e31 2c20 323a 317d 2020 es {0..1, 2:1} │ │ │ │ -000238e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -000238f0: 2020 2030 2020 2031 2020 2020 2020 2020 0 1 │ │ │ │ +00023890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000238a0: 7c6f 3230 203a 206b 6b5b 5820 2e2e 5820 |o20 : kk[X ..X │ │ │ │ +000238b0: 5d2d 6d6f 6475 6c65 2c20 6672 6565 2c20 ]-module, free, │ │ │ │ +000238c0: 6465 6772 6565 7320 7b30 2e2e 312c 2032 degrees {0..1, 2 │ │ │ │ +000238d0: 3a31 7d20 2020 2020 2020 207c 0a7c 2020 :1} |.| │ │ │ │ +000238e0: 2020 2020 2020 2020 3020 2020 3120 2020 0 1 │ │ │ │ +000238f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023920: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00023910: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00023920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023960: 2b0a 7c69 3231 203a 2061 7070 6c79 2874 +.|i21 : apply(t │ │ │ │ -00023970: 6f4c 6973 7428 302e 2e35 292c 2069 2d3e oList(0..5), i-> │ │ │ │ -00023980: 6869 6c62 6572 7446 756e 6374 696f 6e28 hilbertFunction( │ │ │ │ -00023990: 692c 2045 6576 656e 2929 2020 207c 0a7c i, Eeven)) |.| │ │ │ │ +00023950: 2d2d 2d2d 2d2b 0a7c 6932 3120 3a20 6170 -----+.|i21 : ap │ │ │ │ +00023960: 706c 7928 746f 4c69 7374 2830 2e2e 3529 ply(toList(0..5) │ │ │ │ +00023970: 2c20 692d 3e68 696c 6265 7274 4675 6e63 , i->hilbertFunc │ │ │ │ +00023980: 7469 6f6e 2869 2c20 4565 7665 6e29 2920 tion(i, Eeven)) │ │ │ │ +00023990: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000239a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000239b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000239d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3231 |.|o21 │ │ │ │ -000239e0: 203d 207b 312c 2035 2c20 392c 2031 332c = {1, 5, 9, 13, │ │ │ │ -000239f0: 2031 372c 2032 317d 2020 2020 2020 2020 17, 21} │ │ │ │ -00023a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000239c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000239d0: 0a7c 6f32 3120 3d20 7b31 2c20 352c 2039 .|o21 = {1, 5, 9 │ │ │ │ +000239e0: 2c20 3133 2c20 3137 2c20 3231 7d20 2020 , 13, 17, 21} │ │ │ │ +000239f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00023a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a50: 2020 2020 7c0a 7c6f 3231 203a 204c 6973 |.|o21 : Lis │ │ │ │ -00023a60: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ +00023a40: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +00023a50: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +00023a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023a90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00023a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00023ad0: 7c69 3232 203a 2045 6f64 6420 3d20 6f64 |i22 : Eodd = od │ │ │ │ -00023ae0: 6445 7874 4d6f 6475 6c65 204d 3220 2020 dExtModule M2 │ │ │ │ +00023ac0: 2d2d 2d2b 0a7c 6932 3220 3a20 456f 6464 ---+.|i22 : Eodd │ │ │ │ +00023ad0: 203d 206f 6464 4578 744d 6f64 756c 6520 = oddExtModule │ │ │ │ +00023ae0: 4d32 2020 2020 2020 2020 2020 2020 2020 M2 │ │ │ │ 00023af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b00: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00023b00: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00023b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00023b50: 2020 2020 2020 2020 2020 2020 2034 2020 4 │ │ │ │ +00023b30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00023b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00023b50: 2020 3420 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00023b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023b80: 2020 2020 207c 0a7c 6f32 3220 3d20 286b |.|o22 = (k │ │ │ │ -00023b90: 6b5b 5820 2e2e 5820 5d29 2020 2020 2020 k[X ..X ]) │ │ │ │ +00023b70: 2020 2020 2020 2020 2020 7c0a 7c6f 3232 |.|o22 │ │ │ │ +00023b80: 203d 2028 6b6b 5b58 202e 2e58 205d 2920 = (kk[X ..X ]) │ │ │ │ +00023b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bc0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00023bd0: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00023bb0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00023bc0: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00023bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023bf0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00023c00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00023bf0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00023c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023c30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00023c40: 3232 203a 206b 6b5b 5820 2e2e 5820 5d2d 22 : kk[X ..X ]- │ │ │ │ -00023c50: 6d6f 6475 6c65 2c20 6672 6565 2c20 6465 module, free, de │ │ │ │ -00023c60: 6772 6565 7320 7b33 3a30 2c20 317d 2020 grees {3:0, 1} │ │ │ │ -00023c70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00023c80: 2020 2020 2020 3020 2020 3120 2020 2020 0 1 │ │ │ │ +00023c30: 207c 0a7c 6f32 3220 3a20 6b6b 5b58 202e |.|o22 : kk[X . │ │ │ │ +00023c40: 2e58 205d 2d6d 6f64 756c 652c 2066 7265 .X ]-module, fre │ │ │ │ +00023c50: 652c 2064 6567 7265 6573 207b 333a 302c e, degrees {3:0, │ │ │ │ +00023c60: 2031 7d20 2020 2020 2020 2020 2020 7c0a 1} |. │ │ │ │ +00023c70: 7c20 2020 2020 2020 2020 2030 2020 2031 | 0 1 │ │ │ │ +00023c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023cb0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00023ca0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00023cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023cf0: 2d2d 2d2b 0a7c 6932 3320 3a20 6170 706c ---+.|i23 : appl │ │ │ │ -00023d00: 7928 746f 4c69 7374 2830 2e2e 3529 2c20 y(toList(0..5), │ │ │ │ -00023d10: 692d 3e68 696c 6265 7274 4675 6e63 7469 i->hilbertFuncti │ │ │ │ -00023d20: 6f6e 2869 2c20 456f 6464 2929 2020 2020 on(i, Eodd)) │ │ │ │ -00023d30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00023ce0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a --------+.|i23 : │ │ │ │ +00023cf0: 2061 7070 6c79 2874 6f4c 6973 7428 302e apply(toList(0. │ │ │ │ +00023d00: 2e35 292c 2069 2d3e 6869 6c62 6572 7446 .5), i->hilbertF │ │ │ │ +00023d10: 756e 6374 696f 6e28 692c 2045 6f64 6429 unction(i, Eodd) │ │ │ │ +00023d20: 2920 2020 207c 0a7c 2020 2020 2020 2020 ) |.| │ │ │ │ +00023d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023d60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00023d70: 6f32 3320 3d20 7b33 2c20 372c 2031 312c o23 = {3, 7, 11, │ │ │ │ -00023d80: 2031 352c 2031 392c 2032 337d 2020 2020 15, 19, 23} │ │ │ │ -00023d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023da0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00023d60: 2020 7c0a 7c6f 3233 203d 207b 332c 2037 |.|o23 = {3, 7 │ │ │ │ +00023d70: 2c20 3131 2c20 3135 2c20 3139 2c20 3233 , 11, 15, 19, 23 │ │ │ │ +00023d80: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00023d90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00023da0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00023db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023de0: 2020 2020 2020 207c 0a7c 6f32 3320 3a20 |.|o23 : │ │ │ │ -00023df0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ +00023dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00023de0: 3233 203a 204c 6973 7420 2020 2020 2020 23 : List │ │ │ │ +00023df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00023e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00023e20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00023e10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00023e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00023e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00023e60: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -00023e70: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -00023e80: 2065 7665 6e45 7874 4d6f 6475 6c65 3a20 evenExtModule: │ │ │ │ -00023e90: 6576 656e 4578 744d 6f64 756c 652c 202d evenExtModule, - │ │ │ │ -00023ea0: 2d20 6576 656e 2070 6172 7420 6f66 2045 - even part of E │ │ │ │ -00023eb0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -00023ec0: 0a20 2020 2063 6f6d 706c 6574 6520 696e . complete in │ │ │ │ -00023ed0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ -00023ee0: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -00023ef0: 7261 746f 7220 7269 6e67 0a20 202a 202a rator ring. * * │ │ │ │ -00023f00: 6e6f 7465 206f 6464 4578 744d 6f64 756c note oddExtModul │ │ │ │ -00023f10: 653a 206f 6464 4578 744d 6f64 756c 652c e: oddExtModule, │ │ │ │ -00023f20: 202d 2d20 6f64 6420 7061 7274 206f 6620 -- odd part of │ │ │ │ -00023f30: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ -00023f40: 6120 636f 6d70 6c65 7465 0a20 2020 2069 a complete. i │ │ │ │ -00023f50: 6e74 6572 7365 6374 696f 6e20 6173 206d ntersection as m │ │ │ │ -00023f60: 6f64 756c 6520 6f76 6572 2043 4920 6f70 odule over CI op │ │ │ │ -00023f70: 6572 6174 6f72 2072 696e 670a 0a57 6179 erator ring..Way │ │ │ │ -00023f80: 7320 746f 2075 7365 2045 7874 4d6f 6475 s to use ExtModu │ │ │ │ -00023f90: 6c65 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d le:.============ │ │ │ │ -00023fa0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00023fb0: 2245 7874 4d6f 6475 6c65 284d 6f64 756c "ExtModule(Modul │ │ │ │ -00023fc0: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ -00023fd0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00023fe0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00023ff0: 6f62 6a65 6374 202a 6e6f 7465 2045 7874 object *note Ext │ │ │ │ -00024000: 4d6f 6475 6c65 3a20 4578 744d 6f64 756c Module: ExtModul │ │ │ │ -00024010: 652c 2069 7320 6120 2a6e 6f74 6520 6d65 e, is a *note me │ │ │ │ -00024020: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -00024030: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -00024040: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +00023e50: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +00023e60: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +00023e70: 2a6e 6f74 6520 6576 656e 4578 744d 6f64 *note evenExtMod │ │ │ │ +00023e80: 756c 653a 2065 7665 6e45 7874 4d6f 6475 ule: evenExtModu │ │ │ │ +00023e90: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274 le, -- even part │ │ │ │ +00023ea0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +00023eb0: 7665 7220 610a 2020 2020 636f 6d70 6c65 ver a. comple │ │ │ │ +00023ec0: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ +00023ed0: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ +00023ee0: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +00023ef0: 2020 2a20 2a6e 6f74 6520 6f64 6445 7874 * *note oddExt │ │ │ │ +00023f00: 4d6f 6475 6c65 3a20 6f64 6445 7874 4d6f Module: oddExtMo │ │ │ │ +00023f10: 6475 6c65 2c20 2d2d 206f 6464 2070 6172 dule, -- odd par │ │ │ │ +00023f20: 7420 6f66 2045 7874 5e2a 284d 2c6b 2920 t of Ext^*(M,k) │ │ │ │ +00023f30: 6f76 6572 2061 2063 6f6d 706c 6574 650a over a complete. │ │ │ │ +00023f40: 2020 2020 696e 7465 7273 6563 7469 6f6e intersection │ │ │ │ +00023f50: 2061 7320 6d6f 6475 6c65 206f 7665 7220 as module over │ │ │ │ +00023f60: 4349 206f 7065 7261 746f 7220 7269 6e67 CI operator ring │ │ │ │ +00023f70: 0a0a 5761 7973 2074 6f20 7573 6520 4578 ..Ways to use Ex │ │ │ │ +00023f80: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d tModule:.======= │ │ │ │ +00023f90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00023fa0: 0a20 202a 2022 4578 744d 6f64 756c 6528 . * "ExtModule( │ │ │ │ +00023fb0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ +00023fc0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +00023fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00023fe0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00023ff0: 6520 4578 744d 6f64 756c 653a 2045 7874 e ExtModule: Ext │ │ │ │ +00024000: 4d6f 6475 6c65 2c20 6973 2061 202a 6e6f Module, is a *no │ │ │ │ +00024010: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00024020: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +00024030: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +00024040: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 00024050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -000240a0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -000240b0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -000240c0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -000240d0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -000240e0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -000240f0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00024100: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ -00024110: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00024120: 7574 696f 6e73 2e6d 323a 3335 3936 3a30 utions.m2:3596:0 │ │ │ │ -00024130: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ -00024140: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -00024150: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -00024160: 4e6f 6465 3a20 4578 744d 6f64 756c 6544 Node: ExtModuleD │ │ │ │ -00024170: 6174 612c 204e 6578 743a 2065 7874 5673 ata, Next: extVs │ │ │ │ -00024180: 436f 686f 6d6f 6c6f 6779 2c20 5072 6576 Cohomology, Prev │ │ │ │ -00024190: 3a20 4578 744d 6f64 756c 652c 2055 703a : ExtModule, Up: │ │ │ │ -000241a0: 2054 6f70 0a0a 4578 744d 6f64 756c 6544 Top..ExtModuleD │ │ │ │ -000241b0: 6174 6120 2d2d 2045 7665 6e20 616e 6420 ata -- Even and │ │ │ │ -000241c0: 6f64 6420 4578 7420 6d6f 6475 6c65 7320 odd Ext modules │ │ │ │ -000241d0: 616e 6420 7468 6569 7220 7265 6775 6c61 and their regula │ │ │ │ -000241e0: 7269 7479 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a rity.*********** │ │ │ │ +00024090: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +000240a0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +000240b0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +000240c0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +000240d0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +000240e0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +000240f0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +00024100: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00024110: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ +00024120: 3539 363a 302e 0a1f 0a46 696c 653a 2043 596:0....File: C │ │ │ │ +00024130: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +00024140: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +00024150: 6e66 6f2c 204e 6f64 653a 2045 7874 4d6f nfo, Node: ExtMo │ │ │ │ +00024160: 6475 6c65 4461 7461 2c20 4e65 7874 3a20 duleData, Next: │ │ │ │ +00024170: 6578 7456 7343 6f68 6f6d 6f6c 6f67 792c extVsCohomology, │ │ │ │ +00024180: 2050 7265 763a 2045 7874 4d6f 6475 6c65 Prev: ExtModule │ │ │ │ +00024190: 2c20 5570 3a20 546f 700a 0a45 7874 4d6f , Up: Top..ExtMo │ │ │ │ +000241a0: 6475 6c65 4461 7461 202d 2d20 4576 656e duleData -- Even │ │ │ │ +000241b0: 2061 6e64 206f 6464 2045 7874 206d 6f64 and odd Ext mod │ │ │ │ +000241c0: 756c 6573 2061 6e64 2074 6865 6972 2072 ules and their r │ │ │ │ +000241d0: 6567 756c 6172 6974 790a 2a2a 2a2a 2a2a egularity.****** │ │ │ │ +000241e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000241f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00024200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00024220: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -00024230: 0a20 2020 2020 2020 204c 203d 2045 7874 . L = Ext │ │ │ │ -00024240: 4d6f 6475 6c65 4461 7461 204d 0a20 202a ModuleData M. * │ │ │ │ -00024250: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00024260: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -00024270: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00024280: 6329 4d6f 6475 6c65 2c2c 204d 6f64 756c c)Module,, Modul │ │ │ │ -00024290: 6520 6f76 6572 2061 2063 6f6d 706c 6574 e over a complet │ │ │ │ -000242a0: 650a 2020 2020 2020 2020 696e 7465 7273 e. inters │ │ │ │ -000242b0: 6563 7469 6f6e 2053 0a20 202a 204f 7574 ection S. * Out │ │ │ │ -000242c0: 7075 7473 3a0a 2020 2020 2020 2a20 4c2c puts:. * L, │ │ │ │ -000242d0: 2061 202a 6e6f 7465 206c 6973 743a 2028 a *note list: ( │ │ │ │ -000242e0: 4d61 6361 756c 6179 3244 6f63 294c 6973 Macaulay2Doc)Lis │ │ │ │ -000242f0: 742c 2c20 4c20 3d20 5c7b 6576 656e 4578 t,, L = \{evenEx │ │ │ │ -00024300: 744d 6f64 756c 652c 0a20 2020 2020 2020 tModule,. │ │ │ │ -00024310: 206f 6464 4578 744d 6f64 756c 652c 2072 oddExtModule, r │ │ │ │ -00024320: 6567 302c 2072 6567 315c 7d0a 0a44 6573 eg0, reg1\}..Des │ │ │ │ -00024330: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00024340: 3d3d 3d3d 0a0a 5375 7070 6f73 6520 7468 ====..Suppose th │ │ │ │ -00024350: 6174 204d 2069 7320 6120 6d6f 6475 6c65 at M is a module │ │ │ │ -00024360: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ -00024370: 2069 6e74 6572 7365 6374 696f 6e20 5220 intersection R │ │ │ │ -00024380: 736f 2074 6861 740a 0a45 203a 3d20 4578 so that..E := Ex │ │ │ │ -00024390: 744d 6f64 756c 6520 4d0a 0a69 7320 6120 tModule M..is a │ │ │ │ -000243a0: 6d6f 6475 6c65 2067 656e 6572 6174 6564 module generated │ │ │ │ -000243b0: 2069 6e20 6465 6772 6565 7320 3e3d 3020 in degrees >=0 │ │ │ │ -000243c0: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961 over a polynomia │ │ │ │ -000243d0: 6c20 7269 6e67 2054 2720 6765 6e65 7261 l ring T' genera │ │ │ │ -000243e0: 7465 6420 696e 0a64 6567 7265 6520 322c ted in.degree 2, │ │ │ │ -000243f0: 2061 6e64 0a0a 4530 203a 3d20 6576 656e and..E0 := even │ │ │ │ -00024400: 4578 744d 6f64 756c 6520 4d20 616e 6420 ExtModule M and │ │ │ │ -00024410: 4531 203a 3d20 6f64 6445 7874 4d6f 6475 E1 := oddExtModu │ │ │ │ -00024420: 6c65 204d 0a0a 6172 6520 6d6f 6475 6c65 le M..are module │ │ │ │ -00024430: 7320 6765 6e65 7261 7465 6420 696e 2064 s generated in d │ │ │ │ -00024440: 6567 7265 6520 3e3d 2030 206f 7665 7220 egree >= 0 over │ │ │ │ -00024450: 6120 706f 6c79 6e6f 6d69 616c 2072 696e a polynomial rin │ │ │ │ -00024460: 6720 5420 7769 7468 2067 656e 6572 6174 g T with generat │ │ │ │ -00024470: 6f72 730a 696e 2064 6567 7265 6520 312e ors.in degree 1. │ │ │ │ -00024480: 0a0a 5468 6520 7363 7269 7074 2072 6574 ..The script ret │ │ │ │ -00024490: 7572 6e73 0a0a 4c20 3d20 5c7b 4530 2c45 urns..L = \{E0,E │ │ │ │ -000244a0: 312c 2072 6567 756c 6172 6974 7920 4530 1, regularity E0 │ │ │ │ -000244b0: 2c20 7265 6775 6c61 7269 7479 2045 315c , regularity E1\ │ │ │ │ -000244c0: 7d0a 0a61 6e64 2070 7269 6e74 7320 6120 }..and prints a │ │ │ │ -000244d0: 6d65 7373 6167 6520 6966 207c 7265 6730 message if |reg0 │ │ │ │ -000244e0: 2d72 6567 317c 3e31 2e0a 0a49 6620 7765 -reg1|>1...If we │ │ │ │ -000244f0: 2073 6574 2072 203d 206d 6178 2832 2a72 set r = max(2*r │ │ │ │ -00024500: 6567 302c 2031 2b32 2a72 6567 3129 2c20 eg0, 1+2*reg1), │ │ │ │ -00024510: 616e 6420 4620 6973 2061 2072 6573 6f6c and F is a resol │ │ │ │ -00024520: 7574 696f 6e20 6f66 204d 2c20 7468 656e ution of M, then │ │ │ │ -00024530: 2063 6f6b 6572 0a46 2e64 645f 7b28 722b coker.F.dd_{(r+ │ │ │ │ -00024540: 3129 7d20 6973 2074 6865 2066 6972 7374 1)} is the first │ │ │ │ -00024550: 2073 7a79 6779 206d 6f64 756c 6520 6f66 szygy module of │ │ │ │ -00024560: 204d 2073 7563 6820 7468 6174 2072 6567 M such that reg │ │ │ │ -00024570: 756c 6172 6974 7920 6576 656e 4578 744d ularity evenExtM │ │ │ │ -00024580: 6f64 756c 650a 4d20 3d30 2041 4e44 2072 odule.M =0 AND r │ │ │ │ -00024590: 6567 756c 6172 6974 7920 6f64 6445 7874 egularity oddExt │ │ │ │ -000245a0: 4d6f 6475 6c65 204d 203d 300a 0a57 6520 Module M =0..We │ │ │ │ -000245b0: 6861 7665 2062 6565 6e20 7573 696e 6720 have been using │ │ │ │ -000245c0: 7265 6775 6c61 7269 7479 2045 7874 4d6f regularity ExtMo │ │ │ │ -000245d0: 6475 6c65 204d 2061 7320 6120 7375 6273 dule M as a subs │ │ │ │ -000245e0: 7469 7475 7465 2066 6f72 2072 2c20 6275 titute for r, bu │ │ │ │ -000245f0: 7420 7468 6174 2773 206e 6f74 0a61 6c77 t that's not.alw │ │ │ │ -00024600: 6179 7320 7468 6520 7361 6d65 2e0a 0a54 ays the same...T │ │ │ │ -00024610: 6865 2072 6567 756c 6172 6974 6965 7320 he regularities │ │ │ │ -00024620: 6f66 2074 6865 2065 7665 6e20 616e 6420 of the even and │ │ │ │ -00024630: 6f64 6420 4578 7420 6d6f 6475 6c65 7320 odd Ext modules │ │ │ │ -00024640: 2a63 616e 2a20 6469 6666 6572 2062 7920 *can* differ by │ │ │ │ -00024650: 6d6f 7265 2074 6861 6e20 312e 0a41 6e20 more than 1..An │ │ │ │ -00024660: 6578 616d 706c 6520 6361 6e20 6265 2070 example can be p │ │ │ │ -00024670: 726f 6475 6365 6420 7769 7468 2073 6574 roduced with set │ │ │ │ -00024680: 5261 6e64 6f6d 5365 6564 2030 2053 203d RandomSeed 0 S = │ │ │ │ -00024690: 205a 5a2f 3130 315b 612c 622c 632c 645d ZZ/101[a,b,c,d] │ │ │ │ -000246a0: 2066 660a 3d6d 6174 7269 7822 6134 2c62 ff.=matrix"a4,b │ │ │ │ -000246b0: 342c 6334 2c64 3422 2052 203d 2053 2f69 4,c4,d4" R = S/i │ │ │ │ -000246c0: 6465 616c 2066 6620 4e20 3d20 636f 6b65 deal ff N = coke │ │ │ │ -000246d0: 7220 7261 6e64 6f6d 2852 5e7b 302c 317d r random(R^{0,1} │ │ │ │ -000246e0: 2c20 525e 7b20 2d31 2c2d 322c 2d33 2c2d , R^{ -1,-2,-3,- │ │ │ │ -000246f0: 347d 290a 2d2d 6769 7665 7320 7265 6720 4}).--gives reg │ │ │ │ -00024700: 4578 745e 6576 656e 203d 2034 2c20 7265 Ext^even = 4, re │ │ │ │ -00024710: 6720 4578 745e 6f64 6420 3d20 3320 4c20 g Ext^odd = 3 L │ │ │ │ -00024720: 3d20 4578 744d 6f64 756c 6544 6174 6120 = ExtModuleData │ │ │ │ -00024730: 4e3b 2062 7574 2074 616b 6573 2073 6f6d N; but takes som │ │ │ │ -00024740: 650a 7469 6d65 2074 6f20 636f 6d70 7574 e.time to comput │ │ │ │ -00024750: 652e 0a0a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d e.....+--------- │ │ │ │ +00024210: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +00024220: 6167 653a 200a 2020 2020 2020 2020 4c20 age: . L │ │ │ │ +00024230: 3d20 4578 744d 6f64 756c 6544 6174 6120 = ExtModuleData │ │ │ │ +00024240: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ +00024250: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00024260: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +00024270: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +00024280: 4d6f 6475 6c65 206f 7665 7220 6120 636f Module over a co │ │ │ │ +00024290: 6d70 6c65 7465 0a20 2020 2020 2020 2069 mplete. i │ │ │ │ +000242a0: 6e74 6572 7365 6374 696f 6e20 530a 2020 ntersection S. │ │ │ │ +000242b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +000242c0: 202a 204c 2c20 6120 2a6e 6f74 6520 6c69 * L, a *note li │ │ │ │ +000242d0: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ +000242e0: 6329 4c69 7374 2c2c 204c 203d 205c 7b65 c)List,, L = \{e │ │ │ │ +000242f0: 7665 6e45 7874 4d6f 6475 6c65 2c0a 2020 venExtModule,. │ │ │ │ +00024300: 2020 2020 2020 6f64 6445 7874 4d6f 6475 oddExtModu │ │ │ │ +00024310: 6c65 2c20 7265 6730 2c20 7265 6731 5c7d le, reg0, reg1\} │ │ │ │ +00024320: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00024330: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 7570 706f =========..Suppo │ │ │ │ +00024340: 7365 2074 6861 7420 4d20 6973 2061 206d se that M is a m │ │ │ │ +00024350: 6f64 756c 6520 6f76 6572 2061 2063 6f6d odule over a com │ │ │ │ +00024360: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +00024370: 6f6e 2052 2073 6f20 7468 6174 0a0a 4520 on R so that..E │ │ │ │ +00024380: 3a3d 2045 7874 4d6f 6475 6c65 204d 0a0a := ExtModule M.. │ │ │ │ +00024390: 6973 2061 206d 6f64 756c 6520 6765 6e65 is a module gene │ │ │ │ +000243a0: 7261 7465 6420 696e 2064 6567 7265 6573 rated in degrees │ │ │ │ +000243b0: 203e 3d30 206f 7665 7220 6120 706f 6c79 >=0 over a poly │ │ │ │ +000243c0: 6e6f 6d69 616c 2072 696e 6720 5427 2067 nomial ring T' g │ │ │ │ +000243d0: 656e 6572 6174 6564 2069 6e0a 6465 6772 enerated in.degr │ │ │ │ +000243e0: 6565 2032 2c20 616e 640a 0a45 3020 3a3d ee 2, and..E0 := │ │ │ │ +000243f0: 2065 7665 6e45 7874 4d6f 6475 6c65 204d evenExtModule M │ │ │ │ +00024400: 2061 6e64 2045 3120 3a3d 206f 6464 4578 and E1 := oddEx │ │ │ │ +00024410: 744d 6f64 756c 6520 4d0a 0a61 7265 206d tModule M..are m │ │ │ │ +00024420: 6f64 756c 6573 2067 656e 6572 6174 6564 odules generated │ │ │ │ +00024430: 2069 6e20 6465 6772 6565 203e 3d20 3020 in degree >= 0 │ │ │ │ +00024440: 6f76 6572 2061 2070 6f6c 796e 6f6d 6961 over a polynomia │ │ │ │ +00024450: 6c20 7269 6e67 2054 2077 6974 6820 6765 l ring T with ge │ │ │ │ +00024460: 6e65 7261 746f 7273 0a69 6e20 6465 6772 nerators.in degr │ │ │ │ +00024470: 6565 2031 2e0a 0a54 6865 2073 6372 6970 ee 1...The scrip │ │ │ │ +00024480: 7420 7265 7475 726e 730a 0a4c 203d 205c t returns..L = \ │ │ │ │ +00024490: 7b45 302c 4531 2c20 7265 6775 6c61 7269 {E0,E1, regulari │ │ │ │ +000244a0: 7479 2045 302c 2072 6567 756c 6172 6974 ty E0, regularit │ │ │ │ +000244b0: 7920 4531 5c7d 0a0a 616e 6420 7072 696e y E1\}..and prin │ │ │ │ +000244c0: 7473 2061 206d 6573 7361 6765 2069 6620 ts a message if │ │ │ │ +000244d0: 7c72 6567 302d 7265 6731 7c3e 312e 0a0a |reg0-reg1|>1... │ │ │ │ +000244e0: 4966 2077 6520 7365 7420 7220 3d20 6d61 If we set r = ma │ │ │ │ +000244f0: 7828 322a 7265 6730 2c20 312b 322a 7265 x(2*reg0, 1+2*re │ │ │ │ +00024500: 6731 292c 2061 6e64 2046 2069 7320 6120 g1), and F is a │ │ │ │ +00024510: 7265 736f 6c75 7469 6f6e 206f 6620 4d2c resolution of M, │ │ │ │ +00024520: 2074 6865 6e20 636f 6b65 720a 462e 6464 then coker.F.dd │ │ │ │ +00024530: 5f7b 2872 2b31 297d 2069 7320 7468 6520 _{(r+1)} is the │ │ │ │ +00024540: 6669 7273 7420 737a 7967 7920 6d6f 6475 first szygy modu │ │ │ │ +00024550: 6c65 206f 6620 4d20 7375 6368 2074 6861 le of M such tha │ │ │ │ +00024560: 7420 7265 6775 6c61 7269 7479 2065 7665 t regularity eve │ │ │ │ +00024570: 6e45 7874 4d6f 6475 6c65 0a4d 203d 3020 nExtModule.M =0 │ │ │ │ +00024580: 414e 4420 7265 6775 6c61 7269 7479 206f AND regularity o │ │ │ │ +00024590: 6464 4578 744d 6f64 756c 6520 4d20 3d30 ddExtModule M =0 │ │ │ │ +000245a0: 0a0a 5765 2068 6176 6520 6265 656e 2075 ..We have been u │ │ │ │ +000245b0: 7369 6e67 2072 6567 756c 6172 6974 7920 sing regularity │ │ │ │ +000245c0: 4578 744d 6f64 756c 6520 4d20 6173 2061 ExtModule M as a │ │ │ │ +000245d0: 2073 7562 7374 6974 7574 6520 666f 7220 substitute for │ │ │ │ +000245e0: 722c 2062 7574 2074 6861 7427 7320 6e6f r, but that's no │ │ │ │ +000245f0: 740a 616c 7761 7973 2074 6865 2073 616d t.always the sam │ │ │ │ +00024600: 652e 0a0a 5468 6520 7265 6775 6c61 7269 e...The regulari │ │ │ │ +00024610: 7469 6573 206f 6620 7468 6520 6576 656e ties of the even │ │ │ │ +00024620: 2061 6e64 206f 6464 2045 7874 206d 6f64 and odd Ext mod │ │ │ │ +00024630: 756c 6573 202a 6361 6e2a 2064 6966 6665 ules *can* diffe │ │ │ │ +00024640: 7220 6279 206d 6f72 6520 7468 616e 2031 r by more than 1 │ │ │ │ +00024650: 2e0a 416e 2065 7861 6d70 6c65 2063 616e ..An example can │ │ │ │ +00024660: 2062 6520 7072 6f64 7563 6564 2077 6974 be produced wit │ │ │ │ +00024670: 6820 7365 7452 616e 646f 6d53 6565 6420 h setRandomSeed │ │ │ │ +00024680: 3020 5320 3d20 5a5a 2f31 3031 5b61 2c62 0 S = ZZ/101[a,b │ │ │ │ +00024690: 2c63 2c64 5d20 6666 0a3d 6d61 7472 6978 ,c,d] ff.=matrix │ │ │ │ +000246a0: 2261 342c 6234 2c63 342c 6434 2220 5220 "a4,b4,c4,d4" R │ │ │ │ +000246b0: 3d20 532f 6964 6561 6c20 6666 204e 203d = S/ideal ff N = │ │ │ │ +000246c0: 2063 6f6b 6572 2072 616e 646f 6d28 525e coker random(R^ │ │ │ │ +000246d0: 7b30 2c31 7d2c 2052 5e7b 202d 312c 2d32 {0,1}, R^{ -1,-2 │ │ │ │ +000246e0: 2c2d 332c 2d34 7d29 0a2d 2d67 6976 6573 ,-3,-4}).--gives │ │ │ │ +000246f0: 2072 6567 2045 7874 5e65 7665 6e20 3d20 reg Ext^even = │ │ │ │ +00024700: 342c 2072 6567 2045 7874 5e6f 6464 203d 4, reg Ext^odd = │ │ │ │ +00024710: 2033 204c 203d 2045 7874 4d6f 6475 6c65 3 L = ExtModule │ │ │ │ +00024720: 4461 7461 204e 3b20 6275 7420 7461 6b65 Data N; but take │ │ │ │ +00024730: 7320 736f 6d65 0a74 696d 6520 746f 2063 s some.time to c │ │ │ │ +00024740: 6f6d 7075 7465 2e0a 0a0a 0a2b 2d2d 2d2d ompute.....+---- │ │ │ │ +00024750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00024790: 203a 2073 6574 5261 6e64 6f6d 5365 6564 : setRandomSeed │ │ │ │ -000247a0: 2031 3030 2020 2020 2020 2020 2020 2020 100 │ │ │ │ -000247b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000247c0: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ -000247d0: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ -000247e0: 3130 3020 2020 2020 2020 2020 2020 2020 100 │ │ │ │ -000247f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00024780: 2b0a 7c69 3120 3a20 7365 7452 616e 646f +.|i1 : setRando │ │ │ │ +00024790: 6d53 6565 6420 3130 3020 2020 2020 2020 mSeed 100 │ │ │ │ +000247a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000247b0: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ +000247c0: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ +000247d0: 6420 746f 2031 3030 2020 2020 2020 2020 d to 100 │ │ │ │ +000247e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000247f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00024800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024830: 7c0a 7c6f 3120 3d20 3130 3020 2020 2020 |.|o1 = 100 │ │ │ │ +00024820: 2020 2020 207c 0a7c 6f31 203d 2031 3030 |.|o1 = 100 │ │ │ │ +00024830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024860: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024850: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00024860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -000248a0: 7c69 3220 3a20 5320 3d20 5a5a 2f31 3031 |i2 : S = ZZ/101 │ │ │ │ -000248b0: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020 [a,b,c,d]; │ │ │ │ -000248c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000248d0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00024890: 2d2d 2d2b 0a7c 6932 203a 2053 203d 205a ---+.|i2 : S = Z │ │ │ │ +000248a0: 5a2f 3130 315b 612c 622c 632c 645d 3b20 Z/101[a,b,c,d]; │ │ │ │ +000248b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000248c0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000248d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000248e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000248f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00024910: 3320 3a20 6620 3d20 6d61 7028 535e 312c 3 : f = map(S^1, │ │ │ │ -00024920: 2053 5e34 2c20 2869 2c6a 2920 2d3e 2053 S^4, (i,j) -> S │ │ │ │ -00024930: 5f6a 5e33 2920 2020 2020 2020 2020 2020 _j^3) │ │ │ │ -00024940: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024900: 2d2b 0a7c 6933 203a 2066 203d 206d 6170 -+.|i3 : f = map │ │ │ │ +00024910: 2853 5e31 2c20 535e 342c 2028 692c 6a29 (S^1, S^4, (i,j) │ │ │ │ +00024920: 202d 3e20 535f 6a5e 3329 2020 2020 2020 -> S_j^3) │ │ │ │ +00024930: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00024940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024970: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -00024980: 3d20 7c20 6133 2062 3320 6333 2064 3320 = | a3 b3 c3 d3 │ │ │ │ -00024990: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000249a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024960: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00024970: 0a7c 6f33 203d 207c 2061 3320 6233 2063 .|o3 = | a3 b3 c │ │ │ │ +00024980: 3320 6433 207c 2020 2020 2020 2020 2020 3 d3 | │ │ │ │ +00024990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000249a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000249b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000249c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000249e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -000249f0: 2020 2020 2020 2020 3120 2020 2020 2034 1 4 │ │ │ │ +000249d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000249e0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +000249f0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ 00024a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024a10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00024a20: 0a7c 6f33 203a 204d 6174 7269 7820 5320 .|o3 : Matrix S │ │ │ │ -00024a30: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ -00024a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024a50: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00024a10: 2020 2020 7c0a 7c6f 3320 3a20 4d61 7472 |.|o3 : Matr │ │ │ │ +00024a20: 6978 2053 2020 3c2d 2d20 5320 2020 2020 ix S <-- S │ │ │ │ +00024a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024a40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00024a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00024a90: 6934 203a 2052 203d 2053 2f69 6465 616c i4 : R = S/ideal │ │ │ │ -00024aa0: 2066 3b20 2020 2020 2020 2020 2020 2020 f; │ │ │ │ -00024ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00024a80: 2d2d 2b0a 7c69 3420 3a20 5220 3d20 532f --+.|i4 : R = S/ │ │ │ │ +00024a90: 6964 6561 6c20 663b 2020 2020 2020 2020 ideal f; │ │ │ │ +00024aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024ab0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00024ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ -00024b00: 203a 204d 203d 2052 5e31 2f69 6465 616c : M = R^1/ideal │ │ │ │ -00024b10: 2261 6232 2b63 6432 223b 2020 2020 2020 "ab2+cd2"; │ │ │ │ -00024b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024b30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00024af0: 2b0a 7c69 3520 3a20 4d20 3d20 525e 312f +.|i5 : M = R^1/ │ │ │ │ +00024b00: 6964 6561 6c22 6162 322b 6364 3222 3b20 ideal"ab2+cd2"; │ │ │ │ +00024b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024b20: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024b60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -00024b70: 2062 6574 7469 2028 4620 3d20 6672 6565 betti (F = free │ │ │ │ -00024b80: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65 Resolution(M, Le │ │ │ │ -00024b90: 6e67 7468 4c69 6d69 7420 3d3e 2035 2929 ngthLimit => 5)) │ │ │ │ -00024ba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00024b60: 7c69 3620 3a20 6265 7474 6920 2846 203d |i6 : betti (F = │ │ │ │ +00024b70: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ +00024b80: 4d2c 204c 656e 6774 684c 696d 6974 203d M, LengthLimit = │ │ │ │ +00024b90: 3e20 3529 297c 0a7c 2020 2020 2020 2020 > 5))|.| │ │ │ │ +00024ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024bd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00024be0: 2020 2020 2020 3020 3120 3220 2033 2020 0 1 2 3 │ │ │ │ -00024bf0: 3420 2035 2020 2020 2020 2020 2020 2020 4 5 │ │ │ │ -00024c00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00024c10: 7c6f 3620 3d20 746f 7461 6c3a 2031 2031 |o6 = total: 1 1 │ │ │ │ -00024c20: 2035 2031 3620 3335 2036 3420 2020 2020 5 16 35 64 │ │ │ │ -00024c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024c40: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00024c50: 2030 3a20 3120 2e20 2e20 202e 2020 2e20 0: 1 . . . . │ │ │ │ -00024c60: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00024c70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00024c80: 2020 2020 2020 2020 313a 202e 202e 202e 1: . . . │ │ │ │ -00024c90: 2020 2e20 202e 2020 2e20 2020 2020 2020 . . . │ │ │ │ -00024ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024cb0: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ -00024cc0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e : . 1 . . . . │ │ │ │ -00024cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00024cf0: 2020 2020 2020 333a 202e 202e 2031 2020 3: . . 1 │ │ │ │ -00024d00: 2e20 202e 2020 2e20 2020 2020 2020 2020 . . . │ │ │ │ -00024d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d20: 207c 0a7c 2020 2020 2020 2020 2034 3a20 |.| 4: │ │ │ │ -00024d30: 2e20 2e20 3320 2038 2020 3520 202e 2020 . . 3 8 5 . │ │ │ │ -00024d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024d50: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00024d60: 2020 2020 353a 202e 202e 2031 2020 3820 5: . . 1 8 │ │ │ │ -00024d70: 3235 2033 3220 2020 2020 2020 2020 2020 25 32 │ │ │ │ -00024d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00024d90: 0a7c 2020 2020 2020 2020 2036 3a20 2e20 .| 6: . │ │ │ │ -00024da0: 2e20 2e20 202e 2020 3520 3332 2020 2020 . . . 5 32 │ │ │ │ -00024db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024dc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00024bc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00024bd0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +00024be0: 2020 3320 2034 2020 3520 2020 2020 2020 3 4 5 │ │ │ │ +00024bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c00: 2020 207c 0a7c 6f36 203d 2074 6f74 616c |.|o6 = total │ │ │ │ +00024c10: 3a20 3120 3120 3520 3136 2033 3520 3634 : 1 1 5 16 35 64 │ │ │ │ +00024c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00024c40: 2020 2020 2020 303a 2031 202e 202e 2020 0: 1 . . │ │ │ │ +00024c50: 2e20 202e 2020 2e20 2020 2020 2020 2020 . . . │ │ │ │ +00024c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024c70: 207c 0a7c 2020 2020 2020 2020 2031 3a20 |.| 1: │ │ │ │ +00024c80: 2e20 2e20 2e20 202e 2020 2e20 202e 2020 . . . . . . │ │ │ │ +00024c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024ca0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00024cb0: 2020 2020 323a 202e 2031 202e 2020 2e20 2: . 1 . . │ │ │ │ +00024cc0: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ +00024cd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00024ce0: 0a7c 2020 2020 2020 2020 2033 3a20 2e20 .| 3: . │ │ │ │ +00024cf0: 2e20 3120 202e 2020 2e20 202e 2020 2020 . 1 . . . │ │ │ │ +00024d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00024d20: 2020 343a 202e 202e 2033 2020 3820 2035 4: . . 3 8 5 │ │ │ │ +00024d30: 2020 2e20 2020 2020 2020 2020 2020 2020 . │ │ │ │ +00024d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00024d50: 2020 2020 2020 2020 2035 3a20 2e20 2e20 5: . . │ │ │ │ +00024d60: 3120 2038 2032 3520 3332 2020 2020 2020 1 8 25 32 │ │ │ │ +00024d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024d80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00024d90: 363a 202e 202e 202e 2020 2e20 2035 2033 6: . . . . 5 3 │ │ │ │ +00024da0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00024db0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00024dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024df0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00024e00: 6f36 203a 2042 6574 7469 5461 6c6c 7920 o6 : BettiTally │ │ │ │ +00024df0: 2020 7c0a 7c6f 3620 3a20 4265 7474 6954 |.|o6 : BettiT │ │ │ │ +00024e00: 616c 6c79 2020 2020 2020 2020 2020 2020 ally │ │ │ │ 00024e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024e30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00024e20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00024e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -00024e70: 203a 2045 203d 2045 7874 4d6f 6475 6c65 : E = ExtModule │ │ │ │ -00024e80: 4461 7461 204d 3b20 2020 2020 2020 2020 Data M; │ │ │ │ -00024e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024ea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00024e60: 2b0a 7c69 3720 3a20 4520 3d20 4578 744d +.|i7 : E = ExtM │ │ │ │ +00024e70: 6f64 756c 6544 6174 6120 4d3b 2020 2020 oduleData M; │ │ │ │ +00024e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00024e90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00024ea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024ed0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a ---------+.|i8 : │ │ │ │ -00024ee0: 2045 5f32 2020 2020 2020 2020 2020 2020 E_2 │ │ │ │ +00024ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00024ed0: 7c69 3820 3a20 455f 3220 2020 2020 2020 |i8 : E_2 │ │ │ │ +00024ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f10: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00024f00: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00024f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f40: 2020 2020 2020 207c 0a7c 6f38 203d 2032 |.|o8 = 2 │ │ │ │ +00024f30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00024f40: 3820 3d20 3220 2020 2020 2020 2020 2020 8 = 2 │ │ │ │ 00024f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024f70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00024f80: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00024f70: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00024f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00024f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00024fb0: 2d2d 2d2d 2d2b 0a7c 6939 203a 2045 5f33 -----+.|i9 : E_3 │ │ │ │ +00024fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 ----------+.|i9 │ │ │ │ +00024fb0: 3a20 455f 3320 2020 2020 2020 2020 2020 : E_3 │ │ │ │ 00024fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00024fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00024fe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00024fe0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00024ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025020: 2020 207c 0a7c 6f39 203d 2031 2020 2020 |.|o9 = 1 │ │ │ │ +00025010: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ +00025020: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00025030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025050: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00025040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00025050: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00025060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025090: 2d2b 0a7c 6931 3020 3a20 7220 3d20 6d61 -+.|i10 : r = ma │ │ │ │ -000250a0: 7828 322a 455f 322c 322a 455f 332b 3129 x(2*E_2,2*E_3+1) │ │ │ │ -000250b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00025080: 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a 2072 ------+.|i10 : r │ │ │ │ +00025090: 203d 206d 6178 2832 2a45 5f32 2c32 2a45 = max(2*E_2,2*E │ │ │ │ +000250a0: 5f33 2b31 2920 2020 2020 2020 2020 2020 _3+1) │ │ │ │ +000250b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000250c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000250e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000250f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00025100: 0a7c 6f31 3020 3d20 3420 2020 2020 2020 .|o10 = 4 │ │ │ │ +000250f0: 2020 2020 7c0a 7c6f 3130 203d 2034 2020 |.|o10 = 4 │ │ │ │ +00025100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025130: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00025120: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00025130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00025170: 6931 3120 3a20 4572 203d 2045 7874 4d6f i11 : Er = ExtMo │ │ │ │ -00025180: 6475 6c65 4461 7461 2063 6f6b 6572 2046 duleData coker F │ │ │ │ -00025190: 2e64 645f 723b 2020 2020 2020 2020 2020 .dd_r; │ │ │ │ -000251a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00025160: 2d2d 2b0a 7c69 3131 203a 2045 7220 3d20 --+.|i11 : Er = │ │ │ │ +00025170: 4578 744d 6f64 756c 6544 6174 6120 636f ExtModuleData co │ │ │ │ +00025180: 6b65 7220 462e 6464 5f72 3b20 2020 2020 ker F.dd_r; │ │ │ │ +00025190: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000251a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000251b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000251c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000251d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000251e0: 3220 3a20 7265 6775 6c61 7269 7479 2045 2 : regularity E │ │ │ │ -000251f0: 725f 3020 2020 2020 2020 2020 2020 2020 r_0 │ │ │ │ -00025200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025210: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000251d0: 2b0a 7c69 3132 203a 2072 6567 756c 6172 +.|i12 : regular │ │ │ │ +000251e0: 6974 7920 4572 5f30 2020 2020 2020 2020 ity Er_0 │ │ │ │ +000251f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025200: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00025210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025240: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -00025250: 3d20 3020 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ +00025230: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00025240: 7c6f 3132 203d 2030 2020 2020 2020 2020 |o12 = 0 │ │ │ │ +00025250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025280: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00025270: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00025280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000252b0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 -------+.|i13 : │ │ │ │ -000252c0: 7265 6775 6c61 7269 7479 2045 725f 3120 regularity Er_1 │ │ │ │ +000252a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000252b0: 3133 203a 2072 6567 756c 6172 6974 7920 13 : regularity │ │ │ │ +000252c0: 4572 5f31 2020 2020 2020 2020 2020 2020 Er_1 │ │ │ │ 000252d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000252e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000252f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000252e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000252f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025320: 2020 2020 207c 0a7c 6f31 3320 3d20 3020 |.|o13 = 0 │ │ │ │ +00025310: 2020 2020 2020 2020 2020 7c0a 7c6f 3133 |.|o13 │ │ │ │ +00025320: 203d 2030 2020 2020 2020 2020 2020 2020 = 0 │ │ │ │ 00025330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025350: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00025350: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00025360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025390: 2d2d 2d2b 0a7c 6931 3420 3a20 7265 6775 ---+.|i14 : regu │ │ │ │ -000253a0: 6c61 7269 7479 2065 7665 6e45 7874 4d6f larity evenExtMo │ │ │ │ -000253b0: 6475 6c65 2863 6f6b 6572 2046 2e64 645f dule(coker F.dd_ │ │ │ │ -000253c0: 2872 2d31 2929 2020 2020 7c0a 7c20 2020 (r-1)) |.| │ │ │ │ +00025380: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ +00025390: 2072 6567 756c 6172 6974 7920 6576 656e regularity even │ │ │ │ +000253a0: 4578 744d 6f64 756c 6528 636f 6b65 7220 ExtModule(coker │ │ │ │ +000253b0: 462e 6464 5f28 722d 3129 2920 2020 207c F.dd_(r-1)) | │ │ │ │ +000253c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 000253d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000253e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000253f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025400: 207c 0a7c 6f31 3420 3d20 3120 2020 2020 |.|o14 = 1 │ │ │ │ +000253f0: 2020 2020 2020 7c0a 7c6f 3134 203d 2031 |.|o14 = 1 │ │ │ │ +00025400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00025410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00025420: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00025430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00025470: 0a7c 6931 3520 3a20 6666 203d 2066 2a72 .|i15 : ff = f*r │ │ │ │ -00025480: 616e 646f 6d28 736f 7572 6365 2066 2c20 andom(source f, │ │ │ │ -00025490: 736f 7572 6365 2066 293b 2020 2020 2020 source f); │ │ │ │ -000254a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00025460: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620 ----+.|i15 : ff │ │ │ │ +00025470: 3d20 662a 7261 6e64 6f6d 2873 6f75 7263 = f*random(sourc │ │ │ │ +00025480: 6520 662c 2073 6f75 7263 6520 6629 3b20 e f, source f); │ │ │ │ +00025490: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000254a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000254c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000254d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000254e0: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ -000254f0: 2020 2020 2034 2020 2020 2020 2020 2020 4 │ │ │ │ -00025500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025510: 2020 2020 7c0a 7c6f 3135 203a 204d 6174 |.|o15 : Mat │ │ │ │ -00025520: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +000254d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000254e0: 2020 2031 2020 2020 2020 3420 2020 2020 1 4 │ │ │ │ +000254f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00025500: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +00025510: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ +00025520: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00025530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00025540: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00025540: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00025550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025580: 2d2d 2b0a 7c69 3136 203a 206d 6174 7269 --+.|i16 : matri │ │ │ │ -00025590: 7846 6163 746f 7269 7a61 7469 6f6e 2866 xFactorization(f │ │ │ │ -000255a0: 662c 2063 6f6b 6572 2046 2e64 645f 2872 f, coker F.dd_(r │ │ │ │ -000255b0: 2b31 2929 3b20 2020 207c 0a2b 2d2d 2d2d +1)); |.+---- │ │ │ │ +00025570: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 3a20 -------+.|i16 : │ │ │ │ +00025580: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ +00025590: 696f 6e28 6666 2c20 636f 6b65 7220 462e ion(ff, coker F. │ │ │ │ +000255a0: 6464 5f28 722b 3129 293b 2020 2020 7c0a dd_(r+1)); |. │ │ │ │ +000255b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000255c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000255d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000255f0: 2b0a 0a54 6869 7320 7375 6363 6565 6473 +..This succeeds │ │ │ │ -00025600: 2c20 6275 7420 7765 2063 6f75 6c64 2067 , but we could g │ │ │ │ -00025610: 6574 2061 6e20 6572 726f 7220 6672 6f6d et an error from │ │ │ │ -00025620: 0a0a 6d61 7472 6978 4661 6374 6f72 697a ..matrixFactoriz │ │ │ │ -00025630: 6174 696f 6e28 6666 2c20 636f 6b65 7220 ation(ff, coker │ │ │ │ -00025640: 462e 6464 5f72 290a 0a69 6620 6f6e 6520 F.dd_r)..if one │ │ │ │ -00025650: 6f66 2074 6865 2043 4920 6f70 6572 6174 of the CI operat │ │ │ │ -00025660: 6f72 7320 7765 7265 206e 6f74 2073 7572 ors were not sur │ │ │ │ -00025670: 6a65 6374 6976 652e 0a0a 4361 7665 6174 jective...Caveat │ │ │ │ -00025680: 0a3d 3d3d 3d3d 3d0a 0a45 7874 4d6f 6475 .======..ExtModu │ │ │ │ -00025690: 6c65 2063 7265 6174 6573 2061 2072 696e le creates a rin │ │ │ │ -000256a0: 6720 696e 7369 6465 2074 6865 2073 6372 g inside the scr │ │ │ │ -000256b0: 6970 742c 2073 6f20 6966 2069 7427 7320 ipt, so if it's │ │ │ │ -000256c0: 7275 6e20 7477 6963 6520 796f 7520 6765 run twice you ge │ │ │ │ -000256d0: 740a 6d6f 6475 6c65 7320 6f76 6572 2064 t.modules over d │ │ │ │ -000256e0: 6966 6665 7265 6e74 2072 696e 6773 2e20 ifferent rings. │ │ │ │ -000256f0: 5468 6973 2073 686f 756c 6420 6265 2063 This should be c │ │ │ │ -00025700: 6861 6e67 6564 2e0a 0a53 6565 2061 6c73 hanged...See als │ │ │ │ -00025710: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -00025720: 2a6e 6f74 6520 4578 744d 6f64 756c 653a *note ExtModule: │ │ │ │ -00025730: 2045 7874 4d6f 6475 6c65 2c20 2d2d 2045 ExtModule, -- E │ │ │ │ -00025740: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -00025750: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00025760: 6563 7469 6f6e 2061 730a 2020 2020 6d6f ection as. mo │ │ │ │ -00025770: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -00025780: 7261 746f 7220 7269 6e67 0a20 202a 202a rator ring. * * │ │ │ │ -00025790: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ -000257a0: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ -000257b0: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ -000257c0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ -000257d0: 6572 2061 0a20 2020 2063 6f6d 706c 6574 er a. complet │ │ │ │ -000257e0: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ -000257f0: 7320 6d6f 6475 6c65 206f 7665 7220 4349 s module over CI │ │ │ │ -00025800: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ -00025810: 202a 202a 6e6f 7465 206f 6464 4578 744d * *note oddExtM │ │ │ │ -00025820: 6f64 756c 653a 206f 6464 4578 744d 6f64 odule: oddExtMod │ │ │ │ -00025830: 756c 652c 202d 2d20 6f64 6420 7061 7274 ule, -- odd part │ │ │ │ -00025840: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ -00025850: 7665 7220 6120 636f 6d70 6c65 7465 0a20 ver a complete. │ │ │ │ -00025860: 2020 2069 6e74 6572 7365 6374 696f 6e20 intersection │ │ │ │ -00025870: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ -00025880: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ -00025890: 0a57 6179 7320 746f 2075 7365 2045 7874 .Ways to use Ext │ │ │ │ -000258a0: 4d6f 6475 6c65 4461 7461 3a0a 3d3d 3d3d ModuleData:.==== │ │ │ │ -000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000258c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 7874 ======.. * "Ext │ │ │ │ -000258d0: 4d6f 6475 6c65 4461 7461 284d 6f64 756c ModuleData(Modul │ │ │ │ -000258e0: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ -000258f0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00025900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00025910: 6f62 6a65 6374 202a 6e6f 7465 2045 7874 object *note Ext │ │ │ │ -00025920: 4d6f 6475 6c65 4461 7461 3a20 4578 744d ModuleData: ExtM │ │ │ │ -00025930: 6f64 756c 6544 6174 612c 2069 7320 6120 oduleData, is a │ │ │ │ -00025940: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ -00025950: 6374 696f 6e3a 0a28 4d61 6361 756c 6179 ction:.(Macaulay │ │ │ │ -00025960: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -00025970: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +000255e0: 2d2d 2d2d 2d2b 0a0a 5468 6973 2073 7563 -----+..This suc │ │ │ │ +000255f0: 6365 6564 732c 2062 7574 2077 6520 636f ceeds, but we co │ │ │ │ +00025600: 756c 6420 6765 7420 616e 2065 7272 6f72 uld get an error │ │ │ │ +00025610: 2066 726f 6d0a 0a6d 6174 7269 7846 6163 from..matrixFac │ │ │ │ +00025620: 746f 7269 7a61 7469 6f6e 2866 662c 2063 torization(ff, c │ │ │ │ +00025630: 6f6b 6572 2046 2e64 645f 7229 0a0a 6966 oker F.dd_r)..if │ │ │ │ +00025640: 206f 6e65 206f 6620 7468 6520 4349 206f one of the CI o │ │ │ │ +00025650: 7065 7261 746f 7273 2077 6572 6520 6e6f perators were no │ │ │ │ +00025660: 7420 7375 726a 6563 7469 7665 2e0a 0a43 t surjective...C │ │ │ │ +00025670: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4578 aveat.======..Ex │ │ │ │ +00025680: 744d 6f64 756c 6520 6372 6561 7465 7320 tModule creates │ │ │ │ +00025690: 6120 7269 6e67 2069 6e73 6964 6520 7468 a ring inside th │ │ │ │ +000256a0: 6520 7363 7269 7074 2c20 736f 2069 6620 e script, so if │ │ │ │ +000256b0: 6974 2773 2072 756e 2074 7769 6365 2079 it's run twice y │ │ │ │ +000256c0: 6f75 2067 6574 0a6d 6f64 756c 6573 206f ou get.modules o │ │ │ │ +000256d0: 7665 7220 6469 6666 6572 656e 7420 7269 ver different ri │ │ │ │ +000256e0: 6e67 732e 2054 6869 7320 7368 6f75 6c64 ngs. This should │ │ │ │ +000256f0: 2062 6520 6368 616e 6765 642e 0a0a 5365 be changed...Se │ │ │ │ +00025700: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +00025710: 0a20 202a 202a 6e6f 7465 2045 7874 4d6f . * *note ExtMo │ │ │ │ +00025720: 6475 6c65 3a20 4578 744d 6f64 756c 652c dule: ExtModule, │ │ │ │ +00025730: 202d 2d20 4578 745e 2a28 4d2c 6b29 206f -- Ext^*(M,k) o │ │ │ │ +00025740: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +00025750: 6e74 6572 7365 6374 696f 6e20 6173 0a20 ntersection as. │ │ │ │ +00025760: 2020 206d 6f64 756c 6520 6f76 6572 2043 module over C │ │ │ │ +00025770: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +00025780: 2020 2a20 2a6e 6f74 6520 6576 656e 4578 * *note evenEx │ │ │ │ +00025790: 744d 6f64 756c 653a 2065 7665 6e45 7874 tModule: evenExt │ │ │ │ +000257a0: 4d6f 6475 6c65 2c20 2d2d 2065 7665 6e20 Module, -- even │ │ │ │ +000257b0: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ +000257c0: 6b29 206f 7665 7220 610a 2020 2020 636f k) over a. co │ │ │ │ +000257d0: 6d70 6c65 7465 2069 6e74 6572 7365 6374 mplete intersect │ │ │ │ +000257e0: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ +000257f0: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ +00025800: 696e 670a 2020 2a20 2a6e 6f74 6520 6f64 ing. * *note od │ │ │ │ +00025810: 6445 7874 4d6f 6475 6c65 3a20 6f64 6445 dExtModule: oddE │ │ │ │ +00025820: 7874 4d6f 6475 6c65 2c20 2d2d 206f 6464 xtModule, -- odd │ │ │ │ +00025830: 2070 6172 7420 6f66 2045 7874 5e2a 284d part of Ext^*(M │ │ │ │ +00025840: 2c6b 2920 6f76 6572 2061 2063 6f6d 706c ,k) over a compl │ │ │ │ +00025850: 6574 650a 2020 2020 696e 7465 7273 6563 ete. intersec │ │ │ │ +00025860: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f tion as module o │ │ │ │ +00025870: 7665 7220 4349 206f 7065 7261 746f 7220 ver CI operator │ │ │ │ +00025880: 7269 6e67 0a0a 5761 7973 2074 6f20 7573 ring..Ways to us │ │ │ │ +00025890: 6520 4578 744d 6f64 756c 6544 6174 613a e ExtModuleData: │ │ │ │ +000258a0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +000258b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000258c0: 2022 4578 744d 6f64 756c 6544 6174 6128 "ExtModuleData( │ │ │ │ +000258d0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ +000258e0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +000258f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +00025900: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +00025910: 6520 4578 744d 6f64 756c 6544 6174 613a e ExtModuleData: │ │ │ │ +00025920: 2045 7874 4d6f 6475 6c65 4461 7461 2c20 ExtModuleData, │ │ │ │ +00025930: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +00025940: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ +00025950: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +00025960: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00025970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000259a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000259c0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -000259d0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -000259e0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -000259f0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00025a00: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00025a10: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00025a20: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00025a30: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00025a40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00025a50: 323a 3334 3433 3a30 2e0a 1f0a 4669 6c65 2:3443:0....File │ │ │ │ -00025a60: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -00025a70: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00025a80: 732e 696e 666f 2c20 4e6f 6465 3a20 6578 s.info, Node: ex │ │ │ │ -00025a90: 7456 7343 6f68 6f6d 6f6c 6f67 792c 204e tVsCohomology, N │ │ │ │ -00025aa0: 6578 743a 2066 696e 6974 6542 6574 7469 ext: finiteBetti │ │ │ │ -00025ab0: 4e75 6d62 6572 732c 2050 7265 763a 2045 Numbers, Prev: E │ │ │ │ -00025ac0: 7874 4d6f 6475 6c65 4461 7461 2c20 5570 xtModuleData, Up │ │ │ │ -00025ad0: 3a20 546f 700a 0a65 7874 5673 436f 686f : Top..extVsCoho │ │ │ │ -00025ae0: 6d6f 6c6f 6779 202d 2d20 636f 6d70 6172 mology -- compar │ │ │ │ -00025af0: 6573 2045 7874 5f53 284d 2c6b 2920 6173 es Ext_S(M,k) as │ │ │ │ -00025b00: 2065 7874 6572 696f 7220 6d6f 6475 6c65 exterior module │ │ │ │ -00025b10: 2077 6974 6820 636f 6820 7461 626c 6520 with coh table │ │ │ │ -00025b20: 6f66 2073 6865 6166 2045 7874 5f52 284d of sheaf Ext_R(M │ │ │ │ -00025b30: 2c6b 290a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ,k).************ │ │ │ │ +000259b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +000259c0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +000259d0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +000259e0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +000259f0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00025a00: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +00025a10: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00025a20: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00025a30: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00025a40: 6f6e 732e 6d32 3a33 3434 333a 302e 0a1f ons.m2:3443:0... │ │ │ │ +00025a50: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00025a60: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +00025a70: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00025a80: 653a 2065 7874 5673 436f 686f 6d6f 6c6f e: extVsCohomolo │ │ │ │ +00025a90: 6779 2c20 4e65 7874 3a20 6669 6e69 7465 gy, Next: finite │ │ │ │ +00025aa0: 4265 7474 694e 756d 6265 7273 2c20 5072 BettiNumbers, Pr │ │ │ │ +00025ab0: 6576 3a20 4578 744d 6f64 756c 6544 6174 ev: ExtModuleDat │ │ │ │ +00025ac0: 612c 2055 703a 2054 6f70 0a0a 6578 7456 a, Up: Top..extV │ │ │ │ +00025ad0: 7343 6f68 6f6d 6f6c 6f67 7920 2d2d 2063 sCohomology -- c │ │ │ │ +00025ae0: 6f6d 7061 7265 7320 4578 745f 5328 4d2c ompares Ext_S(M, │ │ │ │ +00025af0: 6b29 2061 7320 6578 7465 7269 6f72 206d k) as exterior m │ │ │ │ +00025b00: 6f64 756c 6520 7769 7468 2063 6f68 2074 odule with coh t │ │ │ │ +00025b10: 6162 6c65 206f 6620 7368 6561 6620 4578 able of sheaf Ex │ │ │ │ +00025b20: 745f 5228 4d2c 6b29 0a2a 2a2a 2a2a 2a2a t_R(M,k).******* │ │ │ │ +00025b30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00025b70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00025b80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00025b90: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -00025ba0: 2020 2020 2020 2845 2c54 2920 3d20 6578 (E,T) = ex │ │ │ │ -00025bb0: 7456 7343 6f68 6f6d 6f6c 6f67 7928 6666 tVsCohomology(ff │ │ │ │ -00025bc0: 2c4e 290a 2020 2a20 496e 7075 7473 3a0a ,N). * Inputs:. │ │ │ │ -00025bd0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ -00025be0: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00025bf0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00025c00: 2c2c 2072 6567 756c 6172 2073 6571 7565 ,, regular seque │ │ │ │ -00025c10: 6e63 6520 696e 2061 0a20 2020 2020 2020 nce in a. │ │ │ │ -00025c20: 2072 6567 756c 6172 2072 696e 6720 530a regular ring S. │ │ │ │ -00025c30: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f * N, a *no │ │ │ │ -00025c40: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -00025c50: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -00025c60: 2c20 6772 6164 6564 206d 6f64 756c 6520 , graded module │ │ │ │ -00025c70: 6f76 6572 2052 203d 0a20 2020 2020 2020 over R =. │ │ │ │ -00025c80: 2053 2f69 6465 616c 2866 6629 2028 7573 S/ideal(ff) (us │ │ │ │ -00025c90: 7561 6c6c 7920 6120 6869 6768 2073 797a ually a high syz │ │ │ │ -00025ca0: 7967 7929 0a20 202a 204f 7574 7075 7473 ygy). * Outputs │ │ │ │ -00025cb0: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -00025cc0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00025cd0: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00025ce0: 652c 2c20 0a20 2020 2020 202a 2054 2c20 e,, . * T, │ │ │ │ -00025cf0: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ -00025d00: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ -00025d10: 6475 6c65 2c2c 2045 7874 2061 6e64 2054 dule,, Ext and T │ │ │ │ -00025d20: 6f72 2061 7320 6578 7465 7269 6f72 0a20 or as exterior. │ │ │ │ -00025d30: 2020 2020 2020 206d 6f64 756c 6573 0a0a modules.. │ │ │ │ -00025d40: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00025d50: 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e 2061 =======..Given a │ │ │ │ -00025d60: 206d 6174 7269 7820 6666 2063 6f6e 7461 matrix ff conta │ │ │ │ -00025d70: 696e 696e 6720 6120 7265 6775 6c61 7220 ining a regular │ │ │ │ -00025d80: 7365 7175 656e 6365 2069 6e20 6120 706f sequence in a po │ │ │ │ -00025d90: 6c79 6e6f 6d69 616c 2072 696e 6720 5320 lynomial ring S │ │ │ │ -00025da0: 6f76 6572 206b 2c0a 7365 7420 5220 3d20 over k,.set R = │ │ │ │ -00025db0: 532f 2869 6465 616c 2066 6629 2e20 4966 S/(ideal ff). If │ │ │ │ -00025dc0: 204e 2069 7320 6120 6772 6164 6564 2052 N is a graded R │ │ │ │ -00025dd0: 2d6d 6f64 756c 652c 2061 6e64 204d 2069 -module, and M i │ │ │ │ -00025de0: 7320 7468 6520 6d6f 6475 6c65 204e 2072 s the module N r │ │ │ │ -00025df0: 6567 6172 6465 640a 6173 2061 6e20 532d egarded.as an S- │ │ │ │ -00025e00: 6d6f 6475 6c65 2c20 7468 6520 7363 7269 module, the scri │ │ │ │ -00025e10: 7074 2072 6574 7572 6e73 2045 203d 2045 pt returns E = E │ │ │ │ -00025e20: 7874 5f53 284d 2c6b 2920 616e 6420 5420 xt_S(M,k) and T │ │ │ │ -00025e30: 3d20 546f 725e 5328 4d2c 6b29 2061 7320 = Tor^S(M,k) as │ │ │ │ -00025e40: 6d6f 6475 6c65 730a 6f76 6572 2061 6e20 modules.over an │ │ │ │ -00025e50: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ -00025e60: 2e0a 0a54 6865 2073 6372 6970 7420 7072 ...The script pr │ │ │ │ -00025e70: 696e 7473 2074 6865 2054 6174 6520 7265 ints the Tate re │ │ │ │ -00025e80: 736f 6c75 7469 6f6e 206f 6620 453b 2061 solution of E; a │ │ │ │ -00025e90: 6e64 2074 6865 2063 6f68 6f6d 6f6c 6f67 nd the cohomolog │ │ │ │ -00025ea0: 7920 7461 626c 6520 6f66 2074 6865 0a73 y table of the.s │ │ │ │ -00025eb0: 6865 6166 2061 7373 6f63 6961 7465 6420 heaf associated │ │ │ │ -00025ec0: 746f 2045 7874 5f52 284e 2c6b 2920 6f76 to Ext_R(N,k) ov │ │ │ │ -00025ed0: 6572 2074 6865 2072 696e 6720 6f66 2043 er the ring of C │ │ │ │ -00025ee0: 4920 6f70 6572 6174 6f72 732c 2077 6869 I operators, whi │ │ │ │ -00025ef0: 6368 2069 7320 610a 706f 6c79 6e6f 6d69 ch is a.polynomi │ │ │ │ -00025f00: 616c 2072 696e 6720 6f76 6572 206b 206f al ring over k o │ │ │ │ -00025f10: 6e20 6320 7661 7269 6162 6c65 732e 0a0a n c variables... │ │ │ │ -00025f20: 5468 6520 6f75 7470 7574 2063 616e 2062 The output can b │ │ │ │ -00025f30: 6520 7573 6564 2074 6f20 2873 6f6d 6574 e used to (somet │ │ │ │ -00025f40: 696d 6573 2920 6368 6563 6b20 7768 6574 imes) check whet │ │ │ │ -00025f50: 6865 7220 7468 6520 7375 626d 6f64 756c her the submodul │ │ │ │ -00025f60: 6520 6f66 2045 7874 5f53 284d 2c6b 290a e of Ext_S(M,k). │ │ │ │ -00025f70: 6765 6e65 7261 7465 6420 696e 2064 6567 generated in deg │ │ │ │ -00025f80: 7265 6520 3020 7370 6c69 7473 2028 6173 ree 0 splits (as │ │ │ │ -00025f90: 2061 6e20 6578 7465 7269 6f72 206d 6f64 an exterior mod │ │ │ │ -00025fa0: 756c 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d ule..+---------- │ │ │ │ +00025b80: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00025b90: 3a20 0a20 2020 2020 2020 2028 452c 5429 : . (E,T) │ │ │ │ +00025ba0: 203d 2065 7874 5673 436f 686f 6d6f 6c6f = extVsCohomolo │ │ │ │ +00025bb0: 6779 2866 662c 4e29 0a20 202a 2049 6e70 gy(ff,N). * Inp │ │ │ │ +00025bc0: 7574 733a 0a20 2020 2020 202a 2066 662c uts:. * ff, │ │ │ │ +00025bd0: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00025be0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00025bf0: 6174 7269 782c 2c20 7265 6775 6c61 7220 atrix,, regular │ │ │ │ +00025c00: 7365 7175 656e 6365 2069 6e20 610a 2020 sequence in a. │ │ │ │ +00025c10: 2020 2020 2020 7265 6775 6c61 7220 7269 regular ri │ │ │ │ +00025c20: 6e67 2053 0a20 2020 2020 202a 204e 2c20 ng S. * N, │ │ │ │ +00025c30: 6120 2a6e 6f74 6520 6d6f 6475 6c65 3a20 a *note module: │ │ │ │ +00025c40: 284d 6163 6175 6c61 7932 446f 6329 4d6f (Macaulay2Doc)Mo │ │ │ │ +00025c50: 6475 6c65 2c2c 2067 7261 6465 6420 6d6f dule,, graded mo │ │ │ │ +00025c60: 6475 6c65 206f 7665 7220 5220 3d0a 2020 dule over R =. │ │ │ │ +00025c70: 2020 2020 2020 532f 6964 6561 6c28 6666 S/ideal(ff │ │ │ │ +00025c80: 2920 2875 7375 616c 6c79 2061 2068 6967 ) (usually a hig │ │ │ │ +00025c90: 6820 7379 7a79 6779 290a 2020 2a20 4f75 h syzygy). * Ou │ │ │ │ +00025ca0: 7470 7574 733a 0a20 2020 2020 202a 2045 tputs:. * E │ │ │ │ +00025cb0: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00025cc0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00025cd0: 4d6f 6475 6c65 2c2c 200a 2020 2020 2020 Module,, . │ │ │ │ +00025ce0: 2a20 542c 2061 202a 6e6f 7465 206d 6f64 * T, a *note mod │ │ │ │ +00025cf0: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ +00025d00: 6f63 294d 6f64 756c 652c 2c20 4578 7420 oc)Module,, Ext │ │ │ │ +00025d10: 616e 6420 546f 7220 6173 2065 7874 6572 and Tor as exter │ │ │ │ +00025d20: 696f 720a 2020 2020 2020 2020 6d6f 6475 ior. modu │ │ │ │ +00025d30: 6c65 730a 0a44 6573 6372 6970 7469 6f6e les..Description │ │ │ │ +00025d40: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4769 .===========..Gi │ │ │ │ +00025d50: 7665 6e20 6120 6d61 7472 6978 2066 6620 ven a matrix ff │ │ │ │ +00025d60: 636f 6e74 6169 6e69 6e67 2061 2072 6567 containing a reg │ │ │ │ +00025d70: 756c 6172 2073 6571 7565 6e63 6520 696e ular sequence in │ │ │ │ +00025d80: 2061 2070 6f6c 796e 6f6d 6961 6c20 7269 a polynomial ri │ │ │ │ +00025d90: 6e67 2053 206f 7665 7220 6b2c 0a73 6574 ng S over k,.set │ │ │ │ +00025da0: 2052 203d 2053 2f28 6964 6561 6c20 6666 R = S/(ideal ff │ │ │ │ +00025db0: 292e 2049 6620 4e20 6973 2061 2067 7261 ). If N is a gra │ │ │ │ +00025dc0: 6465 6420 522d 6d6f 6475 6c65 2c20 616e ded R-module, an │ │ │ │ +00025dd0: 6420 4d20 6973 2074 6865 206d 6f64 756c d M is the modul │ │ │ │ +00025de0: 6520 4e20 7265 6761 7264 6564 0a61 7320 e N regarded.as │ │ │ │ +00025df0: 616e 2053 2d6d 6f64 756c 652c 2074 6865 an S-module, the │ │ │ │ +00025e00: 2073 6372 6970 7420 7265 7475 726e 7320 script returns │ │ │ │ +00025e10: 4520 3d20 4578 745f 5328 4d2c 6b29 2061 E = Ext_S(M,k) a │ │ │ │ +00025e20: 6e64 2054 203d 2054 6f72 5e53 284d 2c6b nd T = Tor^S(M,k │ │ │ │ +00025e30: 2920 6173 206d 6f64 756c 6573 0a6f 7665 ) as modules.ove │ │ │ │ +00025e40: 7220 616e 2065 7874 6572 696f 7220 616c r an exterior al │ │ │ │ +00025e50: 6765 6272 612e 0a0a 5468 6520 7363 7269 gebra...The scri │ │ │ │ +00025e60: 7074 2070 7269 6e74 7320 7468 6520 5461 pt prints the Ta │ │ │ │ +00025e70: 7465 2072 6573 6f6c 7574 696f 6e20 6f66 te resolution of │ │ │ │ +00025e80: 2045 3b20 616e 6420 7468 6520 636f 686f E; and the coho │ │ │ │ +00025e90: 6d6f 6c6f 6779 2074 6162 6c65 206f 6620 mology table of │ │ │ │ +00025ea0: 7468 650a 7368 6561 6620 6173 736f 6369 the.sheaf associ │ │ │ │ +00025eb0: 6174 6564 2074 6f20 4578 745f 5228 4e2c ated to Ext_R(N, │ │ │ │ +00025ec0: 6b29 206f 7665 7220 7468 6520 7269 6e67 k) over the ring │ │ │ │ +00025ed0: 206f 6620 4349 206f 7065 7261 746f 7273 of CI operators │ │ │ │ +00025ee0: 2c20 7768 6963 6820 6973 2061 0a70 6f6c , which is a.pol │ │ │ │ +00025ef0: 796e 6f6d 6961 6c20 7269 6e67 206f 7665 ynomial ring ove │ │ │ │ +00025f00: 7220 6b20 6f6e 2063 2076 6172 6961 626c r k on c variabl │ │ │ │ +00025f10: 6573 2e0a 0a54 6865 206f 7574 7075 7420 es...The output │ │ │ │ +00025f20: 6361 6e20 6265 2075 7365 6420 746f 2028 can be used to ( │ │ │ │ +00025f30: 736f 6d65 7469 6d65 7329 2063 6865 636b sometimes) check │ │ │ │ +00025f40: 2077 6865 7468 6572 2074 6865 2073 7562 whether the sub │ │ │ │ +00025f50: 6d6f 6475 6c65 206f 6620 4578 745f 5328 module of Ext_S( │ │ │ │ +00025f60: 4d2c 6b29 0a67 656e 6572 6174 6564 2069 M,k).generated i │ │ │ │ +00025f70: 6e20 6465 6772 6565 2030 2073 706c 6974 n degree 0 split │ │ │ │ +00025f80: 7320 2861 7320 616e 2065 7874 6572 696f s (as an exterio │ │ │ │ +00025f90: 7220 6d6f 6475 6c65 0a0a 2b2d 2d2d 2d2d r module..+----- │ │ │ │ +00025fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00025fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00025fe0: 2d2d 2d2b 0a7c 6931 203a 2053 203d 205a ---+.|i1 : S = Z │ │ │ │ -00025ff0: 5a2f 3130 315b 612c 622c 635d 2020 2020 Z/101[a,b,c] │ │ │ │ +00025fd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00025fe0: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63 S = ZZ/101[a,b,c │ │ │ │ +00025ff0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 00026000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026010: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026060: 2020 207c 0a7c 6f31 203d 2053 2020 2020 |.|o1 = S │ │ │ │ +00026050: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ +00026060: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00026070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026090: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000260a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000260b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000260c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000260e0: 2020 207c 0a7c 6f31 203a 2050 6f6c 796e |.|o1 : Polyn │ │ │ │ -000260f0: 6f6d 6961 6c52 696e 6720 2020 2020 2020 omialRing │ │ │ │ +000260d0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +000260e0: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +000260f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026120: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026110: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00026120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026160: 2d2d 2d2b 0a7c 6932 203a 2066 6620 3d20 ---+.|i2 : ff = │ │ │ │ -00026170: 6d61 7472 6978 2022 6132 2c62 322c 6332 matrix "a2,b2,c2 │ │ │ │ -00026180: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -00026190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026150: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ +00026160: 6666 203d 206d 6174 7269 7820 2261 322c ff = matrix "a2, │ │ │ │ +00026170: 6232 2c63 3222 2020 2020 2020 2020 2020 b2,c2" │ │ │ │ +00026180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026190: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000261a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000261b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000261c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000261e0: 2020 207c 0a7c 6f32 203d 207c 2061 3220 |.|o2 = | a2 │ │ │ │ -000261f0: 6232 2063 3220 7c20 2020 2020 2020 2020 b2 c2 | │ │ │ │ +000261d0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +000261e0: 7c20 6132 2062 3220 6332 207c 2020 2020 | a2 b2 c2 | │ │ │ │ +000261f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026220: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026210: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026260: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026270: 2020 2031 2020 2020 2020 3320 2020 2020 1 3 │ │ │ │ +00026250: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026260: 2020 2020 2020 2020 3120 2020 2020 2033 1 3 │ │ │ │ +00026270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262a0: 2020 207c 0a7c 6f32 203a 204d 6174 7269 |.|o2 : Matri │ │ │ │ -000262b0: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +00026290: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +000262a0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320 Matrix S <-- S │ │ │ │ +000262b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000262c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000262e0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +000262d0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000262e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000262f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026320: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2053 ---+.|i3 : R = S │ │ │ │ -00026330: 2f28 6964 6561 6c20 6666 2920 2020 2020 /(ideal ff) │ │ │ │ +00026310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00026320: 5220 3d20 532f 2869 6465 616c 2066 6629 R = S/(ideal ff) │ │ │ │ +00026330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026360: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026350: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263a0: 2020 207c 0a7c 6f33 203d 2052 2020 2020 |.|o3 = R │ │ │ │ +00026390: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +000263a0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 000263b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000263c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000263e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000263d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000263e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000263f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026420: 2020 207c 0a7c 6f33 203a 2051 756f 7469 |.|o3 : Quoti │ │ │ │ -00026430: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +00026410: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ +00026420: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +00026430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026460: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026450: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00026460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000264a0: 2d2d 2d2b 0a7c 6934 203a 204e 203d 2068 ---+.|i4 : N = h │ │ │ │ -000264b0: 6967 6853 797a 7967 7928 525e 312f 6964 ighSyzygy(R^1/id │ │ │ │ -000264c0: 6561 6c28 612a 622c 6329 2920 2020 2020 eal(a*b,c)) │ │ │ │ -000264d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000264e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00026490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ +000264a0: 4e20 3d20 6869 6768 5379 7a79 6779 2852 N = highSyzygy(R │ │ │ │ +000264b0: 5e31 2f69 6465 616c 2861 2a62 2c63 2929 ^1/ideal(a*b,c)) │ │ │ │ +000264c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000264d0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000264e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000264f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026520: 2020 207c 0a7c 6f34 203d 2063 6f6b 6572 |.|o4 = coker │ │ │ │ -00026530: 6e65 6c20 7b34 7d20 7c20 6320 2d61 6220 nel {4} | c -ab │ │ │ │ -00026540: 3020 3020 3020 2030 2020 3020 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00026550: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -00026560: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026570: 2020 2020 7b35 7d20 7c20 3020 6320 2020 {5} | 0 c │ │ │ │ -00026580: 6220 6120 3020 2030 2020 3020 2030 2030 b a 0 0 0 0 0 │ │ │ │ -00026590: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -000265a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000265b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -000265c0: 6320 3020 2d62 2061 2020 3020 2030 2030 c 0 -b a 0 0 0 │ │ │ │ -000265d0: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -000265e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000265f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026600: 3020 6320 3020 202d 6220 2d61 2030 2030 0 c 0 -b -a 0 0 │ │ │ │ -00026610: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -00026620: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026630: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026640: 3020 3020 6320 2030 2020 3020 2062 2061 0 0 c 0 0 b a │ │ │ │ -00026650: 2030 2030 2020 3020 3020 2030 2020 3020 0 0 0 0 0 0 │ │ │ │ -00026660: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026670: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026680: 3020 3020 3020 2063 2020 3020 2030 2062 0 0 0 c 0 0 b │ │ │ │ -00026690: 2030 2030 2020 3020 2d61 2030 2020 3020 0 0 0 -a 0 0 │ │ │ │ -000266a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000266b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -000266c0: 3020 3020 3020 2030 2020 6320 2030 2030 0 0 0 0 c 0 0 │ │ │ │ -000266d0: 2030 2030 2020 3020 6220 2030 2020 6120 0 0 0 b 0 a │ │ │ │ -000266e0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000266f0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026700: 3020 3020 3020 2030 2020 3020 2063 2030 0 0 0 0 0 c 0 │ │ │ │ -00026710: 2062 202d 6120 3020 3020 2030 2020 3020 b -a 0 0 0 0 │ │ │ │ -00026720: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026730: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026740: 3020 3020 3020 2030 2020 3020 2030 2063 0 0 0 0 0 0 c │ │ │ │ -00026750: 2030 2062 2020 6120 3020 2030 2020 3020 0 b a 0 0 0 │ │ │ │ -00026760: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -00026770: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -00026780: 3020 3020 3020 2030 2020 3020 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00026790: 2030 2030 2020 6220 6320 202d 6120 3020 0 0 b c -a 0 │ │ │ │ -000267a0: 3020 7c7c 0a7c 2020 2020 2020 2020 2020 0 ||.| │ │ │ │ -000267b0: 2020 2020 7b35 7d20 7c20 3020 3020 2020 {5} | 0 0 │ │ │ │ -000267c0: 3020 3020 3020 2030 2020 3020 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -000267d0: 2030 2030 2020 3020 3020 2062 2020 6320 0 0 0 0 b c │ │ │ │ -000267e0: 6120 7c7c 0a7c 2020 2020 2020 2020 2020 a ||.| │ │ │ │ +00026510: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ +00026520: 636f 6b65 726e 656c 207b 347d 207c 2063 cokernel {4} | c │ │ │ │ +00026530: 202d 6162 2030 2030 2030 2020 3020 2030 -ab 0 0 0 0 0 │ │ │ │ +00026540: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +00026550: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026560: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026570: 2063 2020 2062 2061 2030 2020 3020 2030 c b a 0 0 0 │ │ │ │ +00026580: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +00026590: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +000265a0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000265b0: 2030 2020 2063 2030 202d 6220 6120 2030 0 c 0 -b a 0 │ │ │ │ +000265c0: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +000265d0: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +000265e0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000265f0: 2030 2020 2030 2063 2030 2020 2d62 202d 0 0 c 0 -b - │ │ │ │ +00026600: 6120 3020 3020 3020 3020 2030 2030 2020 a 0 0 0 0 0 0 │ │ │ │ +00026610: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026620: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026630: 2030 2020 2030 2030 2063 2020 3020 2030 0 0 0 c 0 0 │ │ │ │ +00026640: 2020 6220 6120 3020 3020 2030 2030 2020 b a 0 0 0 0 │ │ │ │ +00026650: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026660: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026670: 2030 2020 2030 2030 2030 2020 6320 2030 0 0 0 0 c 0 │ │ │ │ +00026680: 2020 3020 6220 3020 3020 2030 202d 6120 0 b 0 0 0 -a │ │ │ │ +00026690: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +000266a0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000266b0: 2030 2020 2030 2030 2030 2020 3020 2063 0 0 0 0 0 c │ │ │ │ +000266c0: 2020 3020 3020 3020 3020 2030 2062 2020 0 0 0 0 0 b │ │ │ │ +000266d0: 3020 2061 2030 207c 7c0a 7c20 2020 2020 0 a 0 ||.| │ │ │ │ +000266e0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000266f0: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +00026700: 2020 6320 3020 6220 2d61 2030 2030 2020 c 0 b -a 0 0 │ │ │ │ +00026710: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026720: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026730: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +00026740: 2020 3020 6320 3020 6220 2061 2030 2020 0 c 0 b a 0 │ │ │ │ +00026750: 3020 2030 2030 207c 7c0a 7c20 2020 2020 0 0 0 ||.| │ │ │ │ +00026760: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +00026770: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +00026780: 2020 3020 3020 3020 3020 2062 2063 2020 0 0 0 0 b c │ │ │ │ +00026790: 2d61 2030 2030 207c 7c0a 7c20 2020 2020 -a 0 0 ||.| │ │ │ │ +000267a0: 2020 2020 2020 2020 207b 357d 207c 2030 {5} | 0 │ │ │ │ +000267b0: 2030 2020 2030 2030 2030 2020 3020 2030 0 0 0 0 0 0 │ │ │ │ +000267c0: 2020 3020 3020 3020 3020 2030 2030 2020 0 0 0 0 0 0 │ │ │ │ +000267d0: 6220 2063 2061 207c 7c0a 7c20 2020 2020 b c a ||.| │ │ │ │ +000267e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000267f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026820: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00026830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026840: 2020 3131 2020 2020 2020 2020 2020 2020 11 │ │ │ │ -00026850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026860: 2020 207c 0a7c 6f34 203a 2052 2d6d 6f64 |.|o4 : R-mod │ │ │ │ -00026870: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ -00026880: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ -00026890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000268a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026810: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00026820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026830: 2020 2020 2020 2031 3120 2020 2020 2020 11 │ │ │ │ +00026840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026850: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +00026860: 522d 6d6f 6475 6c65 2c20 7175 6f74 6965 R-module, quotie │ │ │ │ +00026870: 6e74 206f 6620 5220 2020 2020 2020 2020 nt of R │ │ │ │ +00026880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026890: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000268a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000268c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000268e0: 2d2d 2d2b 0a7c 6935 203a 2045 203d 2065 ---+.|i5 : E = e │ │ │ │ -000268f0: 7874 5673 436f 686f 6d6f 6c6f 6779 2866 xtVsCohomology(f │ │ │ │ -00026900: 662c 6869 6768 5379 7a79 6779 204e 293b f,highSyzygy N); │ │ │ │ -00026910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026920: 2020 207c 0a7c 5461 7465 2052 6573 6f6c |.|Tate Resol │ │ │ │ -00026930: 7574 696f 6e20 6f66 2045 7874 5f53 284d ution of Ext_S(M │ │ │ │ -00026940: 2c6b 2920 6173 2065 7874 6572 696f 7220 ,k) as exterior │ │ │ │ -00026950: 6d6f 6475 6c65 3a20 2020 2020 2020 2020 module: │ │ │ │ -00026960: 2020 207c 0a7c 4e6f 7465 2074 6861 7420 |.|Note that │ │ │ │ -00026970: 6d61 7073 2067 6f20 6c65 6674 2074 6f20 maps go left to │ │ │ │ -00026980: 7269 6768 7420 2020 2020 2020 2020 2020 right │ │ │ │ -00026990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000269a0: 2020 207c 0a7c 2020 2020 2020 202d 3131 |.| -11 │ │ │ │ -000269b0: 202d 3130 2020 2d39 202d 3820 2d37 202d -10 -9 -8 -7 - │ │ │ │ -000269c0: 3620 2d35 202d 3420 2d33 202d 3220 202d 6 -5 -4 -3 -2 - │ │ │ │ -000269d0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -000269e0: 2020 207c 0a7c 746f 7461 6c3a 2031 3938 |.|total: 198 │ │ │ │ -000269f0: 2031 3436 2031 3032 2036 3620 3338 2031 146 102 66 38 1 │ │ │ │ -00026a00: 3820 2039 2031 3620 3336 2036 3420 3130 8 9 16 36 64 10 │ │ │ │ -00026a10: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ -00026a20: 2020 207c 0a7c 2020 2020 383a 2031 3036 |.| 8: 106 │ │ │ │ -00026a30: 2020 3739 2020 3536 2033 3720 3232 2031 79 56 37 22 1 │ │ │ │ -00026a40: 3120 2034 2020 3120 2031 2020 3120 2020 1 4 1 1 1 │ │ │ │ -00026a50: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00026a60: 2020 207c 0a7c 2020 2020 393a 2020 3932 |.| 9: 92 │ │ │ │ -00026a70: 2020 3637 2020 3436 2032 3920 3136 2020 67 46 29 16 │ │ │ │ -00026a80: 3720 2032 2020 2e20 202e 2020 2e20 2020 7 2 . . . │ │ │ │ -00026a90: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ -00026aa0: 2020 207c 0a7c 2020 2031 303a 2020 202e |.| 10: . │ │ │ │ -00026ab0: 2020 202e 2020 202e 2020 2e20 202e 2020 . . . . │ │ │ │ -00026ac0: 2e20 202e 2020 3520 3134 2032 3720 2034 . . 5 14 27 4 │ │ │ │ -00026ad0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00026ae0: 2020 207c 0a7c 2020 2031 313a 2020 202e |.| 11: . │ │ │ │ -00026af0: 2020 202e 2020 202e 2020 2e20 202e 2020 . . . . │ │ │ │ -00026b00: 2e20 2033 2031 3020 3231 2033 3620 2035 . 3 10 21 36 5 │ │ │ │ -00026b10: 3520 2020 2020 2020 2020 2020 2020 2020 5 │ │ │ │ -00026b20: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020 |.|--- │ │ │ │ +000268d0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 --------+.|i5 : │ │ │ │ +000268e0: 4520 3d20 6578 7456 7343 6f68 6f6d 6f6c E = extVsCohomol │ │ │ │ +000268f0: 6f67 7928 6666 2c68 6967 6853 797a 7967 ogy(ff,highSyzyg │ │ │ │ +00026900: 7920 4e29 3b20 2020 2020 2020 2020 2020 y N); │ │ │ │ +00026910: 2020 2020 2020 2020 7c0a 7c54 6174 6520 |.|Tate │ │ │ │ +00026920: 5265 736f 6c75 7469 6f6e 206f 6620 4578 Resolution of Ex │ │ │ │ +00026930: 745f 5328 4d2c 6b29 2061 7320 6578 7465 t_S(M,k) as exte │ │ │ │ +00026940: 7269 6f72 206d 6f64 756c 653a 2020 2020 rior module: │ │ │ │ +00026950: 2020 2020 2020 2020 7c0a 7c4e 6f74 6520 |.|Note │ │ │ │ +00026960: 7468 6174 206d 6170 7320 676f 206c 6566 that maps go lef │ │ │ │ +00026970: 7420 746f 2072 6967 6874 2020 2020 2020 t to right │ │ │ │ +00026980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026990: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000269a0: 2020 2d31 3120 2d31 3020 202d 3920 2d38 -11 -10 -9 -8 │ │ │ │ +000269b0: 202d 3720 2d36 202d 3520 2d34 202d 3320 -7 -6 -5 -4 -3 │ │ │ │ +000269c0: 2d32 2020 2d31 2020 2020 2020 2020 2020 -2 -1 │ │ │ │ +000269d0: 2020 2020 2020 2020 7c0a 7c74 6f74 616c |.|total │ │ │ │ +000269e0: 3a20 3139 3820 3134 3620 3130 3220 3636 : 198 146 102 66 │ │ │ │ +000269f0: 2033 3820 3138 2020 3920 3136 2033 3620 38 18 9 16 36 │ │ │ │ +00026a00: 3634 2031 3030 2020 2020 2020 2020 2020 64 100 │ │ │ │ +00026a10: 2020 2020 2020 2020 7c0a 7c20 2020 2038 |.| 8 │ │ │ │ +00026a20: 3a20 3130 3620 2037 3920 2035 3620 3337 : 106 79 56 37 │ │ │ │ +00026a30: 2032 3220 3131 2020 3420 2031 2020 3120 22 11 4 1 1 │ │ │ │ +00026a40: 2031 2020 2031 2020 2020 2020 2020 2020 1 1 │ │ │ │ +00026a50: 2020 2020 2020 2020 7c0a 7c20 2020 2039 |.| 9 │ │ │ │ +00026a60: 3a20 2039 3220 2036 3720 2034 3620 3239 : 92 67 46 29 │ │ │ │ +00026a70: 2031 3620 2037 2020 3220 202e 2020 2e20 16 7 2 . . │ │ │ │ +00026a80: 202e 2020 202e 2020 2020 2020 2020 2020 . . │ │ │ │ +00026a90: 2020 2020 2020 2020 7c0a 7c20 2020 3130 |.| 10 │ │ │ │ +00026aa0: 3a20 2020 2e20 2020 2e20 2020 2e20 202e : . . . . │ │ │ │ +00026ab0: 2020 2e20 202e 2020 2e20 2035 2031 3420 . . . 5 14 │ │ │ │ +00026ac0: 3237 2020 3434 2020 2020 2020 2020 2020 27 44 │ │ │ │ +00026ad0: 2020 2020 2020 2020 7c0a 7c20 2020 3131 |.| 11 │ │ │ │ +00026ae0: 3a20 2020 2e20 2020 2e20 2020 2e20 202e : . . . . │ │ │ │ +00026af0: 2020 2e20 202e 2020 3320 3130 2032 3120 . . 3 10 21 │ │ │ │ +00026b00: 3336 2020 3535 2020 2020 2020 2020 2020 36 55 │ │ │ │ +00026b10: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2020 |.|--- │ │ │ │ +00026b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026b60: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779 |.|Cohomology │ │ │ │ -00026b70: 2074 6162 6c65 206f 6620 6576 656e 4578 table of evenEx │ │ │ │ -00026b80: 744d 6f64 756c 6520 4d3a 2020 2020 2020 tModule M: │ │ │ │ -00026b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ba0: 2020 207c 0a7c 2020 202d 3520 2d34 202d |.| -5 -4 - │ │ │ │ -00026bb0: 3320 2d32 202d 3120 2030 2020 3120 2032 3 -2 -1 0 1 2 │ │ │ │ -00026bc0: 2020 3320 2034 2020 2035 2020 2020 2020 3 4 5 │ │ │ │ -00026bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026be0: 2020 207c 0a7c 323a 2033 3620 3231 2031 |.|2: 36 21 1 │ │ │ │ -00026bf0: 3020 2033 2020 2e20 202e 2020 2e20 202e 0 3 . . . . │ │ │ │ -00026c00: 2020 2e20 202e 2020 202e 2020 2020 2020 . . . │ │ │ │ -00026c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c20: 2020 207c 0a7c 313a 2020 2e20 202e 2020 |.|1: . . │ │ │ │ -00026c30: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -00026c40: 2020 2e20 202e 2020 202e 2020 2020 2020 . . . │ │ │ │ -00026c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026c60: 2020 207c 0a7c 303a 2020 3120 2031 2020 |.|0: 1 1 │ │ │ │ -00026c70: 3120 2032 2020 3720 3136 2032 3920 3436 1 2 7 16 29 46 │ │ │ │ -00026c80: 2036 3720 3932 2031 3231 2020 2020 2020 67 92 121 │ │ │ │ -00026c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ca0: 2020 207c 0a7c 2d2d 2d20 2020 2020 2020 |.|--- │ │ │ │ +00026b50: 2020 2020 2020 2020 7c0a 7c43 6f68 6f6d |.|Cohom │ │ │ │ +00026b60: 6f6c 6f67 7920 7461 626c 6520 6f66 2065 ology table of e │ │ │ │ +00026b70: 7665 6e45 7874 4d6f 6475 6c65 204d 3a20 venExtModule M: │ │ │ │ +00026b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026b90: 2020 2020 2020 2020 7c0a 7c20 2020 2d35 |.| -5 │ │ │ │ +00026ba0: 202d 3420 2d33 202d 3220 2d31 2020 3020 -4 -3 -2 -1 0 │ │ │ │ +00026bb0: 2031 2020 3220 2033 2020 3420 2020 3520 1 2 3 4 5 │ │ │ │ +00026bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026bd0: 2020 2020 2020 2020 7c0a 7c32 3a20 3336 |.|2: 36 │ │ │ │ +00026be0: 2032 3120 3130 2020 3320 202e 2020 2e20 21 10 3 . . │ │ │ │ +00026bf0: 202e 2020 2e20 202e 2020 2e20 2020 2e20 . . . . . │ │ │ │ +00026c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c10: 2020 2020 2020 2020 7c0a 7c31 3a20 202e |.|1: . │ │ │ │ +00026c20: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +00026c30: 202e 2020 2e20 202e 2020 2e20 2020 2e20 . . . . . │ │ │ │ +00026c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c50: 2020 2020 2020 2020 7c0a 7c30 3a20 2031 |.|0: 1 │ │ │ │ +00026c60: 2020 3120 2031 2020 3220 2037 2031 3620 1 1 2 7 16 │ │ │ │ +00026c70: 3239 2034 3620 3637 2039 3220 3132 3120 29 46 67 92 121 │ │ │ │ +00026c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026c90: 2020 2020 2020 2020 7c0a 7c2d 2d2d 2020 |.|--- │ │ │ │ +00026ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00026cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026ce0: 2020 207c 0a7c 436f 686f 6d6f 6c6f 6779 |.|Cohomology │ │ │ │ -00026cf0: 2074 6162 6c65 206f 6620 6f64 6445 7874 table of oddExt │ │ │ │ -00026d00: 4d6f 6475 6c65 204d 3a20 2020 2020 2020 Module M: │ │ │ │ -00026d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d20: 2020 207c 0a7c 2020 202d 3520 2d34 202d |.| -5 -4 - │ │ │ │ -00026d30: 3320 2d32 202d 3120 2030 2020 3120 2032 3 -2 -1 0 1 2 │ │ │ │ -00026d40: 2020 3320 2020 3420 2020 3520 2020 2020 3 4 5 │ │ │ │ -00026d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026d60: 2020 207c 0a7c 323a 2032 3820 3135 2020 |.|2: 28 15 │ │ │ │ -00026d70: 3620 2031 2020 2e20 202e 2020 2e20 202e 6 1 . . . . │ │ │ │ -00026d80: 2020 2e20 2020 2e20 2020 2e20 2020 2020 . . . │ │ │ │ -00026d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026da0: 2020 207c 0a7c 313a 2020 2e20 202e 2020 |.|1: . . │ │ │ │ -00026db0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -00026dc0: 2020 2e20 2020 2e20 2020 2e20 2020 2020 . . . │ │ │ │ -00026dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026de0: 2020 207c 0a7c 303a 2020 3120 2031 2020 |.|0: 1 1 │ │ │ │ -00026df0: 3120 2034 2031 3120 3232 2033 3720 3536 1 4 11 22 37 56 │ │ │ │ -00026e00: 2037 3920 3130 3620 3133 3720 2020 2020 79 106 137 │ │ │ │ -00026e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00026e20: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00026cd0: 2020 2020 2020 2020 7c0a 7c43 6f68 6f6d |.|Cohom │ │ │ │ +00026ce0: 6f6c 6f67 7920 7461 626c 6520 6f66 206f ology table of o │ │ │ │ +00026cf0: 6464 4578 744d 6f64 756c 6520 4d3a 2020 ddExtModule M: │ │ │ │ +00026d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d10: 2020 2020 2020 2020 7c0a 7c20 2020 2d35 |.| -5 │ │ │ │ +00026d20: 202d 3420 2d33 202d 3220 2d31 2020 3020 -4 -3 -2 -1 0 │ │ │ │ +00026d30: 2031 2020 3220 2033 2020 2034 2020 2035 1 2 3 4 5 │ │ │ │ +00026d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d50: 2020 2020 2020 2020 7c0a 7c32 3a20 3238 |.|2: 28 │ │ │ │ +00026d60: 2031 3520 2036 2020 3120 202e 2020 2e20 15 6 1 . . │ │ │ │ +00026d70: 202e 2020 2e20 202e 2020 202e 2020 202e . . . . . │ │ │ │ +00026d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026d90: 2020 2020 2020 2020 7c0a 7c31 3a20 202e |.|1: . │ │ │ │ +00026da0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +00026db0: 202e 2020 2e20 202e 2020 202e 2020 202e . . . . . │ │ │ │ +00026dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026dd0: 2020 2020 2020 2020 7c0a 7c30 3a20 2031 |.|0: 1 │ │ │ │ +00026de0: 2020 3120 2031 2020 3420 3131 2032 3220 1 1 4 11 22 │ │ │ │ +00026df0: 3337 2035 3620 3739 2031 3036 2031 3337 37 56 79 106 137 │ │ │ │ +00026e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00026e10: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00026e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00026e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00026e60: 2d2d 2d2b 0a0a 5365 6520 616c 736f 0a3d ---+..See also.= │ │ │ │ -00026e70: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -00026e80: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ -00026e90: 6967 6853 797a 7967 792c 202d 2d20 5265 ighSyzygy, -- Re │ │ │ │ -00026ea0: 7475 726e 7320 6120 7379 7a79 6779 206d turns a syzygy m │ │ │ │ -00026eb0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64 odule one beyond │ │ │ │ -00026ec0: 2074 6865 0a20 2020 2072 6567 756c 6172 the. regular │ │ │ │ -00026ed0: 6974 7920 6f66 2045 7874 284d 2c6b 290a ity of Ext(M,k). │ │ │ │ -00026ee0: 2020 2a20 2a6e 6f74 6520 6578 7465 7269 * *note exteri │ │ │ │ -00026ef0: 6f72 4578 744d 6f64 756c 653a 2065 7874 orExtModule: ext │ │ │ │ -00026f00: 6572 696f 7245 7874 4d6f 6475 6c65 2c20 eriorExtModule, │ │ │ │ -00026f10: 2d2d 2045 7874 284d 2c6b 2920 6f72 2045 -- Ext(M,k) or E │ │ │ │ -00026f20: 7874 284d 2c4e 2920 6173 2061 0a20 2020 xt(M,N) as a. │ │ │ │ -00026f30: 206d 6f64 756c 6520 6f76 6572 2061 6e20 module over an │ │ │ │ -00026f40: 6578 7465 7269 6f72 2061 6c67 6562 7261 exterior algebra │ │ │ │ -00026f50: 0a0a 5761 7973 2074 6f20 7573 6520 6578 ..Ways to use ex │ │ │ │ -00026f60: 7456 7343 6f68 6f6d 6f6c 6f67 793a 0a3d tVsCohomology:.= │ │ │ │ +00026e50: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ +00026e60: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ +00026e70: 2a20 2a6e 6f74 6520 6869 6768 5379 7a79 * *note highSyzy │ │ │ │ +00026e80: 6779 3a20 6869 6768 5379 7a79 6779 2c20 gy: highSyzygy, │ │ │ │ +00026e90: 2d2d 2052 6574 7572 6e73 2061 2073 797a -- Returns a syz │ │ │ │ +00026ea0: 7967 7920 6d6f 6475 6c65 206f 6e65 2062 ygy module one b │ │ │ │ +00026eb0: 6579 6f6e 6420 7468 650a 2020 2020 7265 eyond the. re │ │ │ │ +00026ec0: 6775 6c61 7269 7479 206f 6620 4578 7428 gularity of Ext( │ │ │ │ +00026ed0: 4d2c 6b29 0a20 202a 202a 6e6f 7465 2065 M,k). * *note e │ │ │ │ +00026ee0: 7874 6572 696f 7245 7874 4d6f 6475 6c65 xteriorExtModule │ │ │ │ +00026ef0: 3a20 6578 7465 7269 6f72 4578 744d 6f64 : exteriorExtMod │ │ │ │ +00026f00: 756c 652c 202d 2d20 4578 7428 4d2c 6b29 ule, -- Ext(M,k) │ │ │ │ +00026f10: 206f 7220 4578 7428 4d2c 4e29 2061 7320 or Ext(M,N) as │ │ │ │ +00026f20: 610a 2020 2020 6d6f 6475 6c65 206f 7665 a. module ove │ │ │ │ +00026f30: 7220 616e 2065 7874 6572 696f 7220 616c r an exterior al │ │ │ │ +00026f40: 6765 6272 610a 0a57 6179 7320 746f 2075 gebra..Ways to u │ │ │ │ +00026f50: 7365 2065 7874 5673 436f 686f 6d6f 6c6f se extVsCohomolo │ │ │ │ +00026f60: 6779 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d gy:.============ │ │ │ │ 00026f70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00026f80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00026f90: 2022 6578 7456 7343 6f68 6f6d 6f6c 6f67 "extVsCohomolog │ │ │ │ -00026fa0: 7928 4d61 7472 6978 2c4d 6f64 756c 6529 y(Matrix,Module) │ │ │ │ -00026fb0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -00026fc0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -00026fd0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -00026fe0: 6a65 6374 202a 6e6f 7465 2065 7874 5673 ject *note extVs │ │ │ │ -00026ff0: 436f 686f 6d6f 6c6f 6779 3a20 6578 7456 Cohomology: extV │ │ │ │ -00027000: 7343 6f68 6f6d 6f6c 6f67 792c 2069 7320 sCohomology, is │ │ │ │ -00027010: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ -00027020: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ -00027030: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ -00027040: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +00026f80: 0a0a 2020 2a20 2265 7874 5673 436f 686f .. * "extVsCoho │ │ │ │ +00026f90: 6d6f 6c6f 6779 284d 6174 7269 782c 4d6f mology(Matrix,Mo │ │ │ │ +00026fa0: 6475 6c65 2922 0a0a 466f 7220 7468 6520 dule)"..For the │ │ │ │ +00026fb0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +00026fc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +00026fd0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +00026fe0: 6578 7456 7343 6f68 6f6d 6f6c 6f67 793a extVsCohomology: │ │ │ │ +00026ff0: 2065 7874 5673 436f 686f 6d6f 6c6f 6779 extVsCohomology │ │ │ │ +00027000: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +00027010: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ +00027020: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ +00027030: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +00027040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027090: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -000270a0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -000270b0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -000270c0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -000270d0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -000270e0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -000270f0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00027100: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ -00027110: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -00027120: 2e6d 323a 3238 3236 3a30 2e0a 1f0a 4669 .m2:2826:0....Fi │ │ │ │ -00027130: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ -00027140: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00027150: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ -00027160: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -00027170: 7273 2c20 4e65 7874 3a20 6672 6565 4578 rs, Next: freeEx │ │ │ │ -00027180: 7465 7269 6f72 5375 6d6d 616e 642c 2050 teriorSummand, P │ │ │ │ -00027190: 7265 763a 2065 7874 5673 436f 686f 6d6f rev: extVsCohomo │ │ │ │ -000271a0: 6c6f 6779 2c20 5570 3a20 546f 700a 0a66 logy, Up: Top..f │ │ │ │ -000271b0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -000271c0: 7320 2d2d 2062 6574 7469 206e 756d 6265 s -- betti numbe │ │ │ │ -000271d0: 7273 206f 6620 6669 6e69 7465 2072 6573 rs of finite res │ │ │ │ -000271e0: 6f6c 7574 696f 6e20 636f 6d70 7574 6564 olution computed │ │ │ │ -000271f0: 2066 726f 6d20 6120 6d61 7472 6978 2066 from a matrix f │ │ │ │ -00027200: 6163 746f 7269 7a61 7469 6f6e 0a2a 2a2a actorization.*** │ │ │ │ +00027080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00027090: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +000270a0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +000270b0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +000270c0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +000270d0: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +000270e0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +000270f0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ +00027100: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +00027110: 7469 6f6e 732e 6d32 3a32 3832 363a 302e tions.m2:2826:0. │ │ │ │ +00027120: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ +00027130: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00027140: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ +00027150: 6f64 653a 2066 696e 6974 6542 6574 7469 ode: finiteBetti │ │ │ │ +00027160: 4e75 6d62 6572 732c 204e 6578 743a 2066 Numbers, Next: f │ │ │ │ +00027170: 7265 6545 7874 6572 696f 7253 756d 6d61 reeExteriorSumma │ │ │ │ +00027180: 6e64 2c20 5072 6576 3a20 6578 7456 7343 nd, Prev: extVsC │ │ │ │ +00027190: 6f68 6f6d 6f6c 6f67 792c 2055 703a 2054 ohomology, Up: T │ │ │ │ +000271a0: 6f70 0a0a 6669 6e69 7465 4265 7474 694e op..finiteBettiN │ │ │ │ +000271b0: 756d 6265 7273 202d 2d20 6265 7474 6920 umbers -- betti │ │ │ │ +000271c0: 6e75 6d62 6572 7320 6f66 2066 696e 6974 numbers of finit │ │ │ │ +000271d0: 6520 7265 736f 6c75 7469 6f6e 2063 6f6d e resolution com │ │ │ │ +000271e0: 7075 7465 6420 6672 6f6d 2061 206d 6174 puted from a mat │ │ │ │ +000271f0: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ +00027200: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ 00027210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027220: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027230: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00027240: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00027260: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -00027270: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -00027280: 4c20 3d20 6669 6e69 7465 4265 7474 694e L = finiteBettiN │ │ │ │ -00027290: 756d 6265 7273 204d 460a 2020 2a20 496e umbers MF. * In │ │ │ │ -000272a0: 7075 7473 3a0a 2020 2020 2020 2a20 4d46 puts:. * MF │ │ │ │ -000272b0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -000272c0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -000272d0: 7374 2c2c 204c 6973 7420 6f66 2048 6173 st,, List of Has │ │ │ │ -000272e0: 6854 6162 6c65 7320 6173 2063 6f6d 7075 hTables as compu │ │ │ │ -000272f0: 7465 640a 2020 2020 2020 2020 6279 2022 ted. by " │ │ │ │ -00027300: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -00027310: 696f 6e22 0a20 202a 204f 7574 7075 7473 ion". * Outputs │ │ │ │ -00027320: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a :. * L, a * │ │ │ │ -00027330: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -00027340: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -00027350: 4c69 7374 206f 6620 6265 7474 6920 6e75 List of betti nu │ │ │ │ -00027360: 6d62 6572 730a 0a44 6573 6372 6970 7469 mbers..Descripti │ │ │ │ -00027370: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00027380: 5573 6573 2074 6865 2072 616e 6b73 206f Uses the ranks o │ │ │ │ -00027390: 6620 7468 6520 4220 6d61 7472 6963 6573 f the B matrices │ │ │ │ -000273a0: 2069 6e20 6120 6d61 7472 6978 2066 6163 in a matrix fac │ │ │ │ -000273b0: 746f 7269 7a61 7469 6f6e 2066 6f72 2061 torization for a │ │ │ │ -000273c0: 206d 6f64 756c 6520 4d20 6f76 6572 0a53 module M over.S │ │ │ │ -000273d0: 2f28 665f 312c 2e2e 2c66 5f63 2920 746f /(f_1,..,f_c) to │ │ │ │ -000273e0: 2063 6f6d 7075 7465 2074 6865 2062 6574 compute the bet │ │ │ │ -000273f0: 7469 206e 756d 6265 7273 206f 6620 7468 ti numbers of th │ │ │ │ -00027400: 6520 6d69 6e69 6d61 6c20 7265 736f 6c75 e minimal resolu │ │ │ │ -00027410: 7469 6f6e 206f 6620 4d20 6f76 6572 0a53 tion of M over.S │ │ │ │ -00027420: 2c20 7768 6963 6820 6973 2074 6865 2073 , which is the s │ │ │ │ -00027430: 756d 206f 6620 7468 6520 4b6f 737a 756c um of the Koszul │ │ │ │ -00027440: 2063 6f6d 706c 6578 6573 204b 2866 5f31 complexes K(f_1 │ │ │ │ -00027450: 2e2e 665f 7b6a 2d31 7d29 2074 656e 736f ..f_{j-1}) tenso │ │ │ │ -00027460: 7265 6420 7769 7468 2042 286a 290a 0a2b red with B(j)..+ │ │ │ │ +00027250: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00027260: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +00027270: 2020 2020 204c 203d 2066 696e 6974 6542 L = finiteB │ │ │ │ +00027280: 6574 7469 4e75 6d62 6572 7320 4d46 0a20 ettiNumbers MF. │ │ │ │ +00027290: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +000272a0: 202a 204d 462c 2061 202a 6e6f 7465 206c * MF, a *note l │ │ │ │ +000272b0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +000272c0: 6f63 294c 6973 742c 2c20 4c69 7374 206f oc)List,, List o │ │ │ │ +000272d0: 6620 4861 7368 5461 626c 6573 2061 7320 f HashTables as │ │ │ │ +000272e0: 636f 6d70 7574 6564 0a20 2020 2020 2020 computed. │ │ │ │ +000272f0: 2062 7920 226d 6174 7269 7846 6163 746f by "matrixFacto │ │ │ │ +00027300: 7269 7a61 7469 6f6e 220a 2020 2a20 4f75 rization". * Ou │ │ │ │ +00027310: 7470 7574 733a 0a20 2020 2020 202a 204c tputs:. * L │ │ │ │ +00027320: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +00027330: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +00027340: 7374 2c2c 204c 6973 7420 6f66 2062 6574 st,, List of bet │ │ │ │ +00027350: 7469 206e 756d 6265 7273 0a0a 4465 7363 ti numbers..Desc │ │ │ │ +00027360: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00027370: 3d3d 3d0a 0a55 7365 7320 7468 6520 7261 ===..Uses the ra │ │ │ │ +00027380: 6e6b 7320 6f66 2074 6865 2042 206d 6174 nks of the B mat │ │ │ │ +00027390: 7269 6365 7320 696e 2061 206d 6174 7269 rices in a matri │ │ │ │ +000273a0: 7820 6661 6374 6f72 697a 6174 696f 6e20 x factorization │ │ │ │ +000273b0: 666f 7220 6120 6d6f 6475 6c65 204d 206f for a module M o │ │ │ │ +000273c0: 7665 720a 532f 2866 5f31 2c2e 2e2c 665f ver.S/(f_1,..,f_ │ │ │ │ +000273d0: 6329 2074 6f20 636f 6d70 7574 6520 7468 c) to compute th │ │ │ │ +000273e0: 6520 6265 7474 6920 6e75 6d62 6572 7320 e betti numbers │ │ │ │ +000273f0: 6f66 2074 6865 206d 696e 696d 616c 2072 of the minimal r │ │ │ │ +00027400: 6573 6f6c 7574 696f 6e20 6f66 204d 206f esolution of M o │ │ │ │ +00027410: 7665 720a 532c 2077 6869 6368 2069 7320 ver.S, which is │ │ │ │ +00027420: 7468 6520 7375 6d20 6f66 2074 6865 204b the sum of the K │ │ │ │ +00027430: 6f73 7a75 6c20 636f 6d70 6c65 7865 7320 oszul complexes │ │ │ │ +00027440: 4b28 665f 312e 2e66 5f7b 6a2d 317d 2920 K(f_1..f_{j-1}) │ │ │ │ +00027450: 7465 6e73 6f72 6564 2077 6974 6820 4228 tensored with B( │ │ │ │ +00027460: 6a29 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d j)..+----------- │ │ │ │ 00027470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000274a0: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -000274b0: 646f 6d53 6565 6420 3020 2020 2020 2020 domSeed 0 │ │ │ │ -000274c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000274d0: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ -000274e0: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ -000274f0: 6420 746f 2030 2020 2020 2020 2020 2020 d to 0 │ │ │ │ -00027500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027490: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ +000274a0: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +000274b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000274c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000274d0: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +000274e0: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +000274f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027500: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027540: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +00027530: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +00027540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00027560: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000275a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -000275b0: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020 : kk = ZZ/101 │ │ │ │ +000275a0: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f +.|i2 : kk = ZZ/ │ │ │ │ +000275b0: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ 000275c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000275e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000275d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000275e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000275f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027610: 2020 2020 207c 0a7c 6f32 203d 206b 6b20 |.|o2 = kk │ │ │ │ +00027600: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +00027610: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 00027620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027640: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027630: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027640: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00027650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027680: 0a7c 6f32 203a 2051 756f 7469 656e 7452 .|o2 : QuotientR │ │ │ │ -00027690: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -000276a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000276b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00027670: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74 |.|o2 : Quot │ │ │ │ +00027680: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +00027690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000276a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000276b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000276c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000276e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -000276f0: 2053 203d 206b 6b5b 612c 622c 752c 765d S = kk[a,b,u,v] │ │ │ │ +000276d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000276e0: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62 |i3 : S = kk[a,b │ │ │ │ +000276f0: 2c75 2c76 5d20 2020 2020 2020 2020 2020 ,u,v] │ │ │ │ 00027700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027720: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00027710: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027750: 2020 207c 0a7c 6f33 203d 2053 2020 2020 |.|o3 = S │ │ │ │ +00027740: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +00027750: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00027760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027780: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00027770: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00027780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000277a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000277c0: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52 o3 : PolynomialR │ │ │ │ -000277d0: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -000277e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000277f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000277b0: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f |.|o3 : Polyno │ │ │ │ +000277c0: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +000277d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000277e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000277f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027820: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ -00027830: 6620 3d20 6d61 7472 6978 2261 752c 6276 f = matrix"au,bv │ │ │ │ -00027840: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -00027850: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00027820: 3420 3a20 6666 203d 206d 6174 7269 7822 4 : ff = matrix" │ │ │ │ +00027830: 6175 2c62 7622 2020 2020 2020 2020 2020 au,bv" │ │ │ │ +00027840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027850: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027890: 207c 0a7c 6f34 203d 207c 2061 7520 6276 |.|o4 = | au bv │ │ │ │ -000278a0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000278b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00027880: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20 |.|o4 = | │ │ │ │ +00027890: 6175 2062 7620 7c20 2020 2020 2020 2020 au bv | │ │ │ │ +000278a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000278b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000278c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000278e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000278f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00027900: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -00027910: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00027920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027930: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053 |.|o4 : Matrix S │ │ │ │ -00027940: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ -00027950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027960: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000278f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027900: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +00027910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027920: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ +00027930: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +00027940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027950: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00027960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027990: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -000279a0: 3a20 5220 3d20 532f 6964 6561 6c20 6666 : R = S/ideal ff │ │ │ │ +00027980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00027990: 0a7c 6935 203a 2052 203d 2053 2f69 6465 .|i5 : R = S/ide │ │ │ │ +000279a0: 616c 2066 6620 2020 2020 2020 2020 2020 al ff │ │ │ │ 000279b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000279d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000279c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000279d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000279e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000279f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a00: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +000279f0: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +00027a00: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00027a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027a20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027a30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00027a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027a60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027a70: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269 |o5 : QuotientRi │ │ │ │ -00027a80: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -00027a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027aa0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027a60: 2020 207c 0a7c 6f35 203a 2051 756f 7469 |.|o5 : Quoti │ │ │ │ +00027a70: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +00027a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027a90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00027aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ad0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -00027ae0: 4d30 203d 2052 5e31 2f69 6465 616c 2261 M0 = R^1/ideal"a │ │ │ │ -00027af0: 2c62 2220 2020 2020 2020 2020 2020 2020 ,b" │ │ │ │ -00027b00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00027ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00027ad0: 6936 203a 204d 3020 3d20 525e 312f 6964 i6 : M0 = R^1/id │ │ │ │ +00027ae0: 6561 6c22 612c 6222 2020 2020 2020 2020 eal"a,b" │ │ │ │ +00027af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b00: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b40: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ -00027b50: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ -00027b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027b70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00027b30: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ +00027b40: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20 okernel | a b | │ │ │ │ +00027b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027b60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ba0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00027ba0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00027bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027bc0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -00027bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027be0: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c |.|o6 : R-modul │ │ │ │ -00027bf0: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ -00027c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027c10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00027bc0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00027bd0: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ +00027be0: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +00027bf0: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +00027c00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00027c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -00027c50: 203a 2046 203d 2066 7265 6552 6573 6f6c : F = freeResol │ │ │ │ -00027c60: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468 ution(M0, Length │ │ │ │ -00027c70: 4c69 6d69 7420 3d3e 3329 2020 2020 2020 Limit =>3) │ │ │ │ -00027c80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00027c40: 2b0a 7c69 3720 3a20 4620 3d20 6672 6565 +.|i7 : F = free │ │ │ │ +00027c50: 5265 736f 6c75 7469 6f6e 284d 302c 204c Resolution(M0, L │ │ │ │ +00027c60: 656e 6774 684c 696d 6974 203d 3e33 2920 engthLimit =>3) │ │ │ │ +00027c70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00027c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027cb0: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ -00027cc0: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -00027cd0: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -00027ce0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -00027cf0: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20 = R <-- R <-- │ │ │ │ -00027d00: 5220 203c 2d2d 2052 2020 2020 2020 2020 R <-- R │ │ │ │ -00027d10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00027d20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00027ca0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027cb0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +00027cc0: 2033 2020 2020 2020 3420 2020 2020 2020 3 4 │ │ │ │ +00027cd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00027ce0: 0a7c 6f37 203d 2052 2020 3c2d 2d20 5220 .|o7 = R <-- R │ │ │ │ +00027cf0: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020 <-- R <-- R │ │ │ │ +00027d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027d10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00027d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027d50: 2020 2020 7c0a 7c20 2020 2020 3020 2020 |.| 0 │ │ │ │ -00027d60: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -00027d70: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -00027d80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027d40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00027d50: 2030 2020 2020 2020 3120 2020 2020 2032 0 1 2 │ │ │ │ +00027d60: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +00027d70: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00027d80: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00027d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027db0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00027dc0: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020 |o7 : Complex │ │ │ │ +00027db0: 2020 207c 0a7c 6f37 203a 2043 6f6d 706c |.|o7 : Compl │ │ │ │ +00027dc0: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 00027dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027df0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00027de0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00027df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ -00027e30: 4d20 3d20 636f 6b65 7220 462e 6464 5f33 M = coker F.dd_3 │ │ │ │ -00027e40: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -00027e50: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00027e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00027e20: 6938 203a 204d 203d 2063 6f6b 6572 2046 i8 : M = coker F │ │ │ │ +00027e30: 2e64 645f 333b 2020 2020 2020 2020 2020 .dd_3; │ │ │ │ +00027e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027e50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00027e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027e90: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d --+.|i9 : MF = m │ │ │ │ -00027ea0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -00027eb0: 6f6e 2866 662c 4d29 3b20 2020 2020 2020 on(ff,M); │ │ │ │ -00027ec0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00027e80: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 204d -------+.|i9 : M │ │ │ │ +00027e90: 4620 3d20 6d61 7472 6978 4661 6374 6f72 F = matrixFactor │ │ │ │ +00027ea0: 697a 6174 696f 6e28 6666 2c4d 293b 2020 ization(ff,M); │ │ │ │ +00027eb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00027ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00027ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00027ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00027f00: 3130 203a 2062 6574 7469 2066 7265 6552 10 : betti freeR │ │ │ │ -00027f10: 6573 6f6c 7574 696f 6e20 7075 7368 466f esolution pushFo │ │ │ │ -00027f20: 7277 6172 6428 6d61 7028 522c 5329 2c4d rward(map(R,S),M │ │ │ │ -00027f30: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ +00027ef0: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920 -+.|i10 : betti │ │ │ │ +00027f00: 6672 6565 5265 736f 6c75 7469 6f6e 2070 freeResolution p │ │ │ │ +00027f10: 7573 6846 6f72 7761 7264 286d 6170 2852 ushForward(map(R │ │ │ │ +00027f20: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020 ,S),M)|.| │ │ │ │ +00027f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f60: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00027f70: 2020 2020 2020 3020 3120 3220 2020 2020 0 1 2 │ │ │ │ +00027f50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00027f60: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +00027f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00027f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027f90: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00027fa0: 3020 3d20 746f 7461 6c3a 2033 2035 2032 0 = total: 3 5 2 │ │ │ │ +00027f90: 7c0a 7c6f 3130 203d 2074 6f74 616c 3a20 |.|o10 = total: │ │ │ │ +00027fa0: 3320 3520 3220 2020 2020 2020 2020 2020 3 5 2 │ │ │ │ 00027fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00027fd0: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ -00027fe0: 3320 3420 2e20 2020 2020 2020 2020 2020 3 4 . │ │ │ │ -00027ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028000: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00028010: 2020 333a 202e 2031 2032 2020 2020 2020 3: . 1 2 │ │ │ │ -00028020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028030: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00027fc0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00027fd0: 2020 323a 2033 2034 202e 2020 2020 2020 2: 3 4 . │ │ │ │ +00027fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00027ff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00028000: 2020 2020 2020 2033 3a20 2e20 3120 3220 3: . 1 2 │ │ │ │ +00028010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028020: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00028030: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028060: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028070: 0a7c 6f31 3020 3a20 4265 7474 6954 616c .|o10 : BettiTal │ │ │ │ -00028080: 6c79 2020 2020 2020 2020 2020 2020 2020 ly │ │ │ │ -00028090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000280a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028060: 2020 2020 7c0a 7c6f 3130 203a 2042 6574 |.|o10 : Bet │ │ │ │ +00028070: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +00028080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028090: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000280a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000280b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000280d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ -000280e0: 3a20 6669 6e69 7465 4265 7474 694e 756d : finiteBettiNum │ │ │ │ -000280f0: 6265 7273 204d 4620 2020 2020 2020 2020 bers MF │ │ │ │ -00028100: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00028110: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +000280c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000280d0: 7c69 3131 203a 2066 696e 6974 6542 6574 |i11 : finiteBet │ │ │ │ +000280e0: 7469 4e75 6d62 6572 7320 4d46 2020 2020 tiNumbers MF │ │ │ │ +000280f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028100: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028140: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20 |.|o11 = {3, │ │ │ │ -00028150: 352c 2032 7d20 2020 2020 2020 2020 2020 5, 2} │ │ │ │ -00028160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028170: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00028130: 2020 2020 2020 2020 7c0a 7c6f 3131 203d |.|o11 = │ │ │ │ +00028140: 207b 332c 2035 2c20 327d 2020 2020 2020 {3, 5, 2} │ │ │ │ +00028150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00028170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000281b0: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ +000281a0: 2020 7c0a 7c6f 3131 203a 204c 6973 7420 |.|o11 : List │ │ │ │ +000281b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000281c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000281e0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000281d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000281e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028210: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00028220: 696e 6669 6e69 7465 4265 7474 694e 756d infiniteBettiNum │ │ │ │ -00028230: 6265 7273 284d 462c 3529 2020 2020 2020 bers(MF,5) │ │ │ │ -00028240: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00028210: 3132 203a 2069 6e66 696e 6974 6542 6574 12 : infiniteBet │ │ │ │ +00028220: 7469 4e75 6d62 6572 7328 4d46 2c35 2920 tiNumbers(MF,5) │ │ │ │ +00028230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028240: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00028250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028280: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c |.|o12 = {3, 4, │ │ │ │ -00028290: 2035 2c20 362c 2037 2c20 387d 2020 2020 5, 6, 7, 8} │ │ │ │ -000282a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000282b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00028270: 2020 2020 2020 7c0a 7c6f 3132 203d 207b |.|o12 = { │ │ │ │ +00028280: 332c 2034 2c20 352c 2036 2c20 372c 2038 3, 4, 5, 6, 7, 8 │ │ │ │ +00028290: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000282a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000282b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000282c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000282d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000282e0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -000282f0: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ +000282e0: 7c0a 7c6f 3132 203a 204c 6973 7420 2020 |.|o12 : List │ │ │ │ +000282f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028320: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00028310: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00028320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028350: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265 -----+.|i13 : be │ │ │ │ -00028360: 7474 6920 6672 6565 5265 736f 6c75 7469 tti freeResoluti │ │ │ │ -00028370: 6f6e 2028 4d2c 204c 656e 6774 684c 696d on (M, LengthLim │ │ │ │ -00028380: 6974 203d 3e20 3529 2020 7c0a 7c20 2020 it => 5) |.| │ │ │ │ +00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ +00028350: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ +00028360: 6f6c 7574 696f 6e20 284d 2c20 4c65 6e67 olution (M, Leng │ │ │ │ +00028370: 7468 4c69 6d69 7420 3d3e 2035 2920 207c thLimit => 5) | │ │ │ │ +00028380: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00028390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000283a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000283c0: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -000283d0: 2031 2032 2033 2034 2035 2020 2020 2020 1 2 3 4 5 │ │ │ │ -000283e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000283f0: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74 |.|o13 = tot │ │ │ │ -00028400: 616c 3a20 3320 3420 3520 3620 3720 3820 al: 3 4 5 6 7 8 │ │ │ │ -00028410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028420: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00028430: 2020 2020 2020 323a 2033 2034 2035 2036 2: 3 4 5 6 │ │ │ │ -00028440: 2037 2038 2020 2020 2020 2020 2020 2020 7 8 │ │ │ │ -00028450: 2020 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-00028490: 2020 207c 0a7c 6f31 3320 3a20 4265 7474 |.|o13 : Bett │ │ │ │ -000284a0: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -000284b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000284c0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00028480: 2020 2020 2020 2020 7c0a 7c6f 3133 203a |.|o13 : │ │ │ │ +00028490: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +000284a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000284b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000284c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000284d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000284e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000284f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00028500: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -00028510: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174 =.. * *note mat │ │ │ │ -00028520: 7269 7846 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7269 7a61 matrix factoriza │ │ │ │ -00028600: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573 tion..Ways to us │ │ │ │ -00028610: 6520 6669 6e69 7465 4265 7474 694e 756d e finiteBettiNum │ │ │ │ -00028620: 6265 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d bers:.========== │ │ │ │ -00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00028640: 3d3d 3d3d 3d0a 0a20 202a 2022 6669 6e69 =====.. * "fini │ │ │ │ -00028650: 7465 4265 7474 694e 756d 6265 7273 284c teBettiNumbers(L │ │ │ │ -00028660: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ -00028670: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -00028680: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -00028690: 6520 6f62 6a65 6374 202a 6e6f 7465 2066 e object *note f │ │ │ │ -000286a0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -000286b0: 733a 2066 696e 6974 6542 6574 7469 4e75 s: finiteBettiNu │ │ │ │ -000286c0: 6d62 6572 732c 2069 7320 6120 2a6e 6f74 mbers, is a *not │ │ │ │ -000286d0: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ -000286e0: 6e3a 2028 4d61 6361 756c 6179 3244 6f63 n: (Macaulay2Doc │ │ │ │ -000286f0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -00028700: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +000284f0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ +00028500: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +00028510: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ +00028520: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ +00028530: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ +00028540: 6170 7320 696e 2061 2068 6967 6865 720a aps in a higher. │ │ │ │ +00028550: 2020 2020 636f 6469 6d65 6e73 696f 6e20 codimension │ │ │ │ +00028560: 6d61 7472 6978 2066 6163 746f 7269 7a61 matrix factoriza │ │ │ │ +00028570: 7469 6f6e 0a20 202a 202a 6e6f 7465 2069 tion. * *note i │ │ │ │ +00028580: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ +00028590: 6572 733a 2069 6e66 696e 6974 6542 6574 ers: infiniteBet │ │ │ │ +000285a0: 7469 4e75 6d62 6572 732c 202d 2d20 6265 tiNumbers, -- be │ │ │ │ +000285b0: 7474 6920 6e75 6d62 6572 7320 6f66 0a20 tti numbers of. │ │ │ │ +000285c0: 2020 2066 696e 6974 6520 7265 736f 6c75 finite resolu │ │ │ │ +000285d0: 7469 6f6e 2063 6f6d 7075 7465 6420 6672 tion computed fr │ │ │ │ +000285e0: 6f6d 2061 206d 6174 7269 7820 6661 6374 om a matrix fact │ │ │ │ +000285f0: 6f72 697a 6174 696f 6e0a 0a57 6179 7320 orization..Ways │ │ │ │ +00028600: 746f 2075 7365 2066 696e 6974 6542 6574 to use finiteBet │ │ │ │ +00028610: 7469 4e75 6d62 6572 733a 0a3d 3d3d 3d3d tiNumbers:.===== │ │ │ │ +00028620: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00028630: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00028640: 2266 696e 6974 6542 6574 7469 4e75 6d62 "finiteBettiNumb │ │ │ │ +00028650: 6572 7328 4c69 7374 2922 0a0a 466f 7220 ers(List)"..For │ │ │ │ +00028660: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00028670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00028680: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00028690: 6f74 6520 6669 6e69 7465 4265 7474 694e ote finiteBettiN │ │ │ │ +000286a0: 756d 6265 7273 3a20 6669 6e69 7465 4265 umbers: finiteBe │ │ │ │ +000286b0: 7474 694e 756d 6265 7273 2c20 6973 2061 ttiNumbers, is a │ │ │ │ +000286c0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ +000286d0: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ +000286e0: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ +000286f0: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +00028700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028750: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -00028760: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -00028770: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -00028780: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00028790: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -000287a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -000287b0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -000287c0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -000287d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3430 esolutions.m2:40 │ │ │ │ -000287e0: 3732 3a30 2e0a 1f0a 4669 6c65 3a20 436f 72:0....File: Co │ │ │ │ -000287f0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00028800: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -00028810: 666f 2c20 4e6f 6465 3a20 6672 6565 4578 fo, Node: freeEx │ │ │ │ -00028820: 7465 7269 6f72 5375 6d6d 616e 642c 204e teriorSummand, N │ │ │ │ -00028830: 6578 743a 2047 7261 6469 6e67 2c20 5072 ext: Grading, Pr │ │ │ │ -00028840: 6576 3a20 6669 6e69 7465 4265 7474 694e ev: finiteBettiN │ │ │ │ -00028850: 756d 6265 7273 2c20 5570 3a20 546f 700a umbers, Up: Top. │ │ │ │ -00028860: 0a66 7265 6545 7874 6572 696f 7253 756d .freeExteriorSum │ │ │ │ -00028870: 6d61 6e64 202d 2d20 6669 6e64 2074 6865 mand -- find the │ │ │ │ -00028880: 2066 7265 6520 7375 6d6d 616e 6473 206f free summands o │ │ │ │ -00028890: 6620 6120 6d6f 6475 6c65 206f 7665 7220 f a module over │ │ │ │ -000288a0: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ -000288b0: 6272 610a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a bra.************ │ │ │ │ +00028740: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00028750: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00028760: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00028770: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00028780: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00028790: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +000287a0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +000287b0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +000287c0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +000287d0: 6d32 3a34 3037 323a 302e 0a1f 0a46 696c m2:4072:0....Fil │ │ │ │ +000287e0: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +000287f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00028800: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2066 ns.info, Node: f │ │ │ │ +00028810: 7265 6545 7874 6572 696f 7253 756d 6d61 reeExteriorSumma │ │ │ │ +00028820: 6e64 2c20 4e65 7874 3a20 4772 6164 696e nd, Next: Gradin │ │ │ │ +00028830: 672c 2050 7265 763a 2066 696e 6974 6542 g, Prev: finiteB │ │ │ │ +00028840: 6574 7469 4e75 6d62 6572 732c 2055 703a ettiNumbers, Up: │ │ │ │ +00028850: 2054 6f70 0a0a 6672 6565 4578 7465 7269 Top..freeExteri │ │ │ │ +00028860: 6f72 5375 6d6d 616e 6420 2d2d 2066 696e orSummand -- fin │ │ │ │ +00028870: 6420 7468 6520 6672 6565 2073 756d 6d61 d the free summa │ │ │ │ +00028880: 6e64 7320 6f66 2061 206d 6f64 756c 6520 nds of a module │ │ │ │ +00028890: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ +000288a0: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a algebra.******* │ │ │ │ +000288b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000288c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000288d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000288e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00028900: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00028910: 653a 200a 2020 2020 2020 2020 4620 3d20 e: . F = │ │ │ │ -00028920: 6672 6565 4578 7465 7269 6f72 5375 6d6d freeExteriorSumm │ │ │ │ -00028930: 616e 6420 4d0a 2020 2a20 496e 7075 7473 and M. * Inputs │ │ │ │ -00028940: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ -00028950: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -00028960: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -00028970: 652c 2c20 6f76 6572 2061 6e20 6578 7465 e,, over an exte │ │ │ │ -00028980: 7269 6f72 2061 6c67 6562 7261 0a20 202a rior algebra. * │ │ │ │ -00028990: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -000289a0: 2a20 462c 2061 202a 6e6f 7465 206d 6174 * F, a *note mat │ │ │ │ -000289b0: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ -000289c0: 6f63 294d 6174 7269 782c 2c20 4d61 7020 oc)Matrix,, Map │ │ │ │ -000289d0: 6672 6f6d 2061 2066 7265 6520 6d6f 6475 from a free modu │ │ │ │ -000289e0: 6c65 2074 6f20 4d2e 0a20 2020 2020 2020 le to M.. │ │ │ │ -000289f0: 2049 6d61 6765 2069 7320 7468 6520 6c61 Image is the la │ │ │ │ -00028a00: 7267 6573 7420 6672 6565 2073 756d 6d61 rgest free summa │ │ │ │ -00028a10: 6e64 0a0a 4465 7363 7269 7074 696f 6e0a nd..Description. │ │ │ │ -00028a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 2d2d ===========..+-- │ │ │ │ +000288f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00028900: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00028910: 2046 203d 2066 7265 6545 7874 6572 696f F = freeExterio │ │ │ │ +00028920: 7253 756d 6d61 6e64 204d 0a20 202a 2049 rSummand M. * I │ │ │ │ +00028930: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ +00028940: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +00028950: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00028960: 4d6f 6475 6c65 2c2c 206f 7665 7220 616e Module,, over an │ │ │ │ +00028970: 2065 7874 6572 696f 7220 616c 6765 6272 exterior algebr │ │ │ │ +00028980: 610a 2020 2a20 4f75 7470 7574 733a 0a20 a. * Outputs:. │ │ │ │ +00028990: 2020 2020 202a 2046 2c20 6120 2a6e 6f74 * F, a *not │ │ │ │ +000289a0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +000289b0: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +000289c0: 204d 6170 2066 726f 6d20 6120 6672 6565 Map from a free │ │ │ │ +000289d0: 206d 6f64 756c 6520 746f 204d 2e0a 2020 module to M.. │ │ │ │ +000289e0: 2020 2020 2020 496d 6167 6520 6973 2074 Image is t │ │ │ │ +000289f0: 6865 206c 6172 6765 7374 2066 7265 6520 he largest free │ │ │ │ +00028a00: 7375 6d6d 616e 640a 0a44 6573 6372 6970 summand..Descrip │ │ │ │ +00028a10: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00028a20: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00028a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028a60: 2d2b 0a7c 6931 203a 206b 6b3d 205a 5a2f -+.|i1 : kk= ZZ/ │ │ │ │ -00028a70: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ -00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028a50: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 6b6b ------+.|i1 : kk │ │ │ │ +00028a60: 3d20 5a5a 2f31 3031 2020 2020 2020 2020 = ZZ/101 │ │ │ │ +00028a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028ac0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028ad0: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +00028ac0: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +00028ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b00: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00028af0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b30: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -00028b40: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +00028b20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00028b30: 7c6f 3120 3a20 5175 6f74 6965 6e74 5269 |o1 : QuotientRi │ │ │ │ +00028b40: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00028b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028b70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00028b60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00028b70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028ba0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2045 203d -----+.|i2 : E = │ │ │ │ -00028bb0: 206b 6b5b 652c 662c 672c 2053 6b65 7743 kk[e,f,g, SkewC │ │ │ │ -00028bc0: 6f6d 6d75 7461 7469 7665 203d 3e20 7472 ommutative => tr │ │ │ │ -00028bd0: 7565 5d20 2020 2020 2020 207c 0a7c 2020 ue] |.| │ │ │ │ +00028b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ +00028ba0: 3a20 4520 3d20 6b6b 5b65 2c66 2c67 2c20 : E = kk[e,f,g, │ │ │ │ +00028bb0: 536b 6577 436f 6d6d 7574 6174 6976 6520 SkewCommutative │ │ │ │ +00028bc0: 3d3e 2074 7275 655d 2020 2020 2020 2020 => true] │ │ │ │ +00028bd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00028be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c10: 207c 0a7c 6f32 203d 2045 2020 2020 2020 |.|o2 = E │ │ │ │ +00028c00: 2020 2020 2020 7c0a 7c6f 3220 3d20 4520 |.|o2 = E │ │ │ │ +00028c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00028c30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00028c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00028c80: 6f32 203a 2050 6f6c 796e 6f6d 6961 6c52 o2 : PolynomialR │ │ │ │ -00028c90: 696e 672c 2033 2073 6b65 7720 636f 6d6d ing, 3 skew comm │ │ │ │ -00028ca0: 7574 6174 6976 6520 7661 7269 6162 6c65 utative variable │ │ │ │ -00028cb0: 2873 297c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 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6e67 203d 3e20 3229 0a20 202a 2049 6e70 ng => 2). * Inp │ │ │ │ -00029420: 7574 733a 0a20 2020 2020 202a 2043 6865 uts:. * Che │ │ │ │ -00029430: 636b 2c20 616e 202a 6e6f 7465 2069 6e74 ck, an *note int │ │ │ │ -00029440: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -00029450: 446f 6329 5a5a 2c2c 200a 0a44 6573 6372 Doc)ZZ,, ..Descr │ │ │ │ -00029460: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00029470: 3d3d 0a0a 6966 2047 7261 6469 6e67 203d ==..if Grading = │ │ │ │ -00029480: 3e31 2c20 7468 656e 2074 6865 206f 7574 >1, then the out │ │ │ │ -00029490: 7075 7420 6973 2063 6f6e 7665 7274 6564 put is converted │ │ │ │ -000294a0: 2074 6f20 7369 6e67 6c65 2d67 7261 6469 to single-gradi │ │ │ │ -000294b0: 6e67 2c20 7573 6566 756c 2069 6e20 7468 ng, useful in th │ │ │ │ -000294c0: 650a 7061 636b 6167 6520 436c 6966 666f e.package Cliffo │ │ │ │ -000294d0: 7264 0a0a 5365 6520 616c 736f 0a3d 3d3d rd..See also.=== │ │ │ │ -000294e0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -000294f0: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00029500: 546f 7461 6c3a 2045 6973 656e 6275 6453 Total: EisenbudS │ │ │ │ -00029510: 6861 6d61 7368 546f 7461 6c2c 202d 2d20 hamashTotal, -- │ │ │ │ -00029520: 5072 6563 7572 736f 7220 636f 6d70 6c65 Precursor comple │ │ │ │ -00029530: 7820 6f66 0a20 2020 2074 6f74 616c 2045 x of. total E │ │ │ │ -00029540: 7874 0a20 202a 202a 6e6f 7465 206e 6577 xt. * *note new │ │ │ │ -00029550: 4578 743a 206e 6577 4578 742c 202d 2d20 Ext: newExt, -- │ │ │ │ -00029560: 476c 6f62 616c 2045 7874 2066 6f72 206d Global Ext for m │ │ │ │ -00029570: 6f64 756c 6573 206f 7665 7220 6120 636f odules over a co │ │ │ │ -00029580: 6d70 6c65 7465 0a20 2020 2049 6e74 6572 mplete. Inter │ │ │ │ -00029590: 7365 6374 696f 6e0a 0a46 756e 6374 696f section..Functio │ │ │ │ -000295a0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -000295b0: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ -000295c0: 4772 6164 696e 673a 0a3d 3d3d 3d3d 3d3d Grading:.======= │ │ │ │ +000293d0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +000293e0: 2020 2020 2020 4569 7365 6e62 7564 5368 EisenbudSh │ │ │ │ +000293f0: 616d 6173 6854 6f74 616c 284d 6261 722c amashTotal(Mbar, │ │ │ │ +00029400: 4772 6164 696e 6720 3d3e 2032 290a 2020 Grading => 2). │ │ │ │ +00029410: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00029420: 2a20 4368 6563 6b2c 2061 6e20 2a6e 6f74 * Check, an *not │ │ │ │ +00029430: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00029440: 756c 6179 3244 6f63 295a 5a2c 2c20 0a0a ulay2Doc)ZZ,, .. │ │ │ │ +00029450: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +00029460: 3d3d 3d3d 3d3d 3d0a 0a69 6620 4772 6164 =======..if Grad │ │ │ │ +00029470: 696e 6720 3d3e 312c 2074 6865 6e20 7468 ing =>1, then th │ │ │ │ +00029480: 6520 6f75 7470 7574 2069 7320 636f 6e76 e output is conv │ │ │ │ +00029490: 6572 7465 6420 746f 2073 696e 676c 652d erted to single- │ │ │ │ +000294a0: 6772 6164 696e 672c 2075 7365 6675 6c20 grading, useful │ │ │ │ +000294b0: 696e 2074 6865 0a70 6163 6b61 6765 2043 in the.package C │ │ │ │ +000294c0: 6c69 6666 6f72 640a 0a53 6565 2061 6c73 lifford..See als │ │ │ │ +000294d0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +000294e0: 2a6e 6f74 6520 4569 7365 6e62 7564 5368 *note EisenbudSh │ │ │ │ +000294f0: 616d 6173 6854 6f74 616c 3a20 4569 7365 amashTotal: Eise │ │ │ │ +00029500: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00029510: 2c20 2d2d 2050 7265 6375 7273 6f72 2063 , -- Precursor c │ │ │ │ +00029520: 6f6d 706c 6578 206f 660a 2020 2020 746f omplex of. to │ │ │ │ +00029530: 7461 6c20 4578 740a 2020 2a20 2a6e 6f74 tal Ext. * *not │ │ │ │ +00029540: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ +00029550: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420 , -- Global Ext │ │ │ │ +00029560: 666f 7220 6d6f 6475 6c65 7320 6f76 6572 for modules over │ │ │ │ +00029570: 2061 2063 6f6d 706c 6574 650a 2020 2020 a complete. │ │ │ │ +00029580: 496e 7465 7273 6563 7469 6f6e 0a0a 4675 Intersection..Fu │ │ │ │ +00029590: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +000295a0: 696f 6e61 6c20 6172 6775 6d65 6e74 206e ional argument n │ │ │ │ +000295b0: 616d 6564 2047 7261 6469 6e67 3a0a 3d3d amed Grading:.== │ │ │ │ +000295c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 000295d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000295f0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2245 ========.. * "E │ │ │ │ -00029600: 6973 656e 6275 6453 6861 6d61 7368 546f isenbudShamashTo │ │ │ │ -00029610: 7461 6c28 2e2e 2e2c 4772 6164 696e 673d tal(...,Grading= │ │ │ │ -00029620: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ -00029630: 6f74 6520 4569 7365 6e62 7564 5368 616d ote EisenbudSham │ │ │ │ -00029640: 6173 6854 6f74 616c 3a0a 2020 2020 4569 ashTotal:. Ei │ │ │ │ -00029650: 7365 6e62 7564 5368 616d 6173 6854 6f74 senbudShamashTot │ │ │ │ -00029660: 616c 2c20 2d2d 2050 7265 6375 7273 6f72 al, -- Precursor │ │ │ │ -00029670: 2063 6f6d 706c 6578 206f 6620 746f 7461 complex of tota │ │ │ │ -00029680: 6c20 4578 740a 2020 2a20 226e 6577 4578 l Ext. * "newEx │ │ │ │ -00029690: 7428 2e2e 2e2c 4772 6164 696e 673d 3e2e t(...,Grading=>. │ │ │ │ -000296a0: 2e2e 2922 202d 2d20 7365 6520 2a6e 6f74 ..)" -- see *not │ │ │ │ -000296b0: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ -000296c0: 2c20 2d2d 2047 6c6f 6261 6c20 4578 7420 , -- Global Ext │ │ │ │ -000296d0: 666f 720a 2020 2020 6d6f 6475 6c65 7320 for. modules │ │ │ │ -000296e0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ -000296f0: 496e 7465 7273 6563 7469 6f6e 0a0a 466f Intersection..Fo │ │ │ │ -00029700: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00029710: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00029720: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00029730: 2a6e 6f74 6520 4772 6164 696e 673a 2047 *note Grading: G │ │ │ │ -00029740: 7261 6469 6e67 2c20 6973 2061 202a 6e6f rading, is a *no │ │ │ │ -00029750: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ -00029760: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ -00029770: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +000295e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000295f0: 202a 2022 4569 7365 6e62 7564 5368 616d * "EisenbudSham │ │ │ │ +00029600: 6173 6854 6f74 616c 282e 2e2e 2c47 7261 ashTotal(...,Gra │ │ │ │ +00029610: 6469 6e67 3d3e 2e2e 2e29 2220 2d2d 2073 ding=>...)" -- s │ │ │ │ +00029620: 6565 202a 6e6f 7465 2045 6973 656e 6275 ee *note Eisenbu │ │ │ │ +00029630: 6453 6861 6d61 7368 546f 7461 6c3a 0a20 dShamashTotal:. │ │ │ │ +00029640: 2020 2045 6973 656e 6275 6453 6861 6d61 EisenbudShama │ │ │ │ +00029650: 7368 546f 7461 6c2c 202d 2d20 5072 6563 shTotal, -- Prec │ │ │ │ +00029660: 7572 736f 7220 636f 6d70 6c65 7820 6f66 ursor complex of │ │ │ │ +00029670: 2074 6f74 616c 2045 7874 0a20 202a 2022 total Ext. * " │ │ │ │ +00029680: 6e65 7745 7874 282e 2e2e 2c47 7261 6469 newExt(...,Gradi │ │ │ │ +00029690: 6e67 3d3e 2e2e 2e29 2220 2d2d 2073 6565 ng=>...)" -- see │ │ │ │ +000296a0: 202a 6e6f 7465 206e 6577 4578 743a 206e *note newExt: n │ │ │ │ +000296b0: 6577 4578 742c 202d 2d20 476c 6f62 616c ewExt, -- Global │ │ │ │ +000296c0: 2045 7874 2066 6f72 0a20 2020 206d 6f64 Ext for. mod │ │ │ │ +000296d0: 756c 6573 206f 7665 7220 6120 636f 6d70 ules over a comp │ │ │ │ +000296e0: 6c65 7465 2049 6e74 6572 7365 6374 696f lete Intersectio │ │ │ │ +000296f0: 6e0a 0a46 6f72 2074 6865 2070 726f 6772 n..For the progr │ │ │ │ +00029700: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +00029710: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +00029720: 6a65 6374 202a 6e6f 7465 2047 7261 6469 ject *note Gradi │ │ │ │ +00029730: 6e67 3a20 4772 6164 696e 672c 2069 7320 ng: Grading, is │ │ │ │ +00029740: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20 a *note symbol: │ │ │ │ +00029750: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +00029760: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ +00029770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000297a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000297c0: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ -000297d0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ -000297e0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ -000297f0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ -00029800: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ -00029810: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -00029820: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -00029830: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -00029840: 6573 6f6c 7574 696f 6e73 2e6d 323a 3332 esolutions.m2:32 │ │ │ │ -00029850: 3136 3a30 2e0a 1f0a 4669 6c65 3a20 436f 16:0....File: Co │ │ │ │ -00029860: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00029870: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -00029880: 666f 2c20 4e6f 6465 3a20 6866 2c20 4e65 fo, Node: hf, Ne │ │ │ │ -00029890: 7874 3a20 6866 4d6f 6475 6c65 4173 4578 xt: hfModuleAsEx │ │ │ │ -000298a0: 742c 2050 7265 763a 2047 7261 6469 6e67 t, Prev: Grading │ │ │ │ -000298b0: 2c20 5570 3a20 546f 700a 0a68 6620 2d2d , Up: Top..hf -- │ │ │ │ -000298c0: 2043 6f6d 7075 7465 7320 7468 6520 6869 Computes the hi │ │ │ │ -000298d0: 6c62 6572 7420 6675 6e63 7469 6f6e 2069 lbert function i │ │ │ │ -000298e0: 6e20 6120 7261 6e67 6520 6f66 2064 6567 n a range of deg │ │ │ │ -000298f0: 7265 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a rees.*********** │ │ │ │ +000297b0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +000297c0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +000297d0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +000297e0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +000297f0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00029800: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +00029810: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +00029820: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +00029830: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +00029840: 6d32 3a33 3231 363a 302e 0a1f 0a46 696c m2:3216:0....Fil │ │ │ │ +00029850: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +00029860: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00029870: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2068 ns.info, Node: h │ │ │ │ +00029880: 662c 204e 6578 743a 2068 664d 6f64 756c f, Next: hfModul │ │ │ │ +00029890: 6541 7345 7874 2c20 5072 6576 3a20 4772 eAsExt, Prev: Gr │ │ │ │ +000298a0: 6164 696e 672c 2055 703a 2054 6f70 0a0a ading, Up: Top.. │ │ │ │ +000298b0: 6866 202d 2d20 436f 6d70 7574 6573 2074 hf -- Computes t │ │ │ │ +000298c0: 6865 2068 696c 6265 7274 2066 756e 6374 he hilbert funct │ │ │ │ +000298d0: 696f 6e20 696e 2061 2072 616e 6765 206f ion in a range o │ │ │ │ +000298e0: 6620 6465 6772 6565 730a 2a2a 2a2a 2a2a f degrees.****** │ │ │ │ +000298f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029910: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00029930: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00029940: 2020 2020 4820 3d20 6866 2873 2c50 290a H = hf(s,P). │ │ │ │ -00029950: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ -00029960: 2020 2a20 732c 2061 202a 6e6f 7465 2073 * s, a *note s │ │ │ │ -00029970: 6571 7565 6e63 653a 2028 4d61 6361 756c equence: (Macaul │ │ │ │ -00029980: 6179 3244 6f63 2953 6571 7565 6e63 652c ay2Doc)Sequence, │ │ │ │ -00029990: 2c20 6f72 204c 6973 740a 2020 2020 2020 , or List. │ │ │ │ -000299a0: 2a20 502c 2061 202a 6e6f 7465 206d 6f64 * P, a *note mod │ │ │ │ -000299b0: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -000299c0: 6f63 294d 6f64 756c 652c 2c20 6772 6164 oc)Module,, grad │ │ │ │ -000299d0: 6564 206d 6f64 756c 650a 2020 2a20 4f75 ed module. * Ou │ │ │ │ -000299e0: 7470 7574 733a 0a20 2020 2020 202a 2048 tputs:. * H │ │ │ │ -000299f0: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -00029a00: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -00029a10: 7374 2c2c 200a 0a57 6179 7320 746f 2075 st,, ..Ways to u │ │ │ │ -00029a20: 7365 2068 663a 0a3d 3d3d 3d3d 3d3d 3d3d se hf:.========= │ │ │ │ -00029a30: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2268 6628 ======.. * "hf( │ │ │ │ -00029a40: 4c69 7374 2c4d 6f64 756c 6529 220a 2020 List,Module)". │ │ │ │ -00029a50: 2a20 2268 6628 5365 7175 656e 6365 2c4d * "hf(Sequence,M │ │ │ │ -00029a60: 6f64 756c 6529 220a 0a46 6f72 2074 6865 odule)"..For the │ │ │ │ -00029a70: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00029a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00029a90: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00029aa0: 2068 663a 2068 662c 2069 7320 6120 2a6e hf: hf, is a *n │ │ │ │ -00029ab0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00029ac0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00029ad0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00029ae0: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00029920: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00029930: 0a20 2020 2020 2020 2048 203d 2068 6628 . H = hf( │ │ │ │ +00029940: 732c 5029 0a20 202a 2049 6e70 7574 733a s,P). * Inputs: │ │ │ │ +00029950: 0a20 2020 2020 202a 2073 2c20 6120 2a6e . * s, a *n │ │ │ │ +00029960: 6f74 6520 7365 7175 656e 6365 3a20 284d ote sequence: (M │ │ │ │ +00029970: 6163 6175 6c61 7932 446f 6329 5365 7175 acaulay2Doc)Sequ │ │ │ │ +00029980: 656e 6365 2c2c 206f 7220 4c69 7374 0a20 ence,, or List. │ │ │ │ +00029990: 2020 2020 202a 2050 2c20 6120 2a6e 6f74 * P, a *not │ │ │ │ +000299a0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +000299b0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +000299c0: 2067 7261 6465 6420 6d6f 6475 6c65 0a20 graded module. │ │ │ │ +000299d0: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +000299e0: 2020 2a20 482c 2061 202a 6e6f 7465 206c * H, a *note l │ │ │ │ +000299f0: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00029a00: 6f63 294c 6973 742c 2c20 0a0a 5761 7973 oc)List,, ..Ways │ │ │ │ +00029a10: 2074 6f20 7573 6520 6866 3a0a 3d3d 3d3d to use hf:.==== │ │ │ │ +00029a20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +00029a30: 2022 6866 284c 6973 742c 4d6f 6475 6c65 "hf(List,Module │ │ │ │ +00029a40: 2922 0a20 202a 2022 6866 2853 6571 7565 )". * "hf(Seque │ │ │ │ +00029a50: 6e63 652c 4d6f 6475 6c65 2922 0a0a 466f nce,Module)"..Fo │ │ │ │ +00029a60: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00029a70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00029a80: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00029a90: 2a6e 6f74 6520 6866 3a20 6866 2c20 6973 *note hf: hf, is │ │ │ │ +00029aa0: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00029ab0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00029ac0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00029ad0: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00029ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00029b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00029b30: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00029b40: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00029b50: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00029b60: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00029b70: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00029b80: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00029b90: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ -00029ba0: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -00029bb0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ -00029bc0: 3435 3932 3a30 2e0a 1f0a 4669 6c65 3a20 4592:0....File: │ │ │ │ -00029bd0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00029be0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00029bf0: 696e 666f 2c20 4e6f 6465 3a20 6866 4d6f info, Node: hfMo │ │ │ │ -00029c00: 6475 6c65 4173 4578 742c 204e 6578 743a duleAsExt, Next: │ │ │ │ -00029c10: 2068 6967 6853 797a 7967 792c 2050 7265 highSyzygy, Pre │ │ │ │ -00029c20: 763a 2068 662c 2055 703a 2054 6f70 0a0a v: hf, Up: Top.. │ │ │ │ -00029c30: 6866 4d6f 6475 6c65 4173 4578 7420 2d2d hfModuleAsExt -- │ │ │ │ -00029c40: 2070 7265 6469 6374 2062 6574 7469 206e predict betti n │ │ │ │ -00029c50: 756d 6265 7273 206f 6620 6d6f 6475 6c65 umbers of module │ │ │ │ -00029c60: 4173 4578 7428 4d2c 5229 0a2a 2a2a 2a2a AsExt(M,R).***** │ │ │ │ +00029b20: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00029b30: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00029b40: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00029b50: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00029b60: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00029b70: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +00029b80: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00029b90: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00029ba0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00029bb0: 732e 6d32 3a34 3539 323a 302e 0a1f 0a46 s.m2:4592:0....F │ │ │ │ +00029bc0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00029bd0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00029be0: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00029bf0: 2068 664d 6f64 756c 6541 7345 7874 2c20 hfModuleAsExt, │ │ │ │ +00029c00: 4e65 7874 3a20 6869 6768 5379 7a79 6779 Next: highSyzygy │ │ │ │ +00029c10: 2c20 5072 6576 3a20 6866 2c20 5570 3a20 , Prev: hf, Up: │ │ │ │ +00029c20: 546f 700a 0a68 664d 6f64 756c 6541 7345 Top..hfModuleAsE │ │ │ │ +00029c30: 7874 202d 2d20 7072 6564 6963 7420 6265 xt -- predict be │ │ │ │ +00029c40: 7474 6920 6e75 6d62 6572 7320 6f66 206d tti numbers of m │ │ │ │ +00029c50: 6f64 756c 6541 7345 7874 284d 2c52 290a oduleAsExt(M,R). │ │ │ │ +00029c60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029c70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00029c80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00029ca0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -00029cb0: 3a20 0a20 2020 2020 2020 2073 6571 203d : . seq = │ │ │ │ -00029cc0: 2068 664d 6f64 756c 6541 7345 7874 286e hfModuleAsExt(n │ │ │ │ -00029cd0: 756d 5661 6c75 6573 2c4d 2c6e 756d 6765 umValues,M,numge │ │ │ │ -00029ce0: 6e73 5229 0a20 202a 2049 6e70 7574 733a nsR). * Inputs: │ │ │ │ -00029cf0: 0a20 2020 2020 202a 206e 756d 5661 6c75 . * numValu │ │ │ │ -00029d00: 6573 2c20 616e 202a 6e6f 7465 2069 6e74 es, an *note int │ │ │ │ -00029d10: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -00029d20: 446f 6329 5a5a 2c2c 206e 756d 6265 7220 Doc)ZZ,, number │ │ │ │ -00029d30: 6f66 2076 616c 7565 7320 746f 0a20 2020 of values to. │ │ │ │ -00029d40: 2020 2020 2063 6f6d 7075 7465 0a20 2020 compute. │ │ │ │ -00029d50: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ -00029d60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ -00029d70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206d y2Doc)Module,, m │ │ │ │ -00029d80: 6f64 756c 6520 6f76 6572 2074 6865 2072 odule over the r │ │ │ │ -00029d90: 696e 6720 6f66 0a20 2020 2020 2020 206f ing of. o │ │ │ │ -00029da0: 7065 7261 746f 7273 0a20 2020 2020 202a perators. * │ │ │ │ -00029db0: 206e 756d 6765 6e73 522c 2061 6e20 2a6e numgensR, an *n │ │ │ │ -00029dc0: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -00029dd0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -00029de0: 6e75 6d62 6572 206f 6620 6765 6e65 7261 number of genera │ │ │ │ -00029df0: 746f 7273 206f 660a 2020 2020 2020 2020 tors of. │ │ │ │ -00029e00: 7468 6520 7461 7267 6574 2072 696e 670a the target ring. │ │ │ │ -00029e10: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00029e20: 2020 202a 2073 6571 2c20 6120 2a6e 6f74 * seq, a *not │ │ │ │ -00029e30: 6520 7365 7175 656e 6365 3a20 284d 6163 e sequence: (Mac │ │ │ │ -00029e40: 6175 6c61 7932 446f 6329 5365 7175 656e aulay2Doc)Sequen │ │ │ │ -00029e50: 6365 2c2c 2073 6571 7565 6e63 6520 6f66 ce,, sequence of │ │ │ │ -00029e60: 206e 756d 5661 6c75 6573 0a20 2020 2020 numValues. │ │ │ │ -00029e70: 2020 2069 6e74 6567 6572 732c 2074 6865 integers, the │ │ │ │ -00029e80: 2065 7870 6563 7465 6420 746f 7461 6c20 expected total │ │ │ │ -00029e90: 4265 7474 6920 6e75 6d62 6572 730a 0a44 Betti numbers..D │ │ │ │ -00029ea0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00029eb0: 3d3d 3d3d 3d3d 0a0a 4769 7665 6e20 6120 ======..Given a │ │ │ │ -00029ec0: 6d6f 6475 6c65 204d 206f 7665 7220 7468 module M over th │ │ │ │ -00029ed0: 6520 7269 6e67 206f 6620 6f70 6572 6174 e ring of operat │ │ │ │ -00029ee0: 6f72 7320 246b 5b78 5f31 2e2e 785f 635d ors $k[x_1..x_c] │ │ │ │ -00029ef0: 242c 2074 6865 2063 616c 6c20 244e 203d $, the call $N = │ │ │ │ -00029f00: 0a6d 6f64 756c 6541 7345 7874 284d 2c52 .moduleAsExt(M,R │ │ │ │ -00029f10: 2924 2070 726f 6475 6365 7320 6120 6d6f )$ produces a mo │ │ │ │ -00029f20: 6475 6c65 204e 206f 7665 7220 7468 6520 dule N over the │ │ │ │ -00029f30: 7269 6e67 2052 2077 686f 7365 2065 7874 ring R whose ext │ │ │ │ -00029f40: 206d 6f64 756c 6520 6973 2074 6865 0a65 module is the.e │ │ │ │ -00029f50: 7874 6572 696f 7220 616c 6765 6272 6120 xterior algebra │ │ │ │ -00029f60: 6f6e 206e 3d6e 756d 6765 6e73 5220 6765 on n=numgensR ge │ │ │ │ -00029f70: 6e65 7261 746f 7273 2074 656e 736f 7265 nerators tensore │ │ │ │ -00029f80: 6420 7769 7468 204d 2e20 5468 6973 2073 d with M. This s │ │ │ │ -00029f90: 6372 6970 7420 636f 6d70 7574 6573 0a6e cript computes.n │ │ │ │ -00029fa0: 756d 5661 6c75 6573 2076 616c 7565 7320 umValues values │ │ │ │ -00029fb0: 6f66 2074 6865 2048 696c 6265 7274 2066 of the Hilbert f │ │ │ │ -00029fc0: 756e 6374 696f 6e20 6f66 2024 2420 4d20 unction of $$ M │ │ │ │ -00029fd0: 5c6f 7469 6d65 7320 5c77 6564 6765 206b \otimes \wedge k │ │ │ │ -00029fe0: 5e6e 2c20 2424 2077 6869 6368 0a73 686f ^n, $$ which.sho │ │ │ │ -00029ff0: 756c 6420 6265 2065 7175 616c 2074 6f20 uld be equal to │ │ │ │ -0002a000: 7468 6520 746f 7461 6c20 6265 7474 6920 the total betti │ │ │ │ -0002a010: 6e75 6d62 6572 7320 6f66 204e 2e0a 0a2b numbers of N...+ │ │ │ │ +00029c90: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00029ca0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +00029cb0: 7365 7120 3d20 6866 4d6f 6475 6c65 4173 seq = hfModuleAs │ │ │ │ +00029cc0: 4578 7428 6e75 6d56 616c 7565 732c 4d2c Ext(numValues,M, │ │ │ │ +00029cd0: 6e75 6d67 656e 7352 290a 2020 2a20 496e numgensR). * In │ │ │ │ +00029ce0: 7075 7473 3a0a 2020 2020 2020 2a20 6e75 puts:. * nu │ │ │ │ +00029cf0: 6d56 616c 7565 732c 2061 6e20 2a6e 6f74 mValues, an *not │ │ │ │ +00029d00: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ +00029d10: 756c 6179 3244 6f63 295a 5a2c 2c20 6e75 ulay2Doc)ZZ,, nu │ │ │ │ +00029d20: 6d62 6572 206f 6620 7661 6c75 6573 2074 mber of values t │ │ │ │ +00029d30: 6f0a 2020 2020 2020 2020 636f 6d70 7574 o. comput │ │ │ │ +00029d40: 650a 2020 2020 2020 2a20 4d2c 2061 202a e. * M, a * │ │ │ │ +00029d50: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ +00029d60: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ +00029d70: 652c 2c20 6d6f 6475 6c65 206f 7665 7220 e,, module over │ │ │ │ +00029d80: 7468 6520 7269 6e67 206f 660a 2020 2020 the ring of. │ │ │ │ +00029d90: 2020 2020 6f70 6572 6174 6f72 730a 2020 operators. │ │ │ │ +00029da0: 2020 2020 2a20 6e75 6d67 656e 7352 2c20 * numgensR, │ │ │ │ +00029db0: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +00029dc0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00029dd0: 5a5a 2c2c 206e 756d 6265 7220 6f66 2067 ZZ,, number of g │ │ │ │ +00029de0: 656e 6572 6174 6f72 7320 6f66 0a20 2020 enerators of. │ │ │ │ +00029df0: 2020 2020 2074 6865 2074 6172 6765 7420 the target │ │ │ │ +00029e00: 7269 6e67 0a20 202a 204f 7574 7075 7473 ring. * Outputs │ │ │ │ +00029e10: 3a0a 2020 2020 2020 2a20 7365 712c 2061 :. * seq, a │ │ │ │ +00029e20: 202a 6e6f 7465 2073 6571 7565 6e63 653a *note sequence: │ │ │ │ +00029e30: 2028 4d61 6361 756c 6179 3244 6f63 2953 (Macaulay2Doc)S │ │ │ │ +00029e40: 6571 7565 6e63 652c 2c20 7365 7175 656e equence,, sequen │ │ │ │ +00029e50: 6365 206f 6620 6e75 6d56 616c 7565 730a ce of numValues. │ │ │ │ +00029e60: 2020 2020 2020 2020 696e 7465 6765 7273 integers │ │ │ │ +00029e70: 2c20 7468 6520 6578 7065 6374 6564 2074 , the expected t │ │ │ │ +00029e80: 6f74 616c 2042 6574 7469 206e 756d 6265 otal Betti numbe │ │ │ │ +00029e90: 7273 0a0a 4465 7363 7269 7074 696f 6e0a rs..Description. │ │ │ │ +00029ea0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 ===========..Giv │ │ │ │ +00029eb0: 656e 2061 206d 6f64 756c 6520 4d20 6f76 en a module M ov │ │ │ │ +00029ec0: 6572 2074 6865 2072 696e 6720 6f66 206f er the ring of o │ │ │ │ +00029ed0: 7065 7261 746f 7273 2024 6b5b 785f 312e perators $k[x_1. │ │ │ │ +00029ee0: 2e78 5f63 5d24 2c20 7468 6520 6361 6c6c .x_c]$, the call │ │ │ │ +00029ef0: 2024 4e20 3d0a 6d6f 6475 6c65 4173 4578 $N =.moduleAsEx │ │ │ │ +00029f00: 7428 4d2c 5229 2420 7072 6f64 7563 6573 t(M,R)$ produces │ │ │ │ +00029f10: 2061 206d 6f64 756c 6520 4e20 6f76 6572 a module N over │ │ │ │ +00029f20: 2074 6865 2072 696e 6720 5220 7768 6f73 the ring R whos │ │ │ │ +00029f30: 6520 6578 7420 6d6f 6475 6c65 2069 7320 e ext module is │ │ │ │ +00029f40: 7468 650a 6578 7465 7269 6f72 2061 6c67 the.exterior alg │ │ │ │ +00029f50: 6562 7261 206f 6e20 6e3d 6e75 6d67 656e ebra on n=numgen │ │ │ │ +00029f60: 7352 2067 656e 6572 6174 6f72 7320 7465 sR generators te │ │ │ │ +00029f70: 6e73 6f72 6564 2077 6974 6820 4d2e 2054 nsored with M. T │ │ │ │ +00029f80: 6869 7320 7363 7269 7074 2063 6f6d 7075 his script compu │ │ │ │ +00029f90: 7465 730a 6e75 6d56 616c 7565 7320 7661 tes.numValues va │ │ │ │ +00029fa0: 6c75 6573 206f 6620 7468 6520 4869 6c62 lues of the Hilb │ │ │ │ +00029fb0: 6572 7420 6675 6e63 7469 6f6e 206f 6620 ert function of │ │ │ │ +00029fc0: 2424 204d 205c 6f74 696d 6573 205c 7765 $$ M \otimes \we │ │ │ │ +00029fd0: 6467 6520 6b5e 6e2c 2024 2420 7768 6963 dge k^n, $$ whic │ │ │ │ +00029fe0: 680a 7368 6f75 6c64 2062 6520 6571 7561 h.should be equa │ │ │ │ +00029ff0: 6c20 746f 2074 6865 2074 6f74 616c 2062 l to the total b │ │ │ │ +0002a000: 6574 7469 206e 756d 6265 7273 206f 6620 etti numbers of │ │ │ │ +0002a010: 4e2e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d N...+----------- │ │ │ │ 0002a020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a050: 2b0a 7c69 3120 3a20 6b6b 203d 205a 5a2f +.|i1 : kk = ZZ/ │ │ │ │ -0002a060: 3130 313b 2020 2020 2020 2020 2020 2020 101; │ │ │ │ -0002a070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a080: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002a040: 2d2d 2d2d 2d2b 0a7c 6931 203a 206b 6b20 -----+.|i1 : kk │ │ │ │ +0002a050: 3d20 5a5a 2f31 3031 3b20 2020 2020 2020 = ZZ/101; │ │ │ │ +0002a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a070: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a0b0: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 5320 ------+.|i2 : S │ │ │ │ -0002a0c0: 3d20 6b6b 5b61 2c62 2c63 5d3b 2020 2020 = kk[a,b,c]; │ │ │ │ -0002a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a0e0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +0002a0b0: 203a 2053 203d 206b 6b5b 612c 622c 635d : S = kk[a,b,c] │ │ │ │ +0002a0c0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0002a0d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002a0e0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0002a0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002a120: 3320 3a20 6666 203d 206d 6174 7269 787b 3 : ff = matrix{ │ │ │ │ -0002a130: 7b61 5e34 2c20 625e 342c 635e 347d 7d3b {a^4, b^4,c^4}}; │ │ │ │ -0002a140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002a150: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002a110: 2d2b 0a7c 6933 203a 2066 6620 3d20 6d61 -+.|i3 : ff = ma │ │ │ │ +0002a120: 7472 6978 7b7b 615e 342c 2062 5e34 2c63 trix{{a^4, b^4,c │ │ │ │ +0002a130: 5e34 7d7d 3b20 2020 2020 2020 2020 2020 ^4}}; │ │ │ │ +0002a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a180: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002a190: 2020 3120 2020 2020 2033 2020 2020 2020 1 3 │ │ │ │ -0002a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1b0: 2020 2020 207c 0a7c 6f33 203a 204d 6174 |.|o3 : Mat │ │ │ │ -0002a1c0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ -0002a1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a1e0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002a170: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a180: 2020 2020 2020 2031 2020 2020 2020 3320 1 3 │ │ │ │ +0002a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a1a0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +0002a1b0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ +0002a1c0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0002a1d0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -0002a220: 203a 2052 203d 2053 2f69 6465 616c 2066 : R = S/ideal f │ │ │ │ -0002a230: 663b 2020 2020 2020 2020 2020 2020 2020 f; │ │ │ │ -0002a240: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a250: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002a210: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964 +.|i4 : R = S/id │ │ │ │ +0002a220: 6561 6c20 6666 3b20 2020 2020 2020 2020 eal ff; │ │ │ │ +0002a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a240: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002a250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a280: 2d2b 0a7c 6935 203a 204f 7073 203d 206b -+.|i5 : Ops = k │ │ │ │ -0002a290: 6b5b 785f 312c 785f 322c 785f 335d 3b20 k[x_1,x_2,x_3]; │ │ │ │ -0002a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a2b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002a270: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 ------+.|i5 : Op │ │ │ │ +0002a280: 7320 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 s = kk[x_1,x_2,x │ │ │ │ +0002a290: 5f33 5d3b 2020 2020 2020 2020 2020 2020 _3]; │ │ │ │ +0002a2a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002a2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d -------+.|i6 : M │ │ │ │ -0002a2f0: 4d20 3d20 4f70 735e 312f 2878 5f31 2a69 M = Ops^1/(x_1*i │ │ │ │ -0002a300: 6465 616c 2878 5f32 5e32 2c78 5f33 2929 deal(x_2^2,x_3)) │ │ │ │ -0002a310: 3b20 2020 2020 2020 2020 7c0a 2b2d 2d2d ; |.+--- │ │ │ │ +0002a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002a2e0: 3620 3a20 4d4d 203d 204f 7073 5e31 2f28 6 : MM = Ops^1/( │ │ │ │ +0002a2f0: 785f 312a 6964 6561 6c28 785f 325e 322c x_1*ideal(x_2^2, │ │ │ │ +0002a300: 785f 3329 293b 2020 2020 2020 2020 207c x_3)); | │ │ │ │ +0002a310: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002a350: 6937 203a 204e 203d 206d 6f64 756c 6541 i7 : N = moduleA │ │ │ │ -0002a360: 7345 7874 284d 4d2c 5229 3b20 2020 2020 sExt(MM,R); │ │ │ │ -0002a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a380: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002a340: 2d2d 2b0a 7c69 3720 3a20 4e20 3d20 6d6f --+.|i7 : N = mo │ │ │ │ +0002a350: 6475 6c65 4173 4578 7428 4d4d 2c52 293b duleAsExt(MM,R); │ │ │ │ +0002a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002a370: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002a380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a3b0: 2d2d 2d2b 0a7c 6938 203a 2062 6574 7469 ---+.|i8 : betti │ │ │ │ -0002a3c0: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ -0002a3d0: 204e 2c20 4c65 6e67 7468 4c69 6d69 7420 N, LengthLimit │ │ │ │ -0002a3e0: 3d3e 2031 3029 7c0a 7c20 2020 2020 2020 => 10)|.| │ │ │ │ +0002a3a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ +0002a3b0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ +0002a3c0: 7469 6f6e 2820 4e2c 204c 656e 6774 684c tion( N, LengthL │ │ │ │ +0002a3d0: 696d 6974 203d 3e20 3130 297c 0a7c 2020 imit => 10)|.| │ │ │ │ +0002a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a410: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0002a420: 2020 2020 2020 2020 2030 2020 3120 2032 0 1 2 │ │ │ │ -0002a430: 2020 3320 2034 2020 3520 2036 2020 3720 3 4 5 6 7 │ │ │ │ -0002a440: 2038 2020 3920 3130 2020 2020 7c0a 7c6f 8 9 10 |.|o │ │ │ │ -0002a450: 3820 3d20 746f 7461 6c3a 2033 3620 3237 8 = total: 36 27 │ │ │ │ -0002a460: 2032 3920 3331 2033 3320 3335 2033 3720 29 31 33 35 37 │ │ │ │ -0002a470: 3339 2034 3120 3433 2034 3520 2020 207c 39 41 43 45 | │ │ │ │ -0002a480: 0a7c 2020 2020 2020 2020 2d36 3a20 3138 .| -6: 18 │ │ │ │ -0002a490: 2020 3620 202e 2020 2e20 202e 2020 2e20 6 . . . . │ │ │ │ -0002a4a0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a4b0: 2020 7c0a 7c20 2020 2020 2020 202d 353a |.| -5: │ │ │ │ -0002a4c0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0002a4d0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a4e0: 2e20 2020 207c 0a7c 2020 2020 2020 2020 . |.| │ │ │ │ -0002a4f0: 2d34 3a20 3138 2032 3120 3231 2020 3720 -4: 18 21 21 7 │ │ │ │ -0002a500: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a510: 2e20 202e 2020 2020 7c0a 7c20 2020 2020 . . |.| │ │ │ │ -0002a520: 2020 202d 333a 2020 2e20 202e 2020 2e20 -3: . . . │ │ │ │ -0002a530: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a540: 2e20 202e 2020 2e20 2020 207c 0a7c 2020 . . . |.| │ │ │ │ -0002a550: 2020 2020 2020 2d32 3a20 202e 2020 2e20 -2: . . │ │ │ │ -0002a560: 2038 2032 3420 3234 2020 3820 202e 2020 8 24 24 8 . │ │ │ │ -0002a570: 2e20 202e 2020 2e20 202e 2020 2020 7c0a . . . . |. │ │ │ │ -0002a580: 7c20 2020 2020 2020 202d 313a 2020 2e20 | -1: . │ │ │ │ -0002a590: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0002a5a0: 2e20 202e 2020 2e20 202e 2020 2e20 2020 . . . . . │ │ │ │ -0002a5b0: 207c 0a7c 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ -0002a5c0: 202e 2020 2e20 202e 2020 2e20 2039 2032 . . . . 9 2 │ │ │ │ -0002a5d0: 3720 3237 2020 3920 202e 2020 2e20 202e 7 27 9 . . . │ │ │ │ -0002a5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002a5f0: 313a 2020 2e20 202e 2020 2e20 202e 2020 1: . . . . │ │ │ │ -0002a600: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a610: 2020 2e20 2020 207c 0a7c 2020 2020 2020 . |.| │ │ │ │ -0002a620: 2020 2032 3a20 202e 2020 2e20 202e 2020 2: . . . │ │ │ │ -0002a630: 2e20 202e 2020 2e20 3130 2033 3020 3330 . . . 10 30 30 │ │ │ │ -0002a640: 2031 3020 202e 2020 2020 7c0a 7c20 2020 10 . |.| │ │ │ │ -0002a650: 2020 2020 2020 333a 2020 2e20 202e 2020 3: . . │ │ │ │ -0002a660: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a670: 2020 2e20 202e 2020 2e20 2020 207c 0a7c . . . |.| │ │ │ │ -0002a680: 2020 2020 2020 2020 2034 3a20 202e 2020 4: . │ │ │ │ -0002a690: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a6a0: 2020 2e20 3131 2033 3320 3333 2020 2020 . 11 33 33 │ │ │ │ -0002a6b0: 7c0a 7c20 2020 2020 2020 2020 353a 2020 |.| 5: │ │ │ │ -0002a6c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0002a6d0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0002a6e0: 2020 207c 0a7c 2020 2020 2020 2020 2036 |.| 6 │ │ │ │ -0002a6f0: 3a20 202e 2020 2e20 202e 2020 2e20 202e : . . . . . │ │ │ │ -0002a700: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0002a710: 3132 2020 2020 7c0a 7c20 2020 2020 2020 12 |.| │ │ │ │ +0002a400: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002a410: 7c20 2020 2020 2020 2020 2020 2020 3020 | 0 │ │ │ │ +0002a420: 2031 2020 3220 2033 2020 3420 2035 2020 1 2 3 4 5 │ │ │ │ +0002a430: 3620 2037 2020 3820 2039 2031 3020 2020 6 7 8 9 10 │ │ │ │ +0002a440: 207c 0a7c 6f38 203d 2074 6f74 616c 3a20 |.|o8 = total: │ │ │ │ +0002a450: 3336 2032 3720 3239 2033 3120 3333 2033 36 27 29 31 33 3 │ │ │ │ +0002a460: 3520 3337 2033 3920 3431 2034 3320 3435 5 37 39 41 43 45 │ │ │ │ +0002a470: 2020 2020 7c0a 7c20 2020 2020 2020 202d |.| - │ │ │ │ +0002a480: 363a 2031 3820 2036 2020 2e20 202e 2020 6: 18 6 . . │ │ │ │ +0002a490: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0002a4a0: 2020 2e20 2020 207c 0a7c 2020 2020 2020 . |.| │ │ │ │ +0002a4b0: 2020 2d35 3a20 202e 2020 2e20 202e 2020 -5: . . . │ │ │ │ +0002a4c0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0002a4d0: 2020 2e20 202e 2020 2020 7c0a 7c20 2020 . . |.| │ │ │ │ +0002a4e0: 2020 2020 202d 343a 2031 3820 3231 2032 -4: 18 21 2 │ │ │ │ +0002a4f0: 3120 2037 2020 2e20 202e 2020 2e20 202e 1 7 . . . . │ │ │ │ +0002a500: 2020 2e20 202e 2020 2e20 2020 207c 0a7c . . . |.| │ │ │ │ +0002a510: 2020 2020 2020 2020 2d33 3a20 202e 2020 -3: . │ │ │ │ +0002a520: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0002a530: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ +0002a540: 7c0a 7c20 2020 2020 2020 202d 323a 2020 |.| -2: │ │ │ │ +0002a550: 2e20 202e 2020 3820 3234 2032 3420 2038 . . 8 24 24 8 │ │ │ │ +0002a560: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a570: 2020 207c 0a7c 2020 2020 2020 2020 2d31 |.| -1 │ │ │ │ +0002a580: 3a20 202e 2020 2e20 202e 2020 2e20 202e : . . . . . │ │ │ │ +0002a590: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a5a0: 202e 2020 2020 7c0a 7c20 2020 2020 2020 . |.| │ │ │ │ +0002a5b0: 2020 303a 2020 2e20 202e 2020 2e20 202e 0: . . . . │ │ │ │ +0002a5c0: 2020 3920 3237 2032 3720 2039 2020 2e20 9 27 27 9 . │ │ │ │ +0002a5d0: 202e 2020 2e20 2020 207c 0a7c 2020 2020 . . |.| │ │ │ │ +0002a5e0: 2020 2020 2031 3a20 202e 2020 2e20 202e 1: . . . │ │ │ │ +0002a5f0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a600: 202e 2020 2e20 202e 2020 2020 7c0a 7c20 . . . |.| │ │ │ │ +0002a610: 2020 2020 2020 2020 323a 2020 2e20 202e 2: . . │ │ │ │ +0002a620: 2020 2e20 202e 2020 2e20 202e 2031 3020 . . . . 10 │ │ │ │ +0002a630: 3330 2033 3020 3130 2020 2e20 2020 207c 30 30 10 . | │ │ │ │ +0002a640: 0a7c 2020 2020 2020 2020 2033 3a20 202e .| 3: . │ │ │ │ +0002a650: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a660: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002a670: 2020 7c0a 7c20 2020 2020 2020 2020 343a |.| 4: │ │ │ │ +0002a680: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002a690: 202e 2020 2e20 202e 2031 3120 3333 2033 . . . 11 33 3 │ │ │ │ +0002a6a0: 3320 2020 207c 0a7c 2020 2020 2020 2020 3 |.| │ │ │ │ +0002a6b0: 2035 3a20 202e 2020 2e20 202e 2020 2e20 5: . . . . │ │ │ │ +0002a6c0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002a6d0: 2e20 202e 2020 2020 7c0a 7c20 2020 2020 . . |.| │ │ │ │ +0002a6e0: 2020 2020 363a 2020 2e20 202e 2020 2e20 6: . . . │ │ │ │ +0002a6f0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002a700: 2e20 202e 2031 3220 2020 207c 0a7c 2020 . . 12 |.| │ │ │ │ +0002a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a740: 2020 2020 2020 2020 207c 0a7c 6f38 203a |.|o8 : │ │ │ │ -0002a750: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002a730: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002a740: 7c6f 3820 3a20 4265 7474 6954 616c 6c79 |o8 : BettiTally │ │ │ │ +0002a750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a770: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002a770: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002a780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002a7b0: 0a7c 6939 203a 2068 664d 6f64 756c 6541 .|i9 : hfModuleA │ │ │ │ -0002a7c0: 7345 7874 2831 322c 4d4d 2c33 2920 2020 sExt(12,MM,3) │ │ │ │ -0002a7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a7e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002a7a0: 2d2d 2d2d 2b0a 7c69 3920 3a20 6866 4d6f ----+.|i9 : hfMo │ │ │ │ +0002a7b0: 6475 6c65 4173 4578 7428 3132 2c4d 4d2c duleAsExt(12,MM, │ │ │ │ +0002a7c0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ +0002a7d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a810: 2020 2020 207c 0a7c 6f39 203d 2028 3233 |.|o9 = (23 │ │ │ │ -0002a820: 2c20 3235 2c20 3237 2c20 3239 2c20 3331 , 25, 27, 29, 31 │ │ │ │ -0002a830: 2c20 3333 2c20 3335 2c20 3337 2c20 3339 , 33, 35, 37, 39 │ │ │ │ -0002a840: 2c20 3431 2920 2020 7c0a 7c20 2020 2020 , 41) |.| │ │ │ │ +0002a800: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ +0002a810: 3d20 2832 332c 2032 352c 2032 372c 2032 = (23, 25, 27, 2 │ │ │ │ +0002a820: 392c 2033 312c 2033 332c 2033 352c 2033 9, 31, 33, 35, 3 │ │ │ │ +0002a830: 372c 2033 392c 2034 3129 2020 207c 0a7c 7, 39, 41) |.| │ │ │ │ +0002a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a870: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -0002a880: 203a 2053 6571 7565 6e63 6520 2020 2020 : Sequence │ │ │ │ +0002a870: 7c0a 7c6f 3920 3a20 5365 7175 656e 6365 |.|o9 : Sequence │ │ │ │ +0002a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002a890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002a8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002a8b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002a8a0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002a8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002a8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002a8e0: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ -0002a8f0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0002a900: 206d 6f64 756c 6541 7345 7874 3a20 6d6f moduleAsExt: mo │ │ │ │ -0002a910: 6475 6c65 4173 4578 742c 202d 2d20 4669 duleAsExt, -- Fi │ │ │ │ -0002a920: 6e64 2061 206d 6f64 756c 6520 7769 7468 nd a module with │ │ │ │ -0002a930: 2067 6976 656e 2061 7379 6d70 746f 7469 given asymptoti │ │ │ │ -0002a940: 630a 2020 2020 7265 736f 6c75 7469 6f6e c. resolution │ │ │ │ -0002a950: 0a0a 5761 7973 2074 6f20 7573 6520 6866 ..Ways to use hf │ │ │ │ -0002a960: 4d6f 6475 6c65 4173 4578 743a 0a3d 3d3d ModuleAsExt:.=== │ │ │ │ -0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002a980: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6866 =======.. * "hf │ │ │ │ -0002a990: 4d6f 6475 6c65 4173 4578 7428 5a5a 2c4d ModuleAsExt(ZZ,M │ │ │ │ -0002a9a0: 6f64 756c 652c 5a5a 2922 0a0a 466f 7220 odule,ZZ)"..For │ │ │ │ -0002a9b0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0002a9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002a9d0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0002a9e0: 6f74 6520 6866 4d6f 6475 6c65 4173 4578 ote hfModuleAsEx │ │ │ │ -0002a9f0: 743a 2068 664d 6f64 756c 6541 7345 7874 t: hfModuleAsExt │ │ │ │ -0002aa00: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0002aa10: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0002aa20: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0002aa30: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0002a8d0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +0002a8e0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0002a8f0: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578 *note moduleAsEx │ │ │ │ +0002a900: 743a 206d 6f64 756c 6541 7345 7874 2c20 t: moduleAsExt, │ │ │ │ +0002a910: 2d2d 2046 696e 6420 6120 6d6f 6475 6c65 -- Find a module │ │ │ │ +0002a920: 2077 6974 6820 6769 7665 6e20 6173 796d with given asym │ │ │ │ +0002a930: 7074 6f74 6963 0a20 2020 2072 6573 6f6c ptotic. resol │ │ │ │ +0002a940: 7574 696f 6e0a 0a57 6179 7320 746f 2075 ution..Ways to u │ │ │ │ +0002a950: 7365 2068 664d 6f64 756c 6541 7345 7874 se hfModuleAsExt │ │ │ │ +0002a960: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +0002a970: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0002a980: 2a20 2268 664d 6f64 756c 6541 7345 7874 * "hfModuleAsExt │ │ │ │ +0002a990: 285a 5a2c 4d6f 6475 6c65 2c5a 5a29 220a (ZZ,Module,ZZ)". │ │ │ │ +0002a9a0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0002a9b0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0002a9c0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0002a9d0: 6374 202a 6e6f 7465 2068 664d 6f64 756c ct *note hfModul │ │ │ │ +0002a9e0: 6541 7345 7874 3a20 6866 4d6f 6475 6c65 eAsExt: hfModule │ │ │ │ +0002a9f0: 4173 4578 742c 2069 7320 6120 2a6e 6f74 AsExt, is a *not │ │ │ │ +0002aa00: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0002aa10: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0002aa20: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0002aa30: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0002aa40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002aa70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002aa80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0002aa90: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0002aaa0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0002aab0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0002aac0: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0002aad0: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -0002aae0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0002aaf0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0002ab00: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0002ab10: 7469 6f6e 732e 6d32 3a33 3134 313a 302e tions.m2:3141:0. │ │ │ │ -0002ab20: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0002ab30: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0002ab40: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0002ab50: 6f64 653a 2068 6967 6853 797a 7967 792c ode: highSyzygy, │ │ │ │ -0002ab60: 204e 6578 743a 2068 4d61 7073 2c20 5072 Next: hMaps, Pr │ │ │ │ -0002ab70: 6576 3a20 6866 4d6f 6475 6c65 4173 4578 ev: hfModuleAsEx │ │ │ │ -0002ab80: 742c 2055 703a 2054 6f70 0a0a 6869 6768 t, Up: Top..high │ │ │ │ -0002ab90: 5379 7a79 6779 202d 2d20 5265 7475 726e Syzygy -- Return │ │ │ │ -0002aba0: 7320 6120 7379 7a79 6779 206d 6f64 756c s a syzygy modul │ │ │ │ -0002abb0: 6520 6f6e 6520 6265 796f 6e64 2074 6865 e one beyond the │ │ │ │ -0002abc0: 2072 6567 756c 6172 6974 7920 6f66 2045 regularity of E │ │ │ │ -0002abd0: 7874 284d 2c6b 290a 2a2a 2a2a 2a2a 2a2a xt(M,k).******** │ │ │ │ +0002aa80: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0002aa90: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0002aaa0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0002aab0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0002aac0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +0002aad0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0002aae0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +0002aaf0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0002ab00: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331 esolutions.m2:31 │ │ │ │ +0002ab10: 3431 3a30 2e0a 1f0a 4669 6c65 3a20 436f 41:0....File: Co │ │ │ │ +0002ab20: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0002ab30: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +0002ab40: 666f 2c20 4e6f 6465 3a20 6869 6768 5379 fo, Node: highSy │ │ │ │ +0002ab50: 7a79 6779 2c20 4e65 7874 3a20 684d 6170 zygy, Next: hMap │ │ │ │ +0002ab60: 732c 2050 7265 763a 2068 664d 6f64 756c s, Prev: hfModul │ │ │ │ +0002ab70: 6541 7345 7874 2c20 5570 3a20 546f 700a eAsExt, Up: Top. │ │ │ │ +0002ab80: 0a68 6967 6853 797a 7967 7920 2d2d 2052 .highSyzygy -- R │ │ │ │ +0002ab90: 6574 7572 6e73 2061 2073 797a 7967 7920 eturns a syzygy │ │ │ │ +0002aba0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e module one beyon │ │ │ │ +0002abb0: 6420 7468 6520 7265 6775 6c61 7269 7479 d the regularity │ │ │ │ +0002abc0: 206f 6620 4578 7428 4d2c 6b29 0a2a 2a2a of Ext(M,k).*** │ │ │ │ +0002abd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002abe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002abf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002ac00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ac10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ac20: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -0002ac30: 0a20 2020 2020 2020 204d 203d 2068 6967 . M = hig │ │ │ │ -0002ac40: 6853 797a 7967 7920 4d30 0a20 202a 2049 hSyzygy M0. * I │ │ │ │ -0002ac50: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ -0002ac60: 302c 2061 202a 6e6f 7465 206d 6f64 756c 0, a *note modul │ │ │ │ -0002ac70: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0002ac80: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -0002ac90: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0002aca0: 6563 7469 6f6e 0a20 2020 2020 2020 2072 ection. r │ │ │ │ -0002acb0: 696e 670a 2020 2a20 2a6e 6f74 6520 4f70 ing. * *note Op │ │ │ │ -0002acc0: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -0002acd0: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -0002ace0: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -0002acf0: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -0002ad00: 732c 3a0a 2020 2020 2020 2a20 4f70 7469 s,:. * Opti │ │ │ │ -0002ad10: 6d69 736d 203d 3e20 2e2e 2e2c 2064 6566 mism => ..., def │ │ │ │ -0002ad20: 6175 6c74 2076 616c 7565 2030 0a20 202a ault value 0. * │ │ │ │ -0002ad30: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0002ad40: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64 * M, a *note mod │ │ │ │ -0002ad50: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -0002ad60: 6f63 294d 6f64 756c 652c 2c20 6120 7379 oc)Module,, a sy │ │ │ │ -0002ad70: 7a79 6779 206d 6f64 756c 6520 6f66 204d zygy module of M │ │ │ │ -0002ad80: 300a 0a44 6573 6372 6970 7469 6f6e 0a3d 0..Description.= │ │ │ │ -0002ad90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 2268 ==========..A "h │ │ │ │ -0002ada0: 6967 6820 7379 7a79 6779 2220 6f76 6572 igh syzygy" over │ │ │ │ -0002adb0: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -0002adc0: 7273 6563 7469 6f6e 2069 7320 6f6e 6520 rsection is one │ │ │ │ -0002add0: 7375 6368 2074 6861 7420 6765 6e65 7261 such that genera │ │ │ │ -0002ade0: 6c0a 6369 2d6f 7065 7261 746f 7273 2068 l.ci-operators h │ │ │ │ -0002adf0: 6176 6520 7370 6c69 7420 6b65 726e 656c ave split kernel │ │ │ │ -0002ae00: 7320 7768 656e 2061 7070 6c69 6564 2072 s when applied r │ │ │ │ -0002ae10: 6563 7572 7369 7665 6c79 206f 6e20 636f ecursively on co │ │ │ │ -0002ae20: 7379 7a79 6779 2063 6861 696e 7320 6f66 syzygy chains of │ │ │ │ -0002ae30: 0a70 7265 7669 6f75 7320 6b65 726e 656c .previous kernel │ │ │ │ -0002ae40: 732e 0a0a 4966 2070 203d 206d 6642 6f75 s...If p = mfBou │ │ │ │ -0002ae50: 6e64 204d 302c 2074 6865 6e20 6869 6768 nd M0, then high │ │ │ │ -0002ae60: 5379 7a79 6779 204d 3020 7265 7475 726e Syzygy M0 return │ │ │ │ -0002ae70: 7320 7468 6520 702d 7468 2073 797a 7967 s the p-th syzyg │ │ │ │ -0002ae80: 7920 6f66 204d 302e 2028 6966 2046 2069 y of M0. (if F i │ │ │ │ -0002ae90: 7320 610a 7265 736f 6c75 7469 6f6e 206f s a.resolution o │ │ │ │ -0002aea0: 6620 4d20 7468 6973 2069 7320 7468 6520 f M this is the │ │ │ │ -0002aeb0: 636f 6b65 726e 656c 206f 6620 462e 6464 cokernel of F.dd │ │ │ │ -0002aec0: 5f7b 702b 317d 292e 204f 7074 696d 6973 _{p+1}). Optimis │ │ │ │ -0002aed0: 6d20 3d3e 2072 2061 7320 6f70 7469 6f6e m => r as option │ │ │ │ -0002aee0: 616c 0a61 7267 756d 656e 742c 2068 6967 al.argument, hig │ │ │ │ -0002aef0: 6853 797a 7967 7928 4d30 2c4f 7074 696d hSyzygy(M0,Optim │ │ │ │ -0002af00: 6973 6d3d 3e72 2920 7265 7475 726e 7320 ism=>r) returns │ │ │ │ -0002af10: 7468 6520 2870 2d72 292d 7468 2073 797a the (p-r)-th syz │ │ │ │ -0002af20: 7967 792e 2054 6865 2073 6372 6970 7420 ygy. The script │ │ │ │ -0002af30: 6973 0a75 7365 6675 6c20 7769 7468 206d is.useful with m │ │ │ │ -0002af40: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -0002af50: 6f6e 2866 662c 2068 6967 6853 797a 7967 on(ff, highSyzyg │ │ │ │ -0002af60: 7920 4d30 292e 0a0a 2b2d 2d2d 2d2d 2d2d y M0)...+------- │ │ │ │ +0002ac10: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0002ac20: 6167 653a 200a 2020 2020 2020 2020 4d20 age: . M │ │ │ │ +0002ac30: 3d20 6869 6768 5379 7a79 6779 204d 300a = highSyzygy M0. │ │ │ │ +0002ac40: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002ac50: 2020 2a20 4d30 2c20 6120 2a6e 6f74 6520 * M0, a *note │ │ │ │ +0002ac60: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0002ac70: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +0002ac80: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +0002ac90: 6e74 6572 7365 6374 696f 6e0a 2020 2020 ntersection. │ │ │ │ +0002aca0: 2020 2020 7269 6e67 0a20 202a 202a 6e6f ring. * *no │ │ │ │ +0002acb0: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ +0002acc0: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ +0002acd0: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ +0002ace0: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +0002acf0: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ +0002ad00: 204f 7074 696d 6973 6d20 3d3e 202e 2e2e Optimism => ... │ │ │ │ +0002ad10: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ +0002ad20: 300a 2020 2a20 4f75 7470 7574 733a 0a20 0. * Outputs:. │ │ │ │ +0002ad30: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ +0002ad40: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0002ad50: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0002ad60: 2061 2073 797a 7967 7920 6d6f 6475 6c65 a syzygy module │ │ │ │ +0002ad70: 206f 6620 4d30 0a0a 4465 7363 7269 7074 of M0..Descript │ │ │ │ +0002ad80: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0002ad90: 0a41 2022 6869 6768 2073 797a 7967 7922 .A "high syzygy" │ │ │ │ +0002ada0: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0002adb0: 2069 6e74 6572 7365 6374 696f 6e20 6973 intersection is │ │ │ │ +0002adc0: 206f 6e65 2073 7563 6820 7468 6174 2067 one such that g │ │ │ │ +0002add0: 656e 6572 616c 0a63 692d 6f70 6572 6174 eneral.ci-operat │ │ │ │ +0002ade0: 6f72 7320 6861 7665 2073 706c 6974 206b ors have split k │ │ │ │ +0002adf0: 6572 6e65 6c73 2077 6865 6e20 6170 706c ernels when appl │ │ │ │ +0002ae00: 6965 6420 7265 6375 7273 6976 656c 7920 ied recursively │ │ │ │ +0002ae10: 6f6e 2063 6f73 797a 7967 7920 6368 6169 on cosyzygy chai │ │ │ │ +0002ae20: 6e73 206f 660a 7072 6576 696f 7573 206b ns of.previous k │ │ │ │ +0002ae30: 6572 6e65 6c73 2e0a 0a49 6620 7020 3d20 ernels...If p = │ │ │ │ +0002ae40: 6d66 426f 756e 6420 4d30 2c20 7468 656e mfBound M0, then │ │ │ │ +0002ae50: 2068 6967 6853 797a 7967 7920 4d30 2072 highSyzygy M0 r │ │ │ │ +0002ae60: 6574 7572 6e73 2074 6865 2070 2d74 6820 eturns the p-th │ │ │ │ +0002ae70: 7379 7a79 6779 206f 6620 4d30 2e20 2869 syzygy of M0. (i │ │ │ │ +0002ae80: 6620 4620 6973 2061 0a72 6573 6f6c 7574 f F is a.resolut │ │ │ │ +0002ae90: 696f 6e20 6f66 204d 2074 6869 7320 6973 ion of M this is │ │ │ │ +0002aea0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66 the cokernel of │ │ │ │ +0002aeb0: 2046 2e64 645f 7b70 2b31 7d29 2e20 4f70 F.dd_{p+1}). Op │ │ │ │ +0002aec0: 7469 6d69 736d 203d 3e20 7220 6173 206f timism => r as o │ │ │ │ +0002aed0: 7074 696f 6e61 6c0a 6172 6775 6d65 6e74 ptional.argument │ │ │ │ +0002aee0: 2c20 6869 6768 5379 7a79 6779 284d 302c , highSyzygy(M0, │ │ │ │ +0002aef0: 4f70 7469 6d69 736d 3d3e 7229 2072 6574 Optimism=>r) ret │ │ │ │ +0002af00: 7572 6e73 2074 6865 2028 702d 7229 2d74 urns the (p-r)-t │ │ │ │ +0002af10: 6820 7379 7a79 6779 2e20 5468 6520 7363 h syzygy. The sc │ │ │ │ +0002af20: 7269 7074 2069 730a 7573 6566 756c 2077 ript is.useful w │ │ │ │ +0002af30: 6974 6820 6d61 7472 6978 4661 6374 6f72 ith matrixFactor │ │ │ │ +0002af40: 697a 6174 696f 6e28 6666 2c20 6869 6768 ization(ff, high │ │ │ │ +0002af50: 5379 7a79 6779 204d 3029 2e0a 0a2b 2d2d Syzygy M0)...+-- │ │ │ │ +0002af60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002af70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002af80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002afa0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ -0002afb0: 6574 5261 6e64 6f6d 5365 6564 2031 3030 etRandomSeed 100 │ │ │ │ +0002af90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002afa0: 3120 3a20 7365 7452 616e 646f 6d53 6565 1 : setRandomSee │ │ │ │ +0002afb0: 6420 3130 3020 2020 2020 2020 2020 2020 d 100 │ │ │ │ 0002afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002afe0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073 |.| -- s │ │ │ │ -0002aff0: 6574 7469 6e67 2072 616e 646f 6d20 7365 etting random se │ │ │ │ -0002b000: 6564 2074 6f20 3130 3020 2020 2020 2020 ed to 100 │ │ │ │ -0002b010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b020: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002afd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002afe0: 202d 2d20 7365 7474 696e 6720 7261 6e64 -- setting rand │ │ │ │ +0002aff0: 6f6d 2073 6565 6420 746f 2031 3030 2020 om seed to 100 │ │ │ │ +0002b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b010: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b020: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b060: 2020 2020 2020 2020 2020 7c0a 7c6f 3120 |.|o1 │ │ │ │ -0002b070: 3d20 3130 3020 2020 2020 2020 2020 2020 = 100 │ │ │ │ +0002b050: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b060: 0a7c 6f31 203d 2031 3030 2020 2020 2020 .|o1 = 100 │ │ │ │ +0002b070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b0a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002b0a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0002b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002b0f0: 3220 3a20 5320 3d20 5a5a 2f31 3031 5b78 2 : S = ZZ/101[x │ │ │ │ -0002b100: 2c79 2c7a 5d20 2020 2020 2020 2020 2020 ,y,z] │ │ │ │ +0002b0e0: 2d2b 0a7c 6932 203a 2053 203d 205a 5a2f -+.|i2 : S = ZZ/ │ │ │ │ +0002b0f0: 3130 315b 782c 792c 7a5d 2020 2020 2020 101[x,y,z] │ │ │ │ +0002b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b120: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002b120: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b170: 7c6f 3220 3d20 5320 2020 2020 2020 2020 |o2 = S │ │ │ │ +0002b160: 2020 207c 0a7c 6f32 203d 2053 2020 2020 |.|o2 = S │ │ │ │ +0002b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b1b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b1a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b1f0: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69 |.|o2 : Polynomi │ │ │ │ -0002b200: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +0002b1e0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ +0002b1f0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0002b200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b230: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0002b220: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002b230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b270: 2d2d 2b0a 7c69 3320 3a20 6620 3d20 6d61 --+.|i3 : f = ma │ │ │ │ -0002b280: 7472 6978 2278 332c 7933 2b78 332c 7a33 trix"x3,y3+x3,z3 │ │ │ │ -0002b290: 2b78 332b 7933 2220 2020 2020 2020 2020 +x3+y3" │ │ │ │ -0002b2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b260: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 -------+.|i3 : f │ │ │ │ +0002b270: 203d 206d 6174 7269 7822 7833 2c79 332b = matrix"x3,y3+ │ │ │ │ +0002b280: 7833 2c7a 332b 7833 2b79 3322 2020 2020 x3,z3+x3+y3" │ │ │ │ +0002b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b2a0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b2f0: 2020 2020 7c0a 7c6f 3320 3d20 7c20 7833 |.|o3 = | x3 │ │ │ │ -0002b300: 2078 332b 7933 2078 332b 7933 2b7a 3320 x3+y3 x3+y3+z3 │ │ │ │ -0002b310: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002b2e0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ +0002b2f0: 207c 2078 3320 7833 2b79 3320 7833 2b79 | x3 x3+y3 x3+y │ │ │ │ +0002b300: 332b 7a33 207c 2020 2020 2020 2020 2020 3+z3 | │ │ │ │ +0002b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b370: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002b380: 2020 2020 2020 3120 2020 2020 2033 2020 1 3 │ │ │ │ +0002b360: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002b370: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +0002b380: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0002b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3b0: 2020 2020 2020 207c 0a7c 6f33 203a 204d |.|o3 : M │ │ │ │ -0002b3c0: 6174 7269 7820 5320 203c 2d2d 2053 2020 atrix S <-- S │ │ │ │ +0002b3a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0002b3b0: 3320 3a20 4d61 7472 6978 2053 2020 3c2d 3 : Matrix S <- │ │ │ │ +0002b3c0: 2d20 5320 2020 2020 2020 2020 2020 2020 - S │ │ │ │ 0002b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b3f0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002b3e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002b3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b430: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -0002b440: 2066 6620 3d20 662a 7261 6e64 6f6d 2873 ff = f*random(s │ │ │ │ -0002b450: 6f75 7263 6520 662c 2073 6f75 7263 6520 ource f, source │ │ │ │ -0002b460: 6629 2020 2020 2020 2020 2020 2020 2020 f) │ │ │ │ -0002b470: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002b420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002b430: 7c69 3420 3a20 6666 203d 2066 2a72 616e |i4 : ff = f*ran │ │ │ │ +0002b440: 646f 6d28 736f 7572 6365 2066 2c20 736f dom(source f, so │ │ │ │ +0002b450: 7572 6365 2066 2920 2020 2020 2020 2020 urce f) │ │ │ │ +0002b460: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b470: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b4b0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ -0002b4c0: 203d 207c 2031 3078 332d 3232 7933 2d34 = | 10x3-22y3-4 │ │ │ │ -0002b4d0: 7a33 202d 3230 7833 2d32 3079 332d 367a z3 -20x3-20y3-6z │ │ │ │ -0002b4e0: 3320 2d32 3778 332d 3431 7933 2b7a 3320 3 -27x3-41y3+z3 │ │ │ │ -0002b4f0: 7c20 2020 2020 2020 2020 2020 7c0a 7c20 | |.| │ │ │ │ +0002b4b0: 7c0a 7c6f 3420 3d20 7c20 3130 7833 2d32 |.|o4 = | 10x3-2 │ │ │ │ +0002b4c0: 3279 332d 347a 3320 2d32 3078 332d 3230 2y3-4z3 -20x3-20 │ │ │ │ +0002b4d0: 7933 2d36 7a33 202d 3237 7833 2d34 3179 y3-6z3 -27x3-41y │ │ │ │ +0002b4e0: 332b 7a33 207c 2020 2020 2020 2020 2020 3+z3 | │ │ │ │ +0002b4f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b530: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002b540: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0002b550: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ +0002b530: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b540: 2020 3120 2020 2020 2033 2020 2020 2020 1 3 │ │ │ │ +0002b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b570: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b580: 7c6f 3420 3a20 4d61 7472 6978 2053 2020 |o4 : Matrix S │ │ │ │ -0002b590: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0002b570: 2020 207c 0a7c 6f34 203a 204d 6174 7269 |.|o4 : Matri │ │ │ │ +0002b580: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +0002b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b5b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b5c0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002b5b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002b5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b600: 2b0a 7c69 3520 3a20 5220 3d20 532f 6964 +.|i5 : R = S/id │ │ │ │ -0002b610: 6561 6c20 6620 2020 2020 2020 2020 2020 eal f │ │ │ │ +0002b5f0: 2d2d 2d2d 2d2b 0a7c 6935 203a 2052 203d -----+.|i5 : R = │ │ │ │ +0002b600: 2053 2f69 6465 616c 2066 2020 2020 2020 S/ideal f │ │ │ │ +0002b610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b640: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b630: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b680: 2020 7c0a 7c6f 3520 3d20 5220 2020 2020 |.|o5 = R │ │ │ │ +0002b670: 2020 2020 2020 207c 0a7c 6f35 203d 2052 |.|o5 = R │ │ │ │ +0002b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b6b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002b6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b700: 2020 2020 7c0a 7c6f 3520 3a20 5175 6f74 |.|o5 : Quot │ │ │ │ -0002b710: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002b6f0: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +0002b700: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +0002b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b740: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002b730: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002b740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b780: 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 4d30 ------+.|i6 : M0 │ │ │ │ -0002b790: 203d 2052 5e31 2f69 6465 616c 2278 327a = R^1/ideal"x2z │ │ │ │ -0002b7a0: 322c 7879 7a22 2020 2020 2020 2020 2020 2,xyz" │ │ │ │ -0002b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002b770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 -----------+.|i6 │ │ │ │ +0002b780: 203a 204d 3020 3d20 525e 312f 6964 6561 : M0 = R^1/idea │ │ │ │ +0002b790: 6c22 7832 7a32 2c78 797a 2220 2020 2020 l"x2z2,xyz" │ │ │ │ +0002b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b7b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b800: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ -0002b810: 636f 6b65 726e 656c 207c 2078 327a 3220 cokernel | x2z2 │ │ │ │ -0002b820: 7879 7a20 7c20 2020 2020 2020 2020 2020 xyz | │ │ │ │ -0002b830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b840: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002b7f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002b800: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20 o6 = cokernel | │ │ │ │ +0002b810: 7832 7a32 2078 797a 207c 2020 2020 2020 x2z2 xyz | │ │ │ │ +0002b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002b830: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002b840: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b880: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8a0: 2020 2020 2020 2020 2031 2020 2020 2020 1 │ │ │ │ +0002b870: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002b880: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002b890: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +0002b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b8c0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ -0002b8d0: 203a 2052 2d6d 6f64 756c 652c 2071 756f : R-module, quo │ │ │ │ -0002b8e0: 7469 656e 7420 6f66 2052 2020 2020 2020 tient of R │ │ │ │ +0002b8c0: 7c0a 7c6f 3620 3a20 522d 6d6f 6475 6c65 |.|o6 : R-module │ │ │ │ +0002b8d0: 2c20 7175 6f74 6965 6e74 206f 6620 5220 , quotient of R │ │ │ │ +0002b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002b900: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0002b910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0002b950: 6937 203a 2062 6574 7469 2066 7265 6552 i7 : betti freeR │ │ │ │ -0002b960: 6573 6f6c 7574 696f 6e20 284d 302c 204c esolution (M0, L │ │ │ │ -0002b970: 656e 6774 684c 696d 6974 203d 3e20 3729 engthLimit => 7) │ │ │ │ -0002b980: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002b990: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002b940: 2d2d 2b0a 7c69 3720 3a20 6265 7474 6920 --+.|i7 : betti │ │ │ │ +0002b950: 6672 6565 5265 736f 6c75 7469 6f6e 2028 freeResolution ( │ │ │ │ +0002b960: 4d30 2c20 4c65 6e67 7468 4c69 6d69 7420 M0, LengthLimit │ │ │ │ +0002b970: 3d3e 2037 2920 2020 2020 2020 2020 2020 => 7) │ │ │ │ +0002b980: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002b990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002b9c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002b9d0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ -0002b9e0: 3120 3220 2033 2020 3420 2035 2020 3620 1 2 3 4 5 6 │ │ │ │ -0002b9f0: 2037 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ -0002ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba10: 7c0a 7c6f 3720 3d20 746f 7461 6c3a 2031 |.|o7 = total: 1 │ │ │ │ -0002ba20: 2032 2036 2031 3120 3138 2032 3620 3336 2 6 11 18 26 36 │ │ │ │ -0002ba30: 2034 3720 2020 2020 2020 2020 2020 2020 47 │ │ │ │ -0002ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba50: 207c 0a7c 2020 2020 2020 2020 2030 3a20 |.| 0: │ │ │ │ -0002ba60: 3120 2e20 2e20 202e 2020 2e20 202e 2020 1 . . . . . │ │ │ │ -0002ba70: 2e20 202e 2020 2020 2020 2020 2020 2020 . . │ │ │ │ -0002ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ba90: 2020 7c0a 7c20 2020 2020 2020 2020 313a |.| 1: │ │ │ │ -0002baa0: 202e 202e 202e 2020 2e20 202e 2020 2e20 . . . . . . │ │ │ │ -0002bab0: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ -0002bac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bad0: 2020 207c 0a7c 2020 2020 2020 2020 2032 |.| 2 │ │ │ │ -0002bae0: 3a20 2e20 3120 2e20 202e 2020 2e20 202e : . 1 . . . . │ │ │ │ -0002baf0: 2020 2e20 202e 2020 2020 2020 2020 2020 . . │ │ │ │ -0002bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002bb20: 333a 202e 2031 2036 2020 3620 202e 2020 3: . 1 6 6 . │ │ │ │ -0002bb30: 2e20 202e 2020 2e20 2020 2020 2020 2020 . . . │ │ │ │ -0002bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002bb60: 2034 3a20 2e20 2e20 2e20 2035 2031 3820 4: . . . 5 18 │ │ │ │ -0002bb70: 3134 2020 2e20 202e 2020 2020 2020 2020 14 . . │ │ │ │ -0002bb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bb90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002bba0: 2020 353a 202e 202e 202e 2020 2e20 202e 5: . . . . . │ │ │ │ -0002bbb0: 2031 3220 3336 2032 3520 2020 2020 2020 12 36 25 │ │ │ │ -0002bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002bbe0: 2020 2036 3a20 2e20 2e20 2e20 202e 2020 6: . . . . │ │ │ │ -0002bbf0: 2e20 202e 2020 2e20 3232 2020 2020 2020 . . . 22 │ │ │ │ -0002bc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002b9c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002b9d0: 2020 2030 2031 2032 2020 3320 2034 2020 0 1 2 3 4 │ │ │ │ +0002b9e0: 3520 2036 2020 3720 2020 2020 2020 2020 5 6 7 │ │ │ │ +0002b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba00: 2020 2020 207c 0a7c 6f37 203d 2074 6f74 |.|o7 = tot │ │ │ │ +0002ba10: 616c 3a20 3120 3220 3620 3131 2031 3820 al: 1 2 6 11 18 │ │ │ │ +0002ba20: 3236 2033 3620 3437 2020 2020 2020 2020 26 36 47 │ │ │ │ +0002ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba40: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002ba50: 2020 303a 2031 202e 202e 2020 2e20 202e 0: 1 . . . . │ │ │ │ +0002ba60: 2020 2e20 202e 2020 2e20 2020 2020 2020 . . . │ │ │ │ +0002ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ba80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002ba90: 2020 2031 3a20 2e20 2e20 2e20 202e 2020 1: . . . . │ │ │ │ +0002baa0: 2e20 202e 2020 2e20 202e 2020 2020 2020 . . . . │ │ │ │ +0002bab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bac0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002bad0: 2020 2020 323a 202e 2031 202e 2020 2e20 2: . 1 . . │ │ │ │ +0002bae0: 202e 2020 2e20 202e 2020 2e20 2020 2020 . . . . │ │ │ │ +0002baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002bb10: 2020 2020 2033 3a20 2e20 3120 3620 2036 3: . 1 6 6 │ │ │ │ +0002bb20: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ +0002bb30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb40: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002bb50: 2020 2020 2020 343a 202e 202e 202e 2020 4: . . . │ │ │ │ +0002bb60: 3520 3138 2031 3420 202e 2020 2e20 2020 5 18 14 . . │ │ │ │ +0002bb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bb80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002bb90: 2020 2020 2020 2035 3a20 2e20 2e20 2e20 5: . . . │ │ │ │ +0002bba0: 202e 2020 2e20 3132 2033 3620 3235 2020 . . 12 36 25 │ │ │ │ +0002bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bbd0: 2020 2020 2020 2020 363a 202e 202e 202e 6: . . . │ │ │ │ +0002bbe0: 2020 2e20 202e 2020 2e20 202e 2032 3220 . . . . 22 │ │ │ │ +0002bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bc00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002bc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc50: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ -0002bc60: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002bc40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002bc50: 7c6f 3720 3a20 4265 7474 6954 616c 6c79 |o7 : BettiTally │ │ │ │ +0002bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bc90: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002bc80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002bc90: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ -0002bce0: 203a 206d 6642 6f75 6e64 204d 3020 2020 : mfBound M0 │ │ │ │ +0002bcd0: 2b0a 7c69 3820 3a20 6d66 426f 756e 6420 +.|i8 : mfBound │ │ │ │ +0002bce0: 4d30 2020 2020 2020 2020 2020 2020 2020 M0 │ │ │ │ 0002bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bd10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002bd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002bd60: 6f38 203d 2033 2020 2020 2020 2020 2020 o8 = 3 │ │ │ │ +0002bd50: 2020 7c0a 7c6f 3820 3d20 3320 2020 2020 |.|o8 = 3 │ │ │ │ +0002bd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bd90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002bda0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0002bd90: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002bda0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bdb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002bdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002bdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0002bde0: 0a7c 6939 203a 204d 203d 2062 6574 7469 .|i9 : M = betti │ │ │ │ -0002bdf0: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ -0002be00: 6869 6768 5379 7a79 6779 204d 302c 204c highSyzygy M0, L │ │ │ │ -0002be10: 656e 6774 684c 696d 6974 203d 3e20 3729 engthLimit => 7) │ │ │ │ -0002be20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002bdd0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4d20 3d20 ----+.|i9 : M = │ │ │ │ +0002bde0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ +0002bdf0: 7469 6f6e 2868 6967 6853 797a 7967 7920 tion(highSyzygy │ │ │ │ +0002be00: 4d30 2c20 4c65 6e67 7468 4c69 6d69 7420 M0, LengthLimit │ │ │ │ +0002be10: 3d3e 2037 297c 0a7c 2020 2020 2020 2020 => 7)|.| │ │ │ │ +0002be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002be30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002be60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002be70: 2030 2020 3120 2032 2020 3320 2034 2020 0 1 2 3 4 │ │ │ │ -0002be80: 3520 2036 2020 3720 2020 2020 2020 2020 5 6 7 │ │ │ │ -0002be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bea0: 2020 7c0a 7c6f 3920 3d20 746f 7461 6c3a |.|o9 = total: │ │ │ │ -0002beb0: 2031 3120 3138 2032 3620 3336 2034 3720 11 18 26 36 47 │ │ │ │ -0002bec0: 3630 2037 3420 3930 2020 2020 2020 2020 60 74 90 │ │ │ │ -0002bed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bee0: 2020 207c 0a7c 2020 2020 2020 2020 2036 |.| 6 │ │ │ │ -0002bef0: 3a20 2036 2020 2e20 202e 2020 2e20 202e : 6 . . . . │ │ │ │ -0002bf00: 2020 2e20 202e 2020 2e20 2020 2020 2020 . . . │ │ │ │ -0002bf10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0002bf30: 373a 2020 3520 3138 2031 3420 202e 2020 7: 5 18 14 . │ │ │ │ -0002bf40: 2e20 202e 2020 2e20 202e 2020 2020 2020 . . . . │ │ │ │ -0002bf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bf60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002bf70: 2038 3a20 202e 2020 2e20 3132 2033 3620 8: . . 12 36 │ │ │ │ -0002bf80: 3235 2020 2e20 202e 2020 2e20 2020 2020 25 . . . │ │ │ │ -0002bf90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002bfb0: 2020 393a 2020 2e20 202e 2020 2e20 202e 9: . . . . │ │ │ │ -0002bfc0: 2032 3220 3630 2033 3920 202e 2020 2020 22 60 39 . │ │ │ │ -0002bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002bfe0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0002bff0: 2020 3130 3a20 202e 2020 2e20 202e 2020 10: . . . │ │ │ │ -0002c000: 2e20 202e 2020 2e20 3335 2039 3020 2020 . . . 35 90 │ │ │ │ -0002c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c020: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002be50: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002be60: 2020 2020 2020 3020 2031 2020 3220 2033 0 1 2 3 │ │ │ │ +0002be70: 2020 3420 2035 2020 3620 2037 2020 2020 4 5 6 7 │ │ │ │ +0002be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002be90: 2020 2020 2020 207c 0a7c 6f39 203d 2074 |.|o9 = t │ │ │ │ +0002bea0: 6f74 616c 3a20 3131 2031 3820 3236 2033 otal: 11 18 26 3 │ │ │ │ +0002beb0: 3620 3437 2036 3020 3734 2039 3020 2020 6 47 60 74 90 │ │ │ │ +0002bec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bed0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002bee0: 2020 2020 363a 2020 3620 202e 2020 2e20 6: 6 . . │ │ │ │ +0002bef0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0002bf00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002bf20: 2020 2020 2037 3a20 2035 2031 3820 3134 7: 5 18 14 │ │ │ │ +0002bf30: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0002bf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf50: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002bf60: 2020 2020 2020 383a 2020 2e20 202e 2031 8: . . 1 │ │ │ │ +0002bf70: 3220 3336 2032 3520 202e 2020 2e20 202e 2 36 25 . . . │ │ │ │ +0002bf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002bf90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002bfa0: 2020 2020 2020 2039 3a20 202e 2020 2e20 9: . . │ │ │ │ +0002bfb0: 202e 2020 2e20 3232 2036 3020 3339 2020 . . 22 60 39 │ │ │ │ +0002bfc0: 2e20 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ +0002bfd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002bfe0: 2020 2020 2020 2031 303a 2020 2e20 202e 10: . . │ │ │ │ +0002bff0: 2020 2e20 202e 2020 2e20 202e 2033 3520 . . . . 35 │ │ │ │ +0002c000: 3930 2020 2020 2020 2020 2020 2020 2020 90 │ │ │ │ +0002c010: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002c020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c060: 2020 2020 2020 2020 207c 0a7c 6f39 203a |.|o9 : │ │ │ │ -0002c070: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002c050: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002c060: 7c6f 3920 3a20 4265 7474 6954 616c 6c79 |o9 : BettiTally │ │ │ │ +0002c070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c0a0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002c090: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002c0a0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002c0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -0002c0f0: 3020 3a20 6e65 744c 6973 7420 4252 616e 0 : netList BRan │ │ │ │ -0002c100: 6b73 206d 6174 7269 7846 6163 746f 7269 ks matrixFactori │ │ │ │ -0002c110: 7a61 7469 6f6e 2866 662c 2068 6967 6853 zation(ff, highS │ │ │ │ -0002c120: 797a 7967 7920 4d30 2920 2020 7c0a 7c20 yzygy M0) |.| │ │ │ │ +0002c0e0: 2b0a 7c69 3130 203a 206e 6574 4c69 7374 +.|i10 : netList │ │ │ │ +0002c0f0: 2042 5261 6e6b 7320 6d61 7472 6978 4661 BRanks matrixFa │ │ │ │ +0002c100: 6374 6f72 697a 6174 696f 6e28 6666 2c20 ctorization(ff, │ │ │ │ +0002c110: 6869 6768 5379 7a79 6779 204d 3029 2020 highSyzygy M0) │ │ │ │ +0002c120: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002c130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c160: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002c170: 2020 2020 2020 2b2d 2b2d 2b20 2020 2020 +-+-+ │ │ │ │ +0002c160: 2020 7c0a 7c20 2020 2020 202b 2d2b 2d2b |.| +-+-+ │ │ │ │ +0002c170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002c1b0: 7c6f 3130 203d 207c 367c 367c 2020 2020 |o10 = |6|6| │ │ │ │ +0002c1a0: 2020 207c 0a7c 6f31 3020 3d20 7c36 7c36 |.|o10 = |6|6 │ │ │ │ +0002c1b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002c1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c1e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002c1f0: 0a7c 2020 2020 2020 2b2d 2b2d 2b20 2020 .| +-+-+ │ │ │ │ +0002c1e0: 2020 2020 7c0a 7c20 2020 2020 202b 2d2b |.| +-+ │ │ │ │ +0002c1f0: 2d2b 2020 2020 2020 2020 2020 2020 2020 -+ │ │ │ │ 0002c200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c230: 7c0a 7c20 2020 2020 207c 337c 367c 2020 |.| |3|6| │ │ │ │ +0002c220: 2020 2020 207c 0a7c 2020 2020 2020 7c33 |.| |3 │ │ │ │ +0002c230: 7c36 7c20 2020 2020 2020 2020 2020 2020 |6| │ │ │ │ 0002c240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c270: 207c 0a7c 2020 2020 2020 2b2d 2b2d 2b20 |.| +-+-+ │ │ │ │ +0002c260: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +0002c270: 2d2b 2d2b 2020 2020 2020 2020 2020 2020 -+-+ │ │ │ │ 0002c280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2b0: 2020 7c0a 7c20 2020 2020 207c 327c 367c |.| |2|6| │ │ │ │ +0002c2a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002c2b0: 7c32 7c36 7c20 2020 2020 2020 2020 2020 |2|6| │ │ │ │ 0002c2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c2f0: 2020 207c 0a7c 2020 2020 2020 2b2d 2b2d |.| +-+- │ │ │ │ -0002c300: 2b20 2020 2020 2020 2020 2020 2020 2020 + │ │ │ │ +0002c2e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002c2f0: 202b 2d2b 2d2b 2020 2020 2020 2020 2020 +-+-+ │ │ │ │ +0002c300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002c310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002c330: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002c320: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002c330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c370: 2d2d 2d2d 2d2b 0a0a 496e 2074 6869 7320 -----+..In this │ │ │ │ -0002c380: 6361 7365 2061 7320 696e 2061 6c6c 206f case as in all o │ │ │ │ -0002c390: 7468 6572 7320 7765 2068 6176 6520 6578 thers we have ex │ │ │ │ -0002c3a0: 616d 696e 6564 2c20 6772 6561 7465 7220 amined, greater │ │ │ │ -0002c3b0: 224f 7074 696d 6973 6d22 2069 7320 6e6f "Optimism" is no │ │ │ │ -0002c3c0: 740a 6a75 7374 6966 6965 642c 2061 6e64 t.justified, and │ │ │ │ -0002c3d0: 2074 6875 7320 6d61 7472 6978 4661 6374 thus matrixFact │ │ │ │ -0002c3e0: 6f72 697a 6174 696f 6e28 6666 2c20 6869 orization(ff, hi │ │ │ │ -0002c3f0: 6768 5379 7a79 6779 284d 302c 204f 7074 ghSyzygy(M0, Opt │ │ │ │ -0002c400: 696d 6973 6d3d 3e31 2929 3b20 776f 756c imism=>1)); woul │ │ │ │ -0002c410: 640a 7072 6f64 7563 6520 616e 2065 7272 d.produce an err │ │ │ │ -0002c420: 6f72 2e0a 0a43 6176 6561 740a 3d3d 3d3d or...Caveat.==== │ │ │ │ -0002c430: 3d3d 0a0a 4120 6275 6720 696e 2074 6865 ==..A bug in the │ │ │ │ -0002c440: 2074 6f74 616c 2045 7874 2073 6372 6970 total Ext scrip │ │ │ │ -0002c450: 7420 6d65 616e 7320 7468 6174 2074 6865 t means that the │ │ │ │ -0002c460: 206f 6464 4578 744d 6f64 756c 6520 6973 oddExtModule is │ │ │ │ -0002c470: 2073 6f6d 6574 696d 6573 207a 6572 6f2c sometimes zero, │ │ │ │ -0002c480: 0a61 6e64 2074 6869 7320 6361 6e20 6361 .and this can ca │ │ │ │ -0002c490: 7573 6520 6120 7772 6f6e 6720 7661 6c75 use a wrong valu │ │ │ │ -0002c4a0: 6520 746f 2062 6520 7265 7475 726e 6564 e to be returned │ │ │ │ -0002c4b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -0002c4c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0002c4d0: 6576 656e 4578 744d 6f64 756c 653a 2065 evenExtModule: e │ │ │ │ -0002c4e0: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d venExtModule, -- │ │ │ │ -0002c4f0: 2065 7665 6e20 7061 7274 206f 6620 4578 even part of Ex │ │ │ │ -0002c500: 745e 2a28 4d2c 6b29 206f 7665 7220 610a t^*(M,k) over a. │ │ │ │ -0002c510: 2020 2020 636f 6d70 6c65 7465 2069 6e74 complete int │ │ │ │ -0002c520: 6572 7365 6374 696f 6e20 6173 206d 6f64 ersection as mod │ │ │ │ -0002c530: 756c 6520 6f76 6572 2043 4920 6f70 6572 ule over CI oper │ │ │ │ -0002c540: 6174 6f72 2072 696e 670a 2020 2a20 2a6e ator ring. * *n │ │ │ │ -0002c550: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65 ote oddExtModule │ │ │ │ -0002c560: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ -0002c570: 2d2d 206f 6464 2070 6172 7420 6f66 2045 -- odd part of E │ │ │ │ -0002c580: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -0002c590: 2063 6f6d 706c 6574 650a 2020 2020 696e complete. in │ │ │ │ -0002c5a0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ -0002c5b0: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -0002c5c0: 7261 746f 7220 7269 6e67 0a20 202a 202a rator ring. * * │ │ │ │ -0002c5d0: 6e6f 7465 206d 6642 6f75 6e64 3a20 6d66 note mfBound: mf │ │ │ │ -0002c5e0: 426f 756e 642c 202d 2d20 6465 7465 726d Bound, -- determ │ │ │ │ -0002c5f0: 696e 6573 2068 6f77 2068 6967 6820 6120 ines how high a │ │ │ │ -0002c600: 7379 7a79 6779 2074 6f20 7461 6b65 2066 syzygy to take f │ │ │ │ -0002c610: 6f72 0a20 2020 2022 6d61 7472 6978 4661 or. "matrixFa │ │ │ │ -0002c620: 6374 6f72 697a 6174 696f 6e22 0a20 202a ctorization". * │ │ │ │ -0002c630: 202a 6e6f 7465 206d 6174 7269 7846 6163 *note matrixFac │ │ │ │ -0002c640: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472 torization: matr │ │ │ │ -0002c650: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ -0002c660: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ -0002c670: 6768 6572 0a20 2020 2063 6f64 696d 656e gher. codimen │ │ │ │ -0002c680: 7369 6f6e 206d 6174 7269 7820 6661 6374 sion matrix fact │ │ │ │ -0002c690: 6f72 697a 6174 696f 6e0a 0a57 6179 7320 orization..Ways │ │ │ │ -0002c6a0: 746f 2075 7365 2068 6967 6853 797a 7967 to use highSyzyg │ │ │ │ -0002c6b0: 793a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d y:.============= │ │ │ │ -0002c6c0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0002c6d0: 2268 6967 6853 797a 7967 7928 4d6f 6475 "highSyzygy(Modu │ │ │ │ -0002c6e0: 6c65 2922 0a0a 466f 7220 7468 6520 7072 le)"..For the pr │ │ │ │ -0002c6f0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -0002c700: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -0002c710: 206f 626a 6563 7420 2a6e 6f74 6520 6869 object *note hi │ │ │ │ -0002c720: 6768 5379 7a79 6779 3a20 6869 6768 5379 ghSyzygy: highSy │ │ │ │ -0002c730: 7a79 6779 2c20 6973 2061 202a 6e6f 7465 zygy, is a *note │ │ │ │ -0002c740: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ -0002c750: 2077 6974 680a 6f70 7469 6f6e 733a 2028 with.options: ( │ │ │ │ -0002c760: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -0002c770: 686f 6446 756e 6374 696f 6e57 6974 684f hodFunctionWithO │ │ │ │ -0002c780: 7074 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d ptions,...------ │ │ │ │ +0002c360: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 ----------+..In │ │ │ │ +0002c370: 7468 6973 2063 6173 6520 6173 2069 6e20 this case as in │ │ │ │ +0002c380: 616c 6c20 6f74 6865 7273 2077 6520 6861 all others we ha │ │ │ │ +0002c390: 7665 2065 7861 6d69 6e65 642c 2067 7265 ve examined, gre │ │ │ │ +0002c3a0: 6174 6572 2022 4f70 7469 6d69 736d 2220 ater "Optimism" │ │ │ │ +0002c3b0: 6973 206e 6f74 0a6a 7573 7469 6669 6564 is not.justified │ │ │ │ +0002c3c0: 2c20 616e 6420 7468 7573 206d 6174 7269 , and thus matri │ │ │ │ +0002c3d0: 7846 6163 746f 7269 7a61 7469 6f6e 2866 xFactorization(f │ │ │ │ +0002c3e0: 662c 2068 6967 6853 797a 7967 7928 4d30 f, highSyzygy(M0 │ │ │ │ +0002c3f0: 2c20 4f70 7469 6d69 736d 3d3e 3129 293b , Optimism=>1)); │ │ │ │ +0002c400: 2077 6f75 6c64 0a70 726f 6475 6365 2061 would.produce a │ │ │ │ +0002c410: 6e20 6572 726f 722e 0a0a 4361 7665 6174 n error...Caveat │ │ │ │ +0002c420: 0a3d 3d3d 3d3d 3d0a 0a41 2062 7567 2069 .======..A bug i │ │ │ │ +0002c430: 6e20 7468 6520 746f 7461 6c20 4578 7420 n the total Ext │ │ │ │ +0002c440: 7363 7269 7074 206d 6561 6e73 2074 6861 script means tha │ │ │ │ +0002c450: 7420 7468 6520 6f64 6445 7874 4d6f 6475 t the oddExtModu │ │ │ │ +0002c460: 6c65 2069 7320 736f 6d65 7469 6d65 7320 le is sometimes │ │ │ │ +0002c470: 7a65 726f 2c0a 616e 6420 7468 6973 2063 zero,.and this c │ │ │ │ +0002c480: 616e 2063 6175 7365 2061 2077 726f 6e67 an cause a wrong │ │ │ │ +0002c490: 2076 616c 7565 2074 6f20 6265 2072 6574 value to be ret │ │ │ │ +0002c4a0: 7572 6e65 642e 0a0a 5365 6520 616c 736f urned...See also │ │ │ │ +0002c4b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +0002c4c0: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ +0002c4d0: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ +0002c4e0: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ +0002c4f0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +0002c500: 6572 2061 0a20 2020 2063 6f6d 706c 6574 er a. complet │ │ │ │ +0002c510: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ +0002c520: 7320 6d6f 6475 6c65 206f 7665 7220 4349 s module over CI │ │ │ │ +0002c530: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ +0002c540: 202a 202a 6e6f 7465 206f 6464 4578 744d * *note oddExtM │ │ │ │ +0002c550: 6f64 756c 653a 206f 6464 4578 744d 6f64 odule: oddExtMod │ │ │ │ +0002c560: 756c 652c 202d 2d20 6f64 6420 7061 7274 ule, -- odd part │ │ │ │ +0002c570: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +0002c580: 7665 7220 6120 636f 6d70 6c65 7465 0a20 ver a complete. │ │ │ │ +0002c590: 2020 2069 6e74 6572 7365 6374 696f 6e20 intersection │ │ │ │ +0002c5a0: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ +0002c5b0: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +0002c5c0: 2020 2a20 2a6e 6f74 6520 6d66 426f 756e * *note mfBoun │ │ │ │ +0002c5d0: 643a 206d 6642 6f75 6e64 2c20 2d2d 2064 d: mfBound, -- d │ │ │ │ +0002c5e0: 6574 6572 6d69 6e65 7320 686f 7720 6869 etermines how hi │ │ │ │ +0002c5f0: 6768 2061 2073 797a 7967 7920 746f 2074 gh a syzygy to t │ │ │ │ +0002c600: 616b 6520 666f 720a 2020 2020 226d 6174 ake for. "mat │ │ │ │ +0002c610: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +0002c620: 220a 2020 2a20 2a6e 6f74 6520 6d61 7472 ". * *note matr │ │ │ │ +0002c630: 6978 4661 6374 6f72 697a 6174 696f 6e3a ixFactorization: │ │ │ │ +0002c640: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +0002c650: 7469 6f6e 2c20 2d2d 204d 6170 7320 696e tion, -- Maps in │ │ │ │ +0002c660: 2061 2068 6967 6865 720a 2020 2020 636f a higher. co │ │ │ │ +0002c670: 6469 6d65 6e73 696f 6e20 6d61 7472 6978 dimension matrix │ │ │ │ +0002c680: 2066 6163 746f 7269 7a61 7469 6f6e 0a0a factorization.. │ │ │ │ +0002c690: 5761 7973 2074 6f20 7573 6520 6869 6768 Ways to use high │ │ │ │ +0002c6a0: 5379 7a79 6779 3a0a 3d3d 3d3d 3d3d 3d3d Syzygy:.======== │ │ │ │ +0002c6b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0002c6c0: 0a20 202a 2022 6869 6768 5379 7a79 6779 . * "highSyzygy │ │ │ │ +0002c6d0: 284d 6f64 756c 6529 220a 0a46 6f72 2074 (Module)"..For t │ │ │ │ +0002c6e0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +0002c6f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002c700: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +0002c710: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ +0002c720: 6967 6853 797a 7967 792c 2069 7320 6120 ighSyzygy, is a │ │ │ │ +0002c730: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ +0002c740: 6374 696f 6e20 7769 7468 0a6f 7074 696f ction with.optio │ │ │ │ +0002c750: 6e73 3a20 284d 6163 6175 6c61 7932 446f ns: (Macaulay2Do │ │ │ │ +0002c760: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +0002c770: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ +0002c780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002c7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002c7d0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0002c7e0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0002c7f0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0002c800: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0002c810: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0002c820: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -0002c830: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0002c840: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0002c850: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002c860: 732e 6d32 3a33 3330 393a 302e 0a1f 0a46 s.m2:3309:0....F │ │ │ │ -0002c870: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0002c880: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0002c890: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0002c8a0: 2068 4d61 7073 2c20 4e65 7874 3a20 486f hMaps, Next: Ho │ │ │ │ -0002c8b0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c mWithComponents, │ │ │ │ -0002c8c0: 2050 7265 763a 2068 6967 6853 797a 7967 Prev: highSyzyg │ │ │ │ -0002c8d0: 792c 2055 703a 2054 6f70 0a0a 684d 6170 y, Up: Top..hMap │ │ │ │ -0002c8e0: 7320 2d2d 206c 6973 7420 7468 6520 6d61 s -- list the ma │ │ │ │ -0002c8f0: 7073 2020 6828 7029 3a20 415f 3028 7029 ps h(p): A_0(p) │ │ │ │ -0002c900: 2d2d 3e20 415f 3128 7029 2069 6e20 6120 --> A_1(p) in a │ │ │ │ -0002c910: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -0002c920: 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a ion.************ │ │ │ │ +0002c7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0002c7d0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0002c7e0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0002c7f0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0002c800: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0002c810: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +0002c820: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0002c830: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +0002c840: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0002c850: 7574 696f 6e73 2e6d 323a 3333 3039 3a30 utions.m2:3309:0 │ │ │ │ +0002c860: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +0002c870: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0002c880: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +0002c890: 4e6f 6465 3a20 684d 6170 732c 204e 6578 Node: hMaps, Nex │ │ │ │ +0002c8a0: 743a 2048 6f6d 5769 7468 436f 6d70 6f6e t: HomWithCompon │ │ │ │ +0002c8b0: 656e 7473 2c20 5072 6576 3a20 6869 6768 ents, Prev: high │ │ │ │ +0002c8c0: 5379 7a79 6779 2c20 5570 3a20 546f 700a Syzygy, Up: Top. │ │ │ │ +0002c8d0: 0a68 4d61 7073 202d 2d20 6c69 7374 2074 .hMaps -- list t │ │ │ │ +0002c8e0: 6865 206d 6170 7320 2068 2870 293a 2041 he maps h(p): A │ │ │ │ +0002c8f0: 5f30 2870 292d 2d3e 2041 5f31 2870 2920 _0(p)--> A_1(p) │ │ │ │ +0002c900: 696e 2061 206d 6174 7269 7846 6163 746f in a matrixFacto │ │ │ │ +0002c910: 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a rization.******* │ │ │ │ +0002c920: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002c930: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002c940: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002c950: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002c960: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -0002c970: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0002c980: 2068 4d61 7073 203d 2068 4d61 7073 206d hMaps = hMaps m │ │ │ │ -0002c990: 660a 2020 2a20 496e 7075 7473 3a0a 2020 f. * Inputs:. │ │ │ │ -0002c9a0: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74 * mf, a *not │ │ │ │ -0002c9b0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ -0002c9c0: 7932 446f 6329 4c69 7374 2c2c 206f 7574 y2Doc)List,, out │ │ │ │ -0002c9d0: 7075 7420 6f66 2061 206d 6174 7269 7846 put of a matrixF │ │ │ │ -0002c9e0: 6163 746f 7269 7a61 7469 6f6e 0a20 2020 actorization. │ │ │ │ -0002c9f0: 2020 2020 2063 6f6d 7075 7461 7469 6f6e computation │ │ │ │ -0002ca00: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -0002ca10: 2020 2020 2a20 684d 6170 732c 2061 202a * hMaps, a * │ │ │ │ -0002ca20: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ -0002ca30: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ -0002ca40: 6c69 7374 206d 6174 7269 6365 7320 2468 list matrices $h │ │ │ │ -0002ca50: 5f70 3a20 415f 3028 7029 5c74 6f0a 2020 _p: A_0(p)\to. │ │ │ │ -0002ca60: 2020 2020 2020 415f 3128 7029 242e 2054 A_1(p)$. T │ │ │ │ -0002ca70: 6865 2073 6f75 7263 6573 2061 6e64 2074 he sources and t │ │ │ │ -0002ca80: 6172 6765 7473 206f 6620 7468 6573 6520 argets of these │ │ │ │ -0002ca90: 6d61 7073 2068 6176 6520 7468 6520 636f maps have the co │ │ │ │ -0002caa0: 6d70 6f6e 656e 7473 0a20 2020 2020 2020 mponents. │ │ │ │ -0002cab0: 2042 5f73 2870 292e 0a0a 4465 7363 7269 B_s(p)...Descri │ │ │ │ -0002cac0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0002cad0: 3d0a 0a53 6565 2074 6865 2064 6f63 756d =..See the docum │ │ │ │ -0002cae0: 656e 7461 7469 6f6e 2066 6f72 206d 6174 entation for mat │ │ │ │ -0002caf0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0002cb00: 2066 6f72 2061 6e20 6578 616d 706c 652e for an example. │ │ │ │ -0002cb10: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0002cb20: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -0002cb30: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -0002cb40: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72 on: matrixFactor │ │ │ │ -0002cb50: 697a 6174 696f 6e2c 202d 2d20 4d61 7073 ization, -- Maps │ │ │ │ -0002cb60: 2069 6e20 6120 6869 6768 6572 0a20 2020 in a higher. │ │ │ │ -0002cb70: 2063 6f64 696d 656e 7369 6f6e 206d 6174 codimension mat │ │ │ │ -0002cb80: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -0002cb90: 6e0a 2020 2a20 2a6e 6f74 6520 644d 6170 n. * *note dMap │ │ │ │ -0002cba0: 733a 2064 4d61 7073 2c20 2d2d 206c 6973 s: dMaps, -- lis │ │ │ │ -0002cbb0: 7420 7468 6520 6d61 7073 2020 6428 7029 t the maps d(p) │ │ │ │ -0002cbc0: 3a41 5f31 2870 292d 2d3e 2041 5f30 2870 :A_1(p)--> A_0(p │ │ │ │ -0002cbd0: 2920 696e 2061 0a20 2020 206d 6174 7269 ) in a. matri │ │ │ │ -0002cbe0: 7846 6163 746f 7269 7a61 7469 6f6e 0a20 xFactorization. │ │ │ │ -0002cbf0: 202a 202a 6e6f 7465 2042 5261 6e6b 733a * *note BRanks: │ │ │ │ -0002cc00: 2042 5261 6e6b 732c 202d 2d20 7261 6e6b BRanks, -- rank │ │ │ │ -0002cc10: 7320 6f66 2074 6865 206d 6f64 756c 6573 s of the modules │ │ │ │ -0002cc20: 2042 5f69 2864 2920 696e 2061 0a20 2020 B_i(d) in a. │ │ │ │ -0002cc30: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -0002cc40: 7469 6f6e 0a20 202a 202a 6e6f 7465 2062 tion. * *note b │ │ │ │ -0002cc50: 4d61 7073 3a20 624d 6170 732c 202d 2d20 Maps: bMaps, -- │ │ │ │ -0002cc60: 6c69 7374 2074 6865 206d 6170 7320 2064 list the maps d │ │ │ │ -0002cc70: 5f70 3a42 5f31 2870 292d 2d3e 425f 3028 _p:B_1(p)-->B_0( │ │ │ │ -0002cc80: 7029 2069 6e20 610a 2020 2020 6d61 7472 p) in a. matr │ │ │ │ -0002cc90: 6978 4661 6374 6f72 697a 6174 696f 6e0a ixFactorization. │ │ │ │ -0002cca0: 2020 2a20 2a6e 6f74 6520 7073 694d 6170 * *note psiMap │ │ │ │ -0002ccb0: 733a 2070 7369 4d61 7073 2c20 2d2d 206c s: psiMaps, -- l │ │ │ │ -0002ccc0: 6973 7420 7468 6520 6d61 7073 2020 7073 ist the maps ps │ │ │ │ -0002ccd0: 6928 7029 3a20 425f 3128 7029 202d 2d3e i(p): B_1(p) --> │ │ │ │ -0002cce0: 2041 5f30 2870 2d31 2920 696e 2061 0a20 A_0(p-1) in a. │ │ │ │ -0002ccf0: 2020 206d 6174 7269 7846 6163 746f 7269 matrixFactori │ │ │ │ -0002cd00: 7a61 7469 6f6e 0a0a 5761 7973 2074 6f20 zation..Ways to │ │ │ │ -0002cd10: 7573 6520 684d 6170 733a 0a3d 3d3d 3d3d use hMaps:.===== │ │ │ │ -0002cd20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -0002cd30: 202a 2022 684d 6170 7328 4c69 7374 2922 * "hMaps(List)" │ │ │ │ -0002cd40: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0002cd50: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0002cd60: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0002cd70: 6563 7420 2a6e 6f74 6520 684d 6170 733a ect *note hMaps: │ │ │ │ -0002cd80: 2068 4d61 7073 2c20 6973 2061 202a 6e6f hMaps, is a *no │ │ │ │ -0002cd90: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -0002cda0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ -0002cdb0: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0002cdc0: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0002c960: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0002c970: 2020 2020 2020 684d 6170 7320 3d20 684d hMaps = hM │ │ │ │ +0002c980: 6170 7320 6d66 0a20 202a 2049 6e70 7574 aps mf. * Input │ │ │ │ +0002c990: 733a 0a20 2020 2020 202a 206d 662c 2061 s:. * mf, a │ │ │ │ +0002c9a0: 202a 6e6f 7465 206c 6973 743a 2028 4d61 *note list: (Ma │ │ │ │ +0002c9b0: 6361 756c 6179 3244 6f63 294c 6973 742c caulay2Doc)List, │ │ │ │ +0002c9c0: 2c20 6f75 7470 7574 206f 6620 6120 6d61 , output of a ma │ │ │ │ +0002c9d0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +0002c9e0: 6e0a 2020 2020 2020 2020 636f 6d70 7574 n. comput │ │ │ │ +0002c9f0: 6174 696f 6e0a 2020 2a20 4f75 7470 7574 ation. * Output │ │ │ │ +0002ca00: 733a 0a20 2020 2020 202a 2068 4d61 7073 s:. * hMaps │ │ │ │ +0002ca10: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ +0002ca20: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ +0002ca30: 7374 2c2c 206c 6973 7420 6d61 7472 6963 st,, list matric │ │ │ │ +0002ca40: 6573 2024 685f 703a 2041 5f30 2870 295c es $h_p: A_0(p)\ │ │ │ │ +0002ca50: 746f 0a20 2020 2020 2020 2041 5f31 2870 to. A_1(p │ │ │ │ +0002ca60: 2924 2e20 5468 6520 736f 7572 6365 7320 )$. The sources │ │ │ │ +0002ca70: 616e 6420 7461 7267 6574 7320 6f66 2074 and targets of t │ │ │ │ +0002ca80: 6865 7365 206d 6170 7320 6861 7665 2074 hese maps have t │ │ │ │ +0002ca90: 6865 2063 6f6d 706f 6e65 6e74 730a 2020 he components. │ │ │ │ +0002caa0: 2020 2020 2020 425f 7328 7029 2e0a 0a44 B_s(p)...D │ │ │ │ +0002cab0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0002cac0: 3d3d 3d3d 3d3d 0a0a 5365 6520 7468 6520 ======..See the │ │ │ │ +0002cad0: 646f 6375 6d65 6e74 6174 696f 6e20 666f documentation fo │ │ │ │ +0002cae0: 7220 6d61 7472 6978 4661 6374 6f72 697a r matrixFactoriz │ │ │ │ +0002caf0: 6174 696f 6e20 666f 7220 616e 2065 7861 ation for an exa │ │ │ │ +0002cb00: 6d70 6c65 2e0a 0a53 6565 2061 6c73 6f0a mple...See also. │ │ │ │ +0002cb10: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0002cb20: 6f74 6520 6d61 7472 6978 4661 6374 6f72 ote matrixFactor │ │ │ │ +0002cb30: 697a 6174 696f 6e3a 206d 6174 7269 7846 ization: matrixF │ │ │ │ +0002cb40: 6163 746f 7269 7a61 7469 6f6e 2c20 2d2d actorization, -- │ │ │ │ +0002cb50: 204d 6170 7320 696e 2061 2068 6967 6865 Maps in a highe │ │ │ │ +0002cb60: 720a 2020 2020 636f 6469 6d65 6e73 696f r. codimensio │ │ │ │ +0002cb70: 6e20 6d61 7472 6978 2066 6163 746f 7269 n matrix factori │ │ │ │ +0002cb80: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465 zation. * *note │ │ │ │ +0002cb90: 2064 4d61 7073 3a20 644d 6170 732c 202d dMaps: dMaps, - │ │ │ │ +0002cba0: 2d20 6c69 7374 2074 6865 206d 6170 7320 - list the maps │ │ │ │ +0002cbb0: 2064 2870 293a 415f 3128 7029 2d2d 3e20 d(p):A_1(p)--> │ │ │ │ +0002cbc0: 415f 3028 7029 2069 6e20 610a 2020 2020 A_0(p) in a. │ │ │ │ +0002cbd0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ +0002cbe0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 4252 ion. * *note BR │ │ │ │ +0002cbf0: 616e 6b73 3a20 4252 616e 6b73 2c20 2d2d anks: BRanks, -- │ │ │ │ +0002cc00: 2072 616e 6b73 206f 6620 7468 6520 6d6f ranks of the mo │ │ │ │ +0002cc10: 6475 6c65 7320 425f 6928 6429 2069 6e20 dules B_i(d) in │ │ │ │ +0002cc20: 610a 2020 2020 6d61 7472 6978 4661 6374 a. matrixFact │ │ │ │ +0002cc30: 6f72 697a 6174 696f 6e0a 2020 2a20 2a6e orization. * *n │ │ │ │ +0002cc40: 6f74 6520 624d 6170 733a 2062 4d61 7073 ote bMaps: bMaps │ │ │ │ +0002cc50: 2c20 2d2d 206c 6973 7420 7468 6520 6d61 , -- list the ma │ │ │ │ +0002cc60: 7073 2020 645f 703a 425f 3128 7029 2d2d ps d_p:B_1(p)-- │ │ │ │ +0002cc70: 3e42 5f30 2870 2920 696e 2061 0a20 2020 >B_0(p) in a. │ │ │ │ +0002cc80: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +0002cc90: 7469 6f6e 0a20 202a 202a 6e6f 7465 2070 tion. * *note p │ │ │ │ +0002cca0: 7369 4d61 7073 3a20 7073 694d 6170 732c siMaps: psiMaps, │ │ │ │ +0002ccb0: 202d 2d20 6c69 7374 2074 6865 206d 6170 -- list the map │ │ │ │ +0002ccc0: 7320 2070 7369 2870 293a 2042 5f31 2870 s psi(p): B_1(p │ │ │ │ +0002ccd0: 2920 2d2d 3e20 415f 3028 702d 3129 2069 ) --> A_0(p-1) i │ │ │ │ +0002cce0: 6e20 610a 2020 2020 6d61 7472 6978 4661 n a. matrixFa │ │ │ │ +0002ccf0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179 ctorization..Way │ │ │ │ +0002cd00: 7320 746f 2075 7365 2068 4d61 7073 3a0a s to use hMaps:. │ │ │ │ +0002cd10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002cd20: 3d3d 0a0a 2020 2a20 2268 4d61 7073 284c ==.. * "hMaps(L │ │ │ │ +0002cd30: 6973 7429 220a 0a46 6f72 2074 6865 2070 ist)"..For the p │ │ │ │ +0002cd40: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0002cd50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0002cd60: 6520 6f62 6a65 6374 202a 6e6f 7465 2068 e object *note h │ │ │ │ +0002cd70: 4d61 7073 3a20 684d 6170 732c 2069 7320 Maps: hMaps, is │ │ │ │ +0002cd80: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +0002cd90: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +0002cda0: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0002cdb0: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0002cdc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002ce10: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0002ce20: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0002ce30: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0002ce40: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0002ce50: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0002ce60: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0002ce70: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -0002ce80: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0002ce90: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ -0002cea0: 3435 373a 302e 0a1f 0a46 696c 653a 2043 457:0....File: C │ │ │ │ -0002ceb0: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0002cec0: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -0002ced0: 6e66 6f2c 204e 6f64 653a 2048 6f6d 5769 nfo, Node: HomWi │ │ │ │ -0002cee0: 7468 436f 6d70 6f6e 656e 7473 2c20 4e65 thComponents, Ne │ │ │ │ -0002cef0: 7874 3a20 696e 6669 6e69 7465 4265 7474 xt: infiniteBett │ │ │ │ -0002cf00: 694e 756d 6265 7273 2c20 5072 6576 3a20 iNumbers, Prev: │ │ │ │ -0002cf10: 684d 6170 732c 2055 703a 2054 6f70 0a0a hMaps, Up: Top.. │ │ │ │ -0002cf20: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74 HomWithComponent │ │ │ │ -0002cf30: 7320 2d2d 2063 6f6d 7075 7465 7320 486f s -- computes Ho │ │ │ │ -0002cf40: 6d2c 2070 7265 7365 7276 696e 6720 6469 m, preserving di │ │ │ │ -0002cf50: 7265 6374 2073 756d 2069 6e66 6f72 6d61 rect sum informa │ │ │ │ -0002cf60: 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a tion.*********** │ │ │ │ +0002ce00: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0002ce10: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0002ce20: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0002ce30: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0002ce40: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0002ce50: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0002ce60: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0002ce70: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +0002ce80: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0002ce90: 2e6d 323a 3434 3537 3a30 2e0a 1f0a 4669 .m2:4457:0....Fi │ │ │ │ +0002cea0: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +0002ceb0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0002cec0: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +0002ced0: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74 HomWithComponent │ │ │ │ +0002cee0: 732c 204e 6578 743a 2069 6e66 696e 6974 s, Next: infinit │ │ │ │ +0002cef0: 6542 6574 7469 4e75 6d62 6572 732c 2050 eBettiNumbers, P │ │ │ │ +0002cf00: 7265 763a 2068 4d61 7073 2c20 5570 3a20 rev: hMaps, Up: │ │ │ │ +0002cf10: 546f 700a 0a48 6f6d 5769 7468 436f 6d70 Top..HomWithComp │ │ │ │ +0002cf20: 6f6e 656e 7473 202d 2d20 636f 6d70 7574 onents -- comput │ │ │ │ +0002cf30: 6573 2048 6f6d 2c20 7072 6573 6572 7669 es Hom, preservi │ │ │ │ +0002cf40: 6e67 2064 6972 6563 7420 7375 6d20 696e ng direct sum in │ │ │ │ +0002cf50: 666f 726d 6174 696f 6e0a 2a2a 2a2a 2a2a formation.****** │ │ │ │ +0002cf60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002cf70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002cf80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002cfa0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0002cfb0: 7361 6765 3a20 0a20 2020 2020 2020 2048 sage: . H │ │ │ │ -0002cfc0: 203d 2048 6f6d 284d 2c4e 290a 2020 2a20 = Hom(M,N). * │ │ │ │ -0002cfd0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0002cfe0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -0002cff0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0002d000: 294d 6f64 756c 652c 2c20 0a20 2020 2020 )Module,, . │ │ │ │ -0002d010: 202a 204e 2c20 6120 2a6e 6f74 6520 6d6f * N, a *note mo │ │ │ │ -0002d020: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ -0002d030: 446f 6329 4d6f 6475 6c65 2c2c 200a 2020 Doc)Module,, . │ │ │ │ -0002d040: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ -0002d050: 202a 2048 2c20 6120 2a6e 6f74 6520 6d6f * H, a *note mo │ │ │ │ -0002d060: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ -0002d070: 446f 6329 4d6f 6475 6c65 2c2c 200a 0a44 Doc)Module,, ..D │ │ │ │ -0002d080: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0002d090: 3d3d 3d3d 3d3d 0a0a 4966 204d 2061 6e64 ======..If M and │ │ │ │ -0002d0a0: 2f6f 7220 4e20 6172 6520 6469 7265 6374 /or N are direct │ │ │ │ -0002d0b0: 2073 756d 206d 6f64 756c 6573 2028 6973 sum modules (is │ │ │ │ -0002d0c0: 4469 7265 6374 5375 6d20 4d20 3d3d 2074 DirectSum M == t │ │ │ │ -0002d0d0: 7275 6529 2074 6865 6e20 4820 6973 2074 rue) then H is t │ │ │ │ -0002d0e0: 6865 0a64 6972 6563 7420 7375 6d20 6f66 he.direct sum of │ │ │ │ -0002d0f0: 2074 6865 2048 6f6d 7320 6265 7477 6565 the Homs betwee │ │ │ │ -0002d100: 6e20 7468 6520 636f 6d70 6f6e 656e 7473 n the components │ │ │ │ -0002d110: 2e20 5468 6973 2053 484f 554c 4420 6265 . This SHOULD be │ │ │ │ -0002d120: 2062 7569 6c74 2069 6e74 6f0a 486f 6d28 built into.Hom( │ │ │ │ -0002d130: 4d2c 4e29 2c20 6275 7420 6973 6e27 7420 M,N), but isn't │ │ │ │ -0002d140: 6173 206f 6620 4d32 2c20 762e 2031 2e37 as of M2, v. 1.7 │ │ │ │ -0002d150: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -0002d160: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2074 ===.. * *note t │ │ │ │ -0002d170: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ -0002d180: 6e74 733a 2074 656e 736f 7257 6974 6843 nts: tensorWithC │ │ │ │ -0002d190: 6f6d 706f 6e65 6e74 732c 202d 2d20 666f omponents, -- fo │ │ │ │ -0002d1a0: 726d 7320 7468 6520 7465 6e73 6f72 0a20 rms the tensor. │ │ │ │ -0002d1b0: 2020 2070 726f 6475 6374 2c20 7072 6573 product, pres │ │ │ │ -0002d1c0: 6572 7669 6e67 2064 6972 6563 7420 7375 erving direct su │ │ │ │ -0002d1d0: 6d20 696e 666f 726d 6174 696f 6e0a 2020 m information. │ │ │ │ -0002d1e0: 2a20 2a6e 6f74 6520 6475 616c 5769 7468 * *note dualWith │ │ │ │ -0002d1f0: 436f 6d70 6f6e 656e 7473 3a20 6475 616c Components: dual │ │ │ │ -0002d200: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ -0002d210: 2d2d 2064 7561 6c20 6d6f 6475 6c65 2070 -- dual module p │ │ │ │ -0002d220: 7265 7365 7276 696e 670a 2020 2020 6469 reserving. di │ │ │ │ -0002d230: 7265 6374 2073 756d 2069 6e66 6f72 6d61 rect sum informa │ │ │ │ -0002d240: 7469 6f6e 0a0a 5761 7973 2074 6f20 7573 tion..Ways to us │ │ │ │ -0002d250: 6520 486f 6d57 6974 6843 6f6d 706f 6e65 e HomWithCompone │ │ │ │ -0002d260: 6e74 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d nts:.=========== │ │ │ │ -0002d270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002d280: 3d3d 3d0a 0a20 202a 2022 486f 6d57 6974 ===.. * "HomWit │ │ │ │ -0002d290: 6843 6f6d 706f 6e65 6e74 7328 4d6f 6475 hComponents(Modu │ │ │ │ -0002d2a0: 6c65 2c4d 6f64 756c 6529 220a 0a46 6f72 le,Module)"..For │ │ │ │ -0002d2b0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ -0002d2c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002d2d0: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ -0002d2e0: 6e6f 7465 2048 6f6d 5769 7468 436f 6d70 note HomWithComp │ │ │ │ -0002d2f0: 6f6e 656e 7473 3a20 486f 6d57 6974 6843 onents: HomWithC │ │ │ │ -0002d300: 6f6d 706f 6e65 6e74 732c 2069 7320 6120 omponents, is a │ │ │ │ -0002d310: 2a6e 6f74 6520 6d65 7468 6f64 0a66 756e *note method.fun │ │ │ │ -0002d320: 6374 696f 6e3a 2028 4d61 6361 756c 6179 ction: (Macaulay │ │ │ │ -0002d330: 3244 6f63 294d 6574 686f 6446 756e 6374 2Doc)MethodFunct │ │ │ │ -0002d340: 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d ion,...--------- │ │ │ │ +0002cf90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0002cfa0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0002cfb0: 2020 2020 4820 3d20 486f 6d28 4d2c 4e29 H = Hom(M,N) │ │ │ │ +0002cfc0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0002cfd0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +0002cfe0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0002cff0: 7932 446f 6329 4d6f 6475 6c65 2c2c 200a y2Doc)Module,, . │ │ │ │ +0002d000: 2020 2020 2020 2a20 4e2c 2061 202a 6e6f * N, a *no │ │ │ │ +0002d010: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ +0002d020: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0002d030: 2c20 0a20 202a 204f 7574 7075 7473 3a0a , . * Outputs:. │ │ │ │ +0002d040: 2020 2020 2020 2a20 482c 2061 202a 6e6f * H, a *no │ │ │ │ +0002d050: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ +0002d060: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ +0002d070: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +0002d080: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ +0002d090: 4d20 616e 642f 6f72 204e 2061 7265 2064 M and/or N are d │ │ │ │ +0002d0a0: 6972 6563 7420 7375 6d20 6d6f 6475 6c65 irect sum module │ │ │ │ +0002d0b0: 7320 2869 7344 6972 6563 7453 756d 204d s (isDirectSum M │ │ │ │ +0002d0c0: 203d 3d20 7472 7565 2920 7468 656e 2048 == true) then H │ │ │ │ +0002d0d0: 2069 7320 7468 650a 6469 7265 6374 2073 is the.direct s │ │ │ │ +0002d0e0: 756d 206f 6620 7468 6520 486f 6d73 2062 um of the Homs b │ │ │ │ +0002d0f0: 6574 7765 656e 2074 6865 2063 6f6d 706f etween the compo │ │ │ │ +0002d100: 6e65 6e74 732e 2054 6869 7320 5348 4f55 nents. This SHOU │ │ │ │ +0002d110: 4c44 2062 6520 6275 696c 7420 696e 746f LD be built into │ │ │ │ +0002d120: 0a48 6f6d 284d 2c4e 292c 2062 7574 2069 .Hom(M,N), but i │ │ │ │ +0002d130: 736e 2774 2061 7320 6f66 204d 322c 2076 sn't as of M2, v │ │ │ │ +0002d140: 2e20 312e 370a 0a53 6565 2061 6c73 6f0a . 1.7..See also. │ │ │ │ +0002d150: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +0002d160: 6f74 6520 7465 6e73 6f72 5769 7468 436f ote tensorWithCo │ │ │ │ +0002d170: 6d70 6f6e 656e 7473 3a20 7465 6e73 6f72 mponents: tensor │ │ │ │ +0002d180: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ +0002d190: 2d2d 2066 6f72 6d73 2074 6865 2074 656e -- forms the ten │ │ │ │ +0002d1a0: 736f 720a 2020 2020 7072 6f64 7563 742c sor. product, │ │ │ │ +0002d1b0: 2070 7265 7365 7276 696e 6720 6469 7265 preserving dire │ │ │ │ +0002d1c0: 6374 2073 756d 2069 6e66 6f72 6d61 7469 ct sum informati │ │ │ │ +0002d1d0: 6f6e 0a20 202a 202a 6e6f 7465 2064 7561 on. * *note dua │ │ │ │ +0002d1e0: 6c57 6974 6843 6f6d 706f 6e65 6e74 733a lWithComponents: │ │ │ │ +0002d1f0: 2064 7561 6c57 6974 6843 6f6d 706f 6e65 dualWithCompone │ │ │ │ +0002d200: 6e74 732c 202d 2d20 6475 616c 206d 6f64 nts, -- dual mod │ │ │ │ +0002d210: 756c 6520 7072 6573 6572 7669 6e67 0a20 ule preserving. │ │ │ │ +0002d220: 2020 2064 6972 6563 7420 7375 6d20 696e direct sum in │ │ │ │ +0002d230: 666f 726d 6174 696f 6e0a 0a57 6179 7320 formation..Ways │ │ │ │ +0002d240: 746f 2075 7365 2048 6f6d 5769 7468 436f to use HomWithCo │ │ │ │ +0002d250: 6d70 6f6e 656e 7473 3a0a 3d3d 3d3d 3d3d mponents:.====== │ │ │ │ +0002d260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002d270: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2248 ========.. * "H │ │ │ │ +0002d280: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473 omWithComponents │ │ │ │ +0002d290: 284d 6f64 756c 652c 4d6f 6475 6c65 2922 (Module,Module)" │ │ │ │ +0002d2a0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +0002d2b0: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +0002d2c0: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +0002d2d0: 6563 7420 2a6e 6f74 6520 486f 6d57 6974 ect *note HomWit │ │ │ │ +0002d2e0: 6843 6f6d 706f 6e65 6e74 733a 2048 6f6d hComponents: Hom │ │ │ │ +0002d2f0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ +0002d300: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +0002d310: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ +0002d320: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +0002d330: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0002d340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d390: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -0002d3a0: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -0002d3b0: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -0002d3c0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -0002d3d0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -0002d3e0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -0002d3f0: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -0002d400: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0002d410: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -0002d420: 323a 3236 3435 3a30 2e0a 1f0a 4669 6c65 2:2645:0....File │ │ │ │ -0002d430: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -0002d440: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002d450: 732e 696e 666f 2c20 4e6f 6465 3a20 696e s.info, Node: in │ │ │ │ -0002d460: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -0002d470: 7273 2c20 4e65 7874 3a20 6973 4c69 6e65 rs, Next: isLine │ │ │ │ -0002d480: 6172 2c20 5072 6576 3a20 486f 6d57 6974 ar, Prev: HomWit │ │ │ │ -0002d490: 6843 6f6d 706f 6e65 6e74 732c 2055 703a hComponents, Up: │ │ │ │ -0002d4a0: 2054 6f70 0a0a 696e 6669 6e69 7465 4265 Top..infiniteBe │ │ │ │ -0002d4b0: 7474 694e 756d 6265 7273 202d 2d20 6265 ttiNumbers -- be │ │ │ │ -0002d4c0: 7474 6920 6e75 6d62 6572 7320 6f66 2066 tti numbers of f │ │ │ │ -0002d4d0: 696e 6974 6520 7265 736f 6c75 7469 6f6e inite resolution │ │ │ │ -0002d4e0: 2063 6f6d 7075 7465 6420 6672 6f6d 2061 computed from a │ │ │ │ -0002d4f0: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ -0002d500: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ation.********** │ │ │ │ +0002d380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +0002d390: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +0002d3a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +0002d3b0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +0002d3c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +0002d3d0: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +0002d3e0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +0002d3f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +0002d400: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0002d410: 6f6e 732e 6d32 3a32 3634 353a 302e 0a1f ons.m2:2645:0... │ │ │ │ +0002d420: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +0002d430: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0002d440: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +0002d450: 653a 2069 6e66 696e 6974 6542 6574 7469 e: infiniteBetti │ │ │ │ +0002d460: 4e75 6d62 6572 732c 204e 6578 743a 2069 Numbers, Next: i │ │ │ │ +0002d470: 734c 696e 6561 722c 2050 7265 763a 2048 sLinear, Prev: H │ │ │ │ +0002d480: 6f6d 5769 7468 436f 6d70 6f6e 656e 7473 omWithComponents │ │ │ │ +0002d490: 2c20 5570 3a20 546f 700a 0a69 6e66 696e , Up: Top..infin │ │ │ │ +0002d4a0: 6974 6542 6574 7469 4e75 6d62 6572 7320 iteBettiNumbers │ │ │ │ +0002d4b0: 2d2d 2062 6574 7469 206e 756d 6265 7273 -- betti numbers │ │ │ │ +0002d4c0: 206f 6620 6669 6e69 7465 2072 6573 6f6c of finite resol │ │ │ │ +0002d4d0: 7574 696f 6e20 636f 6d70 7574 6564 2066 ution computed f │ │ │ │ +0002d4e0: 726f 6d20 6120 6d61 7472 6978 2066 6163 rom a matrix fac │ │ │ │ +0002d4f0: 746f 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a torization.***** │ │ │ │ +0002d500: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d510: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d520: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d530: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002d540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002d560: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -0002d570: 3a20 0a20 2020 2020 2020 204c 203d 2066 : . L = f │ │ │ │ -0002d580: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ -0002d590: 7320 284d 462c 6c65 6e29 0a20 202a 2049 s (MF,len). * I │ │ │ │ -0002d5a0: 6e70 7574 733a 0a20 2020 2020 202a 204d nputs:. * M │ │ │ │ -0002d5b0: 462c 2061 202a 6e6f 7465 206c 6973 743a F, a *note list: │ │ │ │ -0002d5c0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0002d5d0: 6973 742c 2c20 4c69 7374 206f 6620 4861 ist,, List of Ha │ │ │ │ -0002d5e0: 7368 5461 626c 6573 2061 7320 636f 6d70 shTables as comp │ │ │ │ -0002d5f0: 7574 6564 0a20 2020 2020 2020 2062 7920 uted. by │ │ │ │ -0002d600: 226d 6174 7269 7846 6163 746f 7269 7a61 "matrixFactoriza │ │ │ │ -0002d610: 7469 6f6e 220a 2020 2020 2020 2a20 6c65 tion". * le │ │ │ │ -0002d620: 6e2c 2061 6e20 2a6e 6f74 6520 696e 7465 n, an *note inte │ │ │ │ -0002d630: 6765 723a 2028 4d61 6361 756c 6179 3244 ger: (Macaulay2D │ │ │ │ -0002d640: 6f63 295a 5a2c 2c20 6c65 6e67 7468 206f oc)ZZ,, length o │ │ │ │ -0002d650: 6620 6265 7474 6920 6e75 6d62 6572 0a20 f betti number. │ │ │ │ -0002d660: 2020 2020 2020 2073 6571 7565 6e63 6520 sequence │ │ │ │ -0002d670: 746f 2070 726f 6475 6365 0a20 202a 204f to produce. * O │ │ │ │ -0002d680: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0002d690: 4c2c 2061 202a 6e6f 7465 206c 6973 743a L, a *note list: │ │ │ │ -0002d6a0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0002d6b0: 6973 742c 2c20 4c69 7374 206f 6620 6265 ist,, List of be │ │ │ │ -0002d6c0: 7474 6920 6e75 6d62 6572 730a 0a44 6573 tti numbers..Des │ │ │ │ -0002d6d0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0002d6e0: 3d3d 3d3d 0a0a 5573 6573 2074 6865 2072 ====..Uses the r │ │ │ │ -0002d6f0: 616e 6b73 206f 6620 7468 6520 4220 6d61 anks of the B ma │ │ │ │ -0002d700: 7472 6963 6573 2069 6e20 6120 6d61 7472 trices in a matr │ │ │ │ -0002d710: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -0002d720: 2066 6f72 2061 206d 6f64 756c 6520 4d20 for a module M │ │ │ │ -0002d730: 6f76 6572 0a53 2f28 665f 312c 2e2e 2c66 over.S/(f_1,..,f │ │ │ │ -0002d740: 5f63 2920 746f 2063 6f6d 7075 7465 2074 _c) to compute t │ │ │ │ -0002d750: 6865 2062 6574 7469 206e 756d 6265 7273 he betti numbers │ │ │ │ -0002d760: 206f 6620 7468 6520 6d69 6e69 6d61 6c20 of the minimal │ │ │ │ -0002d770: 7265 736f 6c75 7469 6f6e 206f 6620 4d20 resolution of M │ │ │ │ -0002d780: 6f76 6572 0a52 2c20 7768 6963 6820 6973 over.R, which is │ │ │ │ -0002d790: 2074 6865 2073 756d 206f 6620 7468 6520 the sum of the │ │ │ │ -0002d7a0: 6469 7669 6465 6420 706f 7765 7220 616c divided power al │ │ │ │ -0002d7b0: 6765 6272 6173 206f 6e20 632d 6a2b 3120 gebras on c-j+1 │ │ │ │ -0002d7c0: 7661 7269 6162 6c65 7320 7465 6e73 6f72 variables tensor │ │ │ │ -0002d7d0: 6564 0a77 6974 6820 4228 6a29 2e0a 0a2b ed.with B(j)...+ │ │ │ │ +0002d550: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0002d560: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +0002d570: 4c20 3d20 6669 6e69 7465 4265 7474 694e L = finiteBettiN │ │ │ │ +0002d580: 756d 6265 7273 2028 4d46 2c6c 656e 290a umbers (MF,len). │ │ │ │ +0002d590: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002d5a0: 2020 2a20 4d46 2c20 6120 2a6e 6f74 6520 * MF, a *note │ │ │ │ +0002d5b0: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0002d5c0: 446f 6329 4c69 7374 2c2c 204c 6973 7420 Doc)List,, List │ │ │ │ +0002d5d0: 6f66 2048 6173 6854 6162 6c65 7320 6173 of HashTables as │ │ │ │ +0002d5e0: 2063 6f6d 7075 7465 640a 2020 2020 2020 computed. │ │ │ │ +0002d5f0: 2020 6279 2022 6d61 7472 6978 4661 6374 by "matrixFact │ │ │ │ +0002d600: 6f72 697a 6174 696f 6e22 0a20 2020 2020 orization". │ │ │ │ +0002d610: 202a 206c 656e 2c20 616e 202a 6e6f 7465 * len, an *note │ │ │ │ +0002d620: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ +0002d630: 6c61 7932 446f 6329 5a5a 2c2c 206c 656e lay2Doc)ZZ,, len │ │ │ │ +0002d640: 6774 6820 6f66 2062 6574 7469 206e 756d gth of betti num │ │ │ │ +0002d650: 6265 720a 2020 2020 2020 2020 7365 7175 ber. sequ │ │ │ │ +0002d660: 656e 6365 2074 6f20 7072 6f64 7563 650a ence to produce. │ │ │ │ +0002d670: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0002d680: 2020 202a 204c 2c20 6120 2a6e 6f74 6520 * L, a *note │ │ │ │ +0002d690: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0002d6a0: 446f 6329 4c69 7374 2c2c 204c 6973 7420 Doc)List,, List │ │ │ │ +0002d6b0: 6f66 2062 6574 7469 206e 756d 6265 7273 of betti numbers │ │ │ │ +0002d6c0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0002d6d0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a55 7365 7320 =========..Uses │ │ │ │ +0002d6e0: 7468 6520 7261 6e6b 7320 6f66 2074 6865 the ranks of the │ │ │ │ +0002d6f0: 2042 206d 6174 7269 6365 7320 696e 2061 B matrices in a │ │ │ │ +0002d700: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +0002d710: 6174 696f 6e20 666f 7220 6120 6d6f 6475 ation for a modu │ │ │ │ +0002d720: 6c65 204d 206f 7665 720a 532f 2866 5f31 le M over.S/(f_1 │ │ │ │ +0002d730: 2c2e 2e2c 665f 6329 2074 6f20 636f 6d70 ,..,f_c) to comp │ │ │ │ +0002d740: 7574 6520 7468 6520 6265 7474 6920 6e75 ute the betti nu │ │ │ │ +0002d750: 6d62 6572 7320 6f66 2074 6865 206d 696e mbers of the min │ │ │ │ +0002d760: 696d 616c 2072 6573 6f6c 7574 696f 6e20 imal resolution │ │ │ │ +0002d770: 6f66 204d 206f 7665 720a 522c 2077 6869 of M over.R, whi │ │ │ │ +0002d780: 6368 2069 7320 7468 6520 7375 6d20 6f66 ch is the sum of │ │ │ │ +0002d790: 2074 6865 2064 6976 6964 6564 2070 6f77 the divided pow │ │ │ │ +0002d7a0: 6572 2061 6c67 6562 7261 7320 6f6e 2063 er algebras on c │ │ │ │ +0002d7b0: 2d6a 2b31 2076 6172 6961 626c 6573 2074 -j+1 variables t │ │ │ │ +0002d7c0: 656e 736f 7265 640a 7769 7468 2042 286a ensored.with B(j │ │ │ │ +0002d7d0: 292e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d )...+----------- │ │ │ │ 0002d7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d810: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -0002d820: 646f 6d53 6565 6420 3020 2020 2020 2020 domSeed 0 │ │ │ │ -0002d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d840: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ -0002d850: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ -0002d860: 6420 746f 2030 2020 2020 2020 2020 2020 d to 0 │ │ │ │ -0002d870: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002d800: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ +0002d810: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +0002d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d830: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002d840: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +0002d850: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d870: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8b0: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +0002d8a0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +0002d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d8e0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002d8d0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002d8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002d900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002d910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0002d920: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020 : kk = ZZ/101 │ │ │ │ +0002d910: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f +.|i2 : kk = ZZ/ │ │ │ │ +0002d920: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ 0002d930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d950: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002d940: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002d950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d980: 2020 2020 207c 0a7c 6f32 203d 206b 6b20 |.|o2 = kk │ │ │ │ +0002d970: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0002d980: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 0002d990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d9b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002d9a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002d9b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d9e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002d9f0: 0a7c 6f32 203a 2051 756f 7469 656e 7452 .|o2 : QuotientR │ │ │ │ -0002da00: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0002da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002da20: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002d9e0: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74 |.|o2 : Quot │ │ │ │ +0002d9f0: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0002da00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da10: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002da50: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0002da60: 2053 203d 206b 6b5b 612c 622c 752c 765d S = kk[a,b,u,v] │ │ │ │ +0002da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002da50: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62 |i3 : S = kk[a,b │ │ │ │ +0002da60: 2c75 2c76 5d20 2020 2020 2020 2020 2020 ,u,v] │ │ │ │ 0002da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002da80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002da90: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002da80: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002da90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dac0: 2020 207c 0a7c 6f33 203d 2053 2020 2020 |.|o3 = S │ │ │ │ +0002dab0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +0002dac0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0002dad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002daf0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002dae0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002db10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002db30: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52 o3 : PolynomialR │ │ │ │ -0002db40: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0002db50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002db60: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002db20: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f |.|o3 : Polyno │ │ │ │ +0002db30: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0002db40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002db50: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002db60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002db70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002db90: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ -0002dba0: 6620 3d20 6d61 7472 6978 2261 752c 6276 f = matrix"au,bv │ │ │ │ -0002dbb0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0002dbc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002db80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002db90: 3420 3a20 6666 203d 206d 6174 7269 7822 4 : ff = matrix" │ │ │ │ +0002dba0: 6175 2c62 7622 2020 2020 2020 2020 2020 au,bv" │ │ │ │ +0002dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dbc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc00: 207c 0a7c 6f34 203d 207c 2061 7520 6276 |.|o4 = | au bv │ │ │ │ -0002dc10: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc30: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002dbf0: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20 |.|o4 = | │ │ │ │ +0002dc00: 6175 2062 7620 7c20 2020 2020 2020 2020 au bv | │ │ │ │ +0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc60: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0002dc70: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0002dc80: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dca0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053 |.|o4 : Matrix S │ │ │ │ -0002dcb0: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcd0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002dc60: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002dc70: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0002dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc90: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ +0002dca0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dcc0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002dce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -0002dd10: 3a20 5220 3d20 532f 6964 6561 6c20 6666 : R = S/ideal ff │ │ │ │ +0002dcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002dd00: 0a7c 6935 203a 2052 203d 2053 2f69 6465 .|i5 : R = S/ide │ │ │ │ +0002dd10: 616c 2066 6620 2020 2020 2020 2020 2020 al ff │ │ │ │ 0002dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002dd40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002dd30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd70: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +0002dd60: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +0002dd70: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0002dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dda0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002dd90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002dda0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002ddd0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002dde0: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269 |o5 : QuotientRi │ │ │ │ -0002ddf0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -0002de00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002de10: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002ddd0: 2020 207c 0a7c 6f35 203a 2051 756f 7469 |.|o5 : Quoti │ │ │ │ +0002dde0: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0002ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de00: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002de10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002de20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002de40: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -0002de50: 4d30 203d 2052 5e31 2f69 6465 616c 2261 M0 = R^1/ideal"a │ │ │ │ -0002de60: 2c62 2220 2020 2020 2020 2020 2020 2020 ,b" │ │ │ │ -0002de70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002de30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002de40: 6936 203a 204d 3020 3d20 525e 312f 6964 i6 : M0 = R^1/id │ │ │ │ +0002de50: 6561 6c22 612c 6222 2020 2020 2020 2020 eal"a,b" │ │ │ │ +0002de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002de70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002deb0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ -0002dec0: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ -0002ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dee0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002dea0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ +0002deb0: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20 okernel | a b | │ │ │ │ +0002dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002ded0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002df10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df30: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0002df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df50: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c |.|o6 : R-modul │ │ │ │ -0002df60: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ -0002df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002df80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002df30: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0002df40: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ +0002df50: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0002df60: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +0002df70: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002df80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002df90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002dfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0002dfc0: 203a 2046 203d 2066 7265 6552 6573 6f6c : F = freeResol │ │ │ │ -0002dfd0: 7574 696f 6e28 4d30 2c20 4c65 6e67 7468 ution(M0, Length │ │ │ │ -0002dfe0: 4c69 6d69 7420 3d3e 3329 2020 2020 2020 Limit =>3) │ │ │ │ -0002dff0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002dfb0: 2b0a 7c69 3720 3a20 4620 3d20 6672 6565 +.|i7 : F = free │ │ │ │ +0002dfc0: 5265 736f 6c75 7469 6f6e 284d 302c 204c Resolution(M0, L │ │ │ │ +0002dfd0: 656e 6774 684c 696d 6974 203d 3e33 2920 engthLimit =>3) │ │ │ │ +0002dfe0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e020: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ -0002e030: 2020 2020 2032 2020 2020 2020 3320 2020 2 3 │ │ │ │ -0002e040: 2020 2034 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0002e050: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0002e060: 3d20 5220 203c 2d2d 2052 2020 3c2d 2d20 = R <-- R <-- │ │ │ │ -0002e070: 5220 203c 2d2d 2052 2020 2020 2020 2020 R <-- R │ │ │ │ -0002e080: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002e090: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0002e010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e020: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +0002e030: 2033 2020 2020 2020 3420 2020 2020 2020 3 4 │ │ │ │ +0002e040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002e050: 0a7c 6f37 203d 2052 2020 3c2d 2d20 5220 .|o7 = R <-- R │ │ │ │ +0002e060: 203c 2d2d 2052 2020 3c2d 2d20 5220 2020 <-- R <-- R │ │ │ │ +0002e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e0c0: 2020 2020 7c0a 7c20 2020 2020 3020 2020 |.| 0 │ │ │ │ -0002e0d0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0002e0e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ -0002e0f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002e0b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0002e0c0: 2030 2020 2020 2020 3120 2020 2020 2032 0 1 2 │ │ │ │ +0002e0d0: 2020 2020 2020 3320 2020 2020 2020 2020 3 │ │ │ │ +0002e0e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002e0f0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0002e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e120: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002e130: 7c6f 3720 3a20 436f 6d70 6c65 7820 2020 |o7 : Complex │ │ │ │ +0002e120: 2020 207c 0a7c 6f37 203a 2043 6f6d 706c |.|o7 : Compl │ │ │ │ +0002e130: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 0002e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e160: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002e150: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002e160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e190: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ -0002e1a0: 4d20 3d20 636f 6b65 7220 462e 6464 5f33 M = coker F.dd_3 │ │ │ │ -0002e1b0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -0002e1c0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002e180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0002e190: 6938 203a 204d 203d 2063 6f6b 6572 2046 i8 : M = coker F │ │ │ │ +0002e1a0: 2e64 645f 333b 2020 2020 2020 2020 2020 .dd_3; │ │ │ │ +0002e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e1c0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0002e1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e200: 2d2d 2b0a 7c69 3920 3a20 4d46 203d 206d --+.|i9 : MF = m │ │ │ │ -0002e210: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -0002e220: 6f6e 2866 662c 4d29 3b20 2020 2020 2020 on(ff,M); │ │ │ │ -0002e230: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002e1f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 204d -------+.|i9 : M │ │ │ │ +0002e200: 4620 3d20 6d61 7472 6978 4661 6374 6f72 F = matrixFactor │ │ │ │ +0002e210: 697a 6174 696f 6e28 6666 2c4d 293b 2020 ization(ff,M); │ │ │ │ +0002e220: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0002e230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0002e270: 3130 203a 2062 6574 7469 2066 7265 6552 10 : betti freeR │ │ │ │ -0002e280: 6573 6f6c 7574 696f 6e20 7075 7368 466f esolution pushFo │ │ │ │ -0002e290: 7277 6172 6428 6d61 7028 522c 5329 2c4d rward(map(R,S),M │ │ │ │ -0002e2a0: 297c 0a7c 2020 2020 2020 2020 2020 2020 )|.| │ │ │ │ +0002e260: 2d2b 0a7c 6931 3020 3a20 6265 7474 6920 -+.|i10 : betti │ │ │ │ +0002e270: 6672 6565 5265 736f 6c75 7469 6f6e 2070 freeResolution p │ │ │ │ +0002e280: 7573 6846 6f72 7761 7264 286d 6170 2852 ushForward(map(R │ │ │ │ +0002e290: 2c53 292c 4d29 7c0a 7c20 2020 2020 2020 ,S),M)|.| │ │ │ │ +0002e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e2d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002e2e0: 2020 2020 2020 3020 3120 3220 2020 2020 0 1 2 │ │ │ │ +0002e2c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e2d0: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +0002e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e300: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002e310: 3020 3d20 746f 7461 6c3a 2033 2035 2032 0 = total: 3 5 2 │ │ │ │ +0002e300: 7c0a 7c6f 3130 203d 2074 6f74 616c 3a20 |.|o10 = total: │ │ │ │ +0002e310: 3320 3520 3220 2020 2020 2020 2020 2020 3 5 2 │ │ │ │ 0002e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e340: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ -0002e350: 3320 3420 2e20 2020 2020 2020 2020 2020 3 4 . │ │ │ │ -0002e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e370: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0002e380: 2020 333a 202e 2031 2032 2020 2020 2020 3: . 1 2 │ │ │ │ -0002e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e330: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0002e340: 2020 323a 2033 2034 202e 2020 2020 2020 2: 3 4 . │ │ │ │ +0002e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e360: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002e370: 2020 2020 2020 2033 3a20 2e20 3120 3220 3: . 1 2 │ │ │ │ +0002e380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e390: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002e3a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e3d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002e3e0: 0a7c 6f31 3020 3a20 4265 7474 6954 616c .|o10 : BettiTal │ │ │ │ -0002e3f0: 6c79 2020 2020 2020 2020 2020 2020 2020 ly │ │ │ │ -0002e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e410: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0002e3d0: 2020 2020 7c0a 7c6f 3130 203a 2042 6574 |.|o10 : Bet │ │ │ │ +0002e3e0: 7469 5461 6c6c 7920 2020 2020 2020 2020 tiTally │ │ │ │ +0002e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e400: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0002e410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e440: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ -0002e450: 3a20 6669 6e69 7465 4265 7474 694e 756d : finiteBettiNum │ │ │ │ -0002e460: 6265 7273 204d 4620 2020 2020 2020 2020 bers MF │ │ │ │ -0002e470: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002e480: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0002e440: 7c69 3131 203a 2066 696e 6974 6542 6574 |i11 : finiteBet │ │ │ │ +0002e450: 7469 4e75 6d62 6572 7320 4d46 2020 2020 tiNumbers MF │ │ │ │ +0002e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e470: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4b0: 2020 207c 0a7c 6f31 3120 3d20 7b33 2c20 |.|o11 = {3, │ │ │ │ -0002e4c0: 352c 2032 7d20 2020 2020 2020 2020 2020 5, 2} │ │ │ │ -0002e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e4e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0002e4a0: 2020 2020 2020 2020 7c0a 7c6f 3131 203d |.|o11 = │ │ │ │ +0002e4b0: 207b 332c 2035 2c20 327d 2020 2020 2020 {3, 5, 2} │ │ │ │ +0002e4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e4d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0002e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e510: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0002e520: 6f31 3120 3a20 4c69 7374 2020 2020 2020 o11 : List │ │ │ │ +0002e510: 2020 7c0a 7c6f 3131 203a 204c 6973 7420 |.|o11 : List │ │ │ │ +0002e520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e550: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002e540: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002e550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e580: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -0002e590: 696e 6669 6e69 7465 4265 7474 694e 756d infiniteBettiNum │ │ │ │ -0002e5a0: 6265 7273 284d 462c 3529 2020 2020 2020 bers(MF,5) │ │ │ │ -0002e5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0002e570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0002e580: 3132 203a 2069 6e66 696e 6974 6542 6574 12 : infiniteBet │ │ │ │ +0002e590: 7469 4e75 6d62 6572 7328 4d46 2c35 2920 tiNumbers(MF,5) │ │ │ │ +0002e5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0002e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e5f0: 207c 0a7c 6f31 3220 3d20 7b33 2c20 342c |.|o12 = {3, 4, │ │ │ │ -0002e600: 2035 2c20 362c 2037 2c20 387d 2020 2020 5, 6, 7, 8} │ │ │ │ -0002e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e620: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002e5e0: 2020 2020 2020 7c0a 7c6f 3132 203d 207b |.|o12 = { │ │ │ │ +0002e5f0: 332c 2034 2c20 352c 2036 2c20 372c 2038 3, 4, 5, 6, 7, 8 │ │ │ │ +0002e600: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0002e610: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002e620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e650: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0002e660: 3220 3a20 4c69 7374 2020 2020 2020 2020 2 : List │ │ │ │ +0002e650: 7c0a 7c6f 3132 203a 204c 6973 7420 2020 |.|o12 : List │ │ │ │ +0002e660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e690: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0002e680: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0002e690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e6c0: 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 6265 -----+.|i13 : be │ │ │ │ -0002e6d0: 7474 6920 6672 6565 5265 736f 6c75 7469 tti freeResoluti │ │ │ │ -0002e6e0: 6f6e 2028 4d2c 204c 656e 6774 684c 696d on (M, LengthLim │ │ │ │ -0002e6f0: 6974 203d 3e20 3529 2020 7c0a 7c20 2020 it => 5) |.| │ │ │ │ +0002e6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 ----------+.|i13 │ │ │ │ +0002e6c0: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ +0002e6d0: 6f6c 7574 696f 6e20 284d 2c20 4c65 6e67 olution (M, Leng │ │ │ │ +0002e6e0: 7468 4c69 6d69 7420 3d3e 2035 2920 207c thLimit => 5) | │ │ │ │ +0002e6f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e720: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002e730: 0a7c 2020 2020 2020 2020 2020 2020 2030 .| 0 │ │ │ │ -0002e740: 2031 2032 2033 2034 2035 2020 2020 2020 1 2 3 4 5 │ │ │ │ -0002e750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e760: 2020 2020 7c0a 7c6f 3133 203d 2074 6f74 |.|o13 = tot │ │ │ │ -0002e770: 616c 3a20 3320 3420 3520 3620 3720 3820 al: 3 4 5 6 7 8 │ │ │ │ -0002e780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e790: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0002e7a0: 2020 2020 2020 323a 2033 2034 2035 2036 2: 3 4 5 6 │ │ │ │ -0002e7b0: 2037 2038 2020 2020 2020 2020 2020 2020 7 8 │ │ │ │ -0002e7c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002e7d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0002e720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0002e730: 2020 2020 3020 3120 3220 3320 3420 3520 0 1 2 3 4 5 │ │ │ │ +0002e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e750: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0002e760: 3d20 746f 7461 6c3a 2033 2034 2035 2036 = total: 3 4 5 6 │ │ │ │ +0002e770: 2037 2038 2020 2020 2020 2020 2020 2020 7 8 │ │ │ │ +0002e780: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0002e790: 7c20 2020 2020 2020 2020 2032 3a20 3320 | 2: 3 │ │ │ │ +0002e7a0: 3420 3520 3620 3720 3820 2020 2020 2020 4 5 6 7 8 │ │ │ │ +0002e7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e7c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e800: 2020 207c 0a7c 6f31 3320 3a20 4265 7474 |.|o13 : Bett │ │ │ │ -0002e810: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -0002e820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002e830: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0002e7f0: 2020 2020 2020 2020 7c0a 7c6f 3133 203a |.|o13 : │ │ │ │ +0002e800: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0002e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002e820: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0002e830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002e850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002e860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -0002e870: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ -0002e880: 3d0a 0a20 202a 202a 6e6f 7465 206d 6174 =.. * *note mat │ │ │ │ -0002e890: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0002e8a0: 3a20 6d61 7472 6978 4661 6374 6f72 697a : matrixFactoriz │ │ │ │ -0002e8b0: 6174 696f 6e2c 202d 2d20 4d61 7073 2069 ation, -- Maps i │ │ │ │ -0002e8c0: 6e20 6120 6869 6768 6572 0a20 2020 2063 n a higher. c │ │ │ │ -0002e8d0: 6f64 696d 656e 7369 6f6e 206d 6174 7269 odimension matri │ │ │ │ -0002e8e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ -0002e8f0: 2020 2a20 2a6e 6f74 6520 6669 6e69 7465 * *note finite │ │ │ │ -0002e900: 4265 7474 694e 756d 6265 7273 3a20 6669 BettiNumbers: fi │ │ │ │ -0002e910: 6e69 7465 4265 7474 694e 756d 6265 7273 niteBettiNumbers │ │ │ │ -0002e920: 2c20 2d2d 2062 6574 7469 206e 756d 6265 , -- betti numbe │ │ │ │ -0002e930: 7273 206f 6620 6669 6e69 7465 0a20 2020 rs of finite. │ │ │ │ -0002e940: 2072 6573 6f6c 7574 696f 6e20 636f 6d70 resolution comp │ │ │ │ -0002e950: 7574 6564 2066 726f 6d20 6120 6d61 7472 uted from a matr │ │ │ │ -0002e960: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -0002e970: 0a0a 5761 7973 2074 6f20 7573 6520 696e ..Ways to use in │ │ │ │ -0002e980: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ -0002e990: 7273 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d rs:.============ │ │ │ │ -0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002e9b0: 3d3d 3d3d 3d0a 0a20 202a 2022 696e 6669 =====.. * "infi │ │ │ │ -0002e9c0: 6e69 7465 4265 7474 694e 756d 6265 7273 niteBettiNumbers │ │ │ │ -0002e9d0: 284c 6973 742c 5a5a 2922 0a0a 466f 7220 (List,ZZ)"..For │ │ │ │ -0002e9e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0002e9f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002ea00: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0002ea10: 6f74 6520 696e 6669 6e69 7465 4265 7474 ote infiniteBett │ │ │ │ -0002ea20: 694e 756d 6265 7273 3a20 696e 6669 6e69 iNumbers: infini │ │ │ │ -0002ea30: 7465 4265 7474 694e 756d 6265 7273 2c20 teBettiNumbers, │ │ │ │ -0002ea40: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0002ea50: 640a 6675 6e63 7469 6f6e 3a20 284d 6163 d.function: (Mac │ │ │ │ -0002ea60: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0002ea70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0002e860: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ +0002e870: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +0002e880: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ +0002e890: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ +0002e8a0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ +0002e8b0: 6170 7320 696e 2061 2068 6967 6865 720a aps in a higher. │ │ │ │ +0002e8c0: 2020 2020 636f 6469 6d65 6e73 696f 6e20 codimension │ │ │ │ +0002e8d0: 6d61 7472 6978 2066 6163 746f 7269 7a61 matrix factoriza │ │ │ │ +0002e8e0: 7469 6f6e 0a20 202a 202a 6e6f 7465 2066 tion. * *note f │ │ │ │ +0002e8f0: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ +0002e900: 733a 2066 696e 6974 6542 6574 7469 4e75 s: finiteBettiNu │ │ │ │ +0002e910: 6d62 6572 732c 202d 2d20 6265 7474 6920 mbers, -- betti │ │ │ │ +0002e920: 6e75 6d62 6572 7320 6f66 2066 696e 6974 numbers of finit │ │ │ │ +0002e930: 650a 2020 2020 7265 736f 6c75 7469 6f6e e. resolution │ │ │ │ +0002e940: 2063 6f6d 7075 7465 6420 6672 6f6d 2061 computed from a │ │ │ │ +0002e950: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +0002e960: 6174 696f 6e0a 0a57 6179 7320 746f 2075 ation..Ways to u │ │ │ │ +0002e970: 7365 2069 6e66 696e 6974 6542 6574 7469 se infiniteBetti │ │ │ │ +0002e980: 4e75 6d62 6572 733a 0a3d 3d3d 3d3d 3d3d Numbers:.======= │ │ │ │ +0002e990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002e9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0002e9b0: 2269 6e66 696e 6974 6542 6574 7469 4e75 "infiniteBettiNu │ │ │ │ +0002e9c0: 6d62 6572 7328 4c69 7374 2c5a 5a29 220a mbers(List,ZZ)". │ │ │ │ +0002e9d0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0002e9e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0002e9f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0002ea00: 6374 202a 6e6f 7465 2069 6e66 696e 6974 ct *note infinit │ │ │ │ +0002ea10: 6542 6574 7469 4e75 6d62 6572 733a 2069 eBettiNumbers: i │ │ │ │ +0002ea20: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ +0002ea30: 6572 732c 2069 7320 6120 2a6e 6f74 6520 ers, is a *note │ │ │ │ +0002ea40: 6d65 7468 6f64 0a66 756e 6374 696f 6e3a method.function: │ │ │ │ +0002ea50: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0002ea60: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0002ea70: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0002ea80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ea90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002eaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002eab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0002ead0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0002eae0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0002eaf0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0002eb00: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0002eb10: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -0002eb20: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0002eb30: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -0002eb40: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0002eb50: 6f6e 732e 6d32 3a34 3131 333a 302e 0a1f ons.m2:4113:0... │ │ │ │ -0002eb60: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -0002eb70: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0002eb80: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -0002eb90: 653a 2069 734c 696e 6561 722c 204e 6578 e: isLinear, Nex │ │ │ │ -0002eba0: 743a 2069 7351 7561 7369 5265 6775 6c61 t: isQuasiRegula │ │ │ │ -0002ebb0: 722c 2050 7265 763a 2069 6e66 696e 6974 r, Prev: infinit │ │ │ │ -0002ebc0: 6542 6574 7469 4e75 6d62 6572 732c 2055 eBettiNumbers, U │ │ │ │ -0002ebd0: 703a 2054 6f70 0a0a 6973 4c69 6e65 6172 p: Top..isLinear │ │ │ │ -0002ebe0: 202d 2d20 6368 6563 6b20 7768 6574 6865 -- check whethe │ │ │ │ -0002ebf0: 7220 6d61 7472 6978 2065 6e74 7269 6573 r matrix entries │ │ │ │ -0002ec00: 2068 6176 6520 6465 6772 6565 2031 0a2a have degree 1.* │ │ │ │ +0002eac0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0002ead0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0002eae0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0002eaf0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0002eb00: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +0002eb10: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0002eb20: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +0002eb30: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0002eb40: 6f6c 7574 696f 6e73 2e6d 323a 3431 3133 olutions.m2:4113 │ │ │ │ +0002eb50: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +0002eb60: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0002eb70: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +0002eb80: 2c20 4e6f 6465 3a20 6973 4c69 6e65 6172 , Node: isLinear │ │ │ │ +0002eb90: 2c20 4e65 7874 3a20 6973 5175 6173 6952 , Next: isQuasiR │ │ │ │ +0002eba0: 6567 756c 6172 2c20 5072 6576 3a20 696e egular, Prev: in │ │ │ │ +0002ebb0: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ +0002ebc0: 7273 2c20 5570 3a20 546f 700a 0a69 734c rs, Up: Top..isL │ │ │ │ +0002ebd0: 696e 6561 7220 2d2d 2063 6865 636b 2077 inear -- check w │ │ │ │ +0002ebe0: 6865 7468 6572 206d 6174 7269 7820 656e hether matrix en │ │ │ │ +0002ebf0: 7472 6965 7320 6861 7665 2064 6567 7265 tries have degre │ │ │ │ +0002ec00: 6520 310a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e 1.************ │ │ │ │ 0002ec10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002ec20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002ec40: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ -0002ec50: 3a20 0a20 2020 2020 2020 2062 203d 2069 : . b = i │ │ │ │ -0002ec60: 734c 696e 6561 7220 4d0a 2020 2a20 496e sLinear M. * In │ │ │ │ -0002ec70: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c puts:. * M, │ │ │ │ -0002ec80: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ -0002ec90: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -0002eca0: 6174 7269 782c 2c20 0a20 202a 204f 7574 atrix,, . * Out │ │ │ │ -0002ecb0: 7075 7473 3a0a 2020 2020 2020 2a20 622c puts:. * b, │ │ │ │ -0002ecc0: 2061 202a 6e6f 7465 2042 6f6f 6c65 616e a *note Boolean │ │ │ │ -0002ecd0: 2076 616c 7565 3a20 284d 6163 6175 6c61 value: (Macaula │ │ │ │ -0002ece0: 7932 446f 6329 426f 6f6c 6561 6e2c 2c20 y2Doc)Boolean,, │ │ │ │ -0002ecf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0002ed00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a4e 6f74 6520 =========..Note │ │ │ │ -0002ed10: 7468 6174 2061 206c 696e 6561 7220 6d61 that a linear ma │ │ │ │ -0002ed20: 7472 6978 2c20 696e 2074 6869 7320 7365 trix, in this se │ │ │ │ -0002ed30: 6e73 652c 2063 616e 2073 7469 6c6c 2068 nse, can still h │ │ │ │ -0002ed40: 6176 6520 6469 6666 6572 656e 7420 7461 ave different ta │ │ │ │ -0002ed50: 7267 6574 0a64 6567 7265 6573 2028 696e rget.degrees (in │ │ │ │ -0002ed60: 2077 6869 6368 2063 6173 6520 7468 6520 which case the │ │ │ │ -0002ed70: 636f 6b65 726e 656c 2064 6563 6f6d 706f cokernel decompo │ │ │ │ -0002ed80: 7365 7320 696e 746f 2061 2064 6972 6563 ses into a direc │ │ │ │ -0002ed90: 7420 7375 6d20 6279 2067 656e 6572 6174 t sum by generat │ │ │ │ -0002eda0: 6f72 0a64 6567 7265 652e 290a 0a57 6179 or.degree.)..Way │ │ │ │ -0002edb0: 7320 746f 2075 7365 2069 734c 696e 6561 s to use isLinea │ │ │ │ -0002edc0: 723a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r:.============= │ │ │ │ -0002edd0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2269 ========.. * "i │ │ │ │ -0002ede0: 734c 696e 6561 7228 4d61 7472 6978 2922 sLinear(Matrix)" │ │ │ │ -0002edf0: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -0002ee00: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -0002ee10: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -0002ee20: 6563 7420 2a6e 6f74 6520 6973 4c69 6e65 ect *note isLine │ │ │ │ -0002ee30: 6172 3a20 6973 4c69 6e65 6172 2c20 6973 ar: isLinear, is │ │ │ │ -0002ee40: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -0002ee50: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -0002ee60: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -0002ee70: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +0002ec30: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0002ec40: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +0002ec50: 6220 3d20 6973 4c69 6e65 6172 204d 0a20 b = isLinear M. │ │ │ │ +0002ec60: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +0002ec70: 202a 204d 2c20 6120 2a6e 6f74 6520 6d61 * M, a *note ma │ │ │ │ +0002ec80: 7472 6978 3a20 284d 6163 6175 6c61 7932 trix: (Macaulay2 │ │ │ │ +0002ec90: 446f 6329 4d61 7472 6978 2c2c 200a 2020 Doc)Matrix,, . │ │ │ │ +0002eca0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0002ecb0: 202a 2062 2c20 6120 2a6e 6f74 6520 426f * b, a *note Bo │ │ │ │ +0002ecc0: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61 olean value: (Ma │ │ │ │ +0002ecd0: 6361 756c 6179 3244 6f63 2942 6f6f 6c65 caulay2Doc)Boole │ │ │ │ +0002ece0: 616e 2c2c 200a 0a44 6573 6372 6970 7469 an,, ..Descripti │ │ │ │ +0002ecf0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0002ed00: 4e6f 7465 2074 6861 7420 6120 6c69 6e65 Note that a line │ │ │ │ +0002ed10: 6172 206d 6174 7269 782c 2069 6e20 7468 ar matrix, in th │ │ │ │ +0002ed20: 6973 2073 656e 7365 2c20 6361 6e20 7374 is sense, can st │ │ │ │ +0002ed30: 696c 6c20 6861 7665 2064 6966 6665 7265 ill have differe │ │ │ │ +0002ed40: 6e74 2074 6172 6765 740a 6465 6772 6565 nt target.degree │ │ │ │ +0002ed50: 7320 2869 6e20 7768 6963 6820 6361 7365 s (in which case │ │ │ │ +0002ed60: 2074 6865 2063 6f6b 6572 6e65 6c20 6465 the cokernel de │ │ │ │ +0002ed70: 636f 6d70 6f73 6573 2069 6e74 6f20 6120 composes into a │ │ │ │ +0002ed80: 6469 7265 6374 2073 756d 2062 7920 6765 direct sum by ge │ │ │ │ +0002ed90: 6e65 7261 746f 720a 6465 6772 6565 2e29 nerator.degree.) │ │ │ │ +0002eda0: 0a0a 5761 7973 2074 6f20 7573 6520 6973 ..Ways to use is │ │ │ │ +0002edb0: 4c69 6e65 6172 3a0a 3d3d 3d3d 3d3d 3d3d Linear:.======== │ │ │ │ +0002edc0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +0002edd0: 202a 2022 6973 4c69 6e65 6172 284d 6174 * "isLinear(Mat │ │ │ │ +0002ede0: 7269 7829 220a 0a46 6f72 2074 6865 2070 rix)"..For the p │ │ │ │ +0002edf0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +0002ee00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +0002ee10: 6520 6f62 6a65 6374 202a 6e6f 7465 2069 e object *note i │ │ │ │ +0002ee20: 734c 696e 6561 723a 2069 734c 696e 6561 sLinear: isLinea │ │ │ │ +0002ee30: 722c 2069 7320 6120 2a6e 6f74 6520 6d65 r, is a *note me │ │ │ │ +0002ee40: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ +0002ee50: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ +0002ee60: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0002ee70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ee80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002ee90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002eec0: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -0002eed0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -0002eee0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -0002eef0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -0002ef00: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -0002ef10: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -0002ef20: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -0002ef30: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -0002ef40: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -0002ef50: 732e 6d32 3a33 3436 333a 302e 0a1f 0a46 s.m2:3463:0....F │ │ │ │ -0002ef60: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -0002ef70: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -0002ef80: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -0002ef90: 2069 7351 7561 7369 5265 6775 6c61 722c isQuasiRegular, │ │ │ │ -0002efa0: 204e 6578 743a 2069 7353 7461 626c 7954 Next: isStablyT │ │ │ │ -0002efb0: 7269 7669 616c 2c20 5072 6576 3a20 6973 rivial, Prev: is │ │ │ │ -0002efc0: 4c69 6e65 6172 2c20 5570 3a20 546f 700a Linear, Up: Top. │ │ │ │ -0002efd0: 0a69 7351 7561 7369 5265 6775 6c61 7220 .isQuasiRegular │ │ │ │ -0002efe0: 2d2d 2074 6573 7473 2061 206d 6174 7269 -- tests a matri │ │ │ │ -0002eff0: 7820 6f72 2073 6571 7565 6e63 6520 6f72 x or sequence or │ │ │ │ -0002f000: 206c 6973 7420 666f 7220 7175 6173 692d list for quasi- │ │ │ │ -0002f010: 7265 6775 6c61 7269 7479 206f 6e20 6120 regularity on a │ │ │ │ -0002f020: 6d6f 6475 6c65 0a2a 2a2a 2a2a 2a2a 2a2a module.********* │ │ │ │ +0002eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +0002eec0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +0002eed0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +0002eee0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +0002eef0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +0002ef00: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +0002ef10: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +0002ef20: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +0002ef30: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +0002ef40: 7574 696f 6e73 2e6d 323a 3334 3633 3a30 utions.m2:3463:0 │ │ │ │ +0002ef50: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +0002ef60: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0002ef70: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +0002ef80: 4e6f 6465 3a20 6973 5175 6173 6952 6567 Node: isQuasiReg │ │ │ │ +0002ef90: 756c 6172 2c20 4e65 7874 3a20 6973 5374 ular, Next: isSt │ │ │ │ +0002efa0: 6162 6c79 5472 6976 6961 6c2c 2050 7265 ablyTrivial, Pre │ │ │ │ +0002efb0: 763a 2069 734c 696e 6561 722c 2055 703a v: isLinear, Up: │ │ │ │ +0002efc0: 2054 6f70 0a0a 6973 5175 6173 6952 6567 Top..isQuasiReg │ │ │ │ +0002efd0: 756c 6172 202d 2d20 7465 7374 7320 6120 ular -- tests a │ │ │ │ +0002efe0: 6d61 7472 6978 206f 7220 7365 7175 656e matrix or sequen │ │ │ │ +0002eff0: 6365 206f 7220 6c69 7374 2066 6f72 2071 ce or list for q │ │ │ │ +0002f000: 7561 7369 2d72 6567 756c 6172 6974 7920 uasi-regularity │ │ │ │ +0002f010: 6f6e 2061 206d 6f64 756c 650a 2a2a 2a2a on a module.**** │ │ │ │ +0002f020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f050: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002f060: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002f070: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -0002f080: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0002f090: 2020 7420 3d20 6973 5175 6173 6952 6567 t = isQuasiReg │ │ │ │ -0002f0a0: 756c 6172 2866 662c 4d29 0a20 202a 2049 ular(ff,M). * I │ │ │ │ -0002f0b0: 6e70 7574 733a 0a20 2020 2020 202a 2066 nputs:. * f │ │ │ │ -0002f0c0: 662c 2061 202a 6e6f 7465 206d 6174 7269 f, a *note matri │ │ │ │ -0002f0d0: 783a 2028 4d61 6361 756c 6179 3244 6f63 x: (Macaulay2Doc │ │ │ │ -0002f0e0: 294d 6174 7269 782c 2c20 0a20 2020 2020 )Matrix,, . │ │ │ │ -0002f0f0: 202a 2066 662c 2061 202a 6e6f 7465 206c * ff, a *note l │ │ │ │ -0002f100: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -0002f110: 6f63 294c 6973 742c 2c20 0a20 2020 2020 oc)List,, . │ │ │ │ -0002f120: 202a 2066 662c 2061 202a 6e6f 7465 2073 * ff, a *note s │ │ │ │ -0002f130: 6571 7565 6e63 653a 2028 4d61 6361 756c equence: (Macaul │ │ │ │ -0002f140: 6179 3244 6f63 2953 6571 7565 6e63 652c ay2Doc)Sequence, │ │ │ │ -0002f150: 2c20 0a20 2020 2020 202a 204d 2c20 6120 , . * M, a │ │ │ │ -0002f160: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -0002f170: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -0002f180: 6c65 2c2c 200a 2020 2a20 4f75 7470 7574 le,, . * Output │ │ │ │ -0002f190: 733a 0a20 2020 2020 202a 2074 2c20 6120 s:. * t, a │ │ │ │ -0002f1a0: 2a6e 6f74 6520 426f 6f6c 6561 6e20 7661 *note Boolean va │ │ │ │ -0002f1b0: 6c75 653a 2028 4d61 6361 756c 6179 3244 lue: (Macaulay2D │ │ │ │ -0002f1c0: 6f63 2942 6f6f 6c65 616e 2c2c 200a 0a44 oc)Boolean,, ..D │ │ │ │ -0002f1d0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -0002f1e0: 3d3d 3d3d 3d3d 0a0a 6666 2069 7320 7175 ======..ff is qu │ │ │ │ -0002f1f0: 6173 692d 7265 6775 6c61 7220 6966 2074 asi-regular if t │ │ │ │ -0002f200: 6865 206c 656e 6774 6820 6f66 2066 6620 he length of ff │ │ │ │ -0002f210: 6973 203c 3d20 6469 6d20 4d20 616e 6420 is <= dim M and │ │ │ │ -0002f220: 7468 6520 616e 6e69 6869 6c61 746f 7220 the annihilator │ │ │ │ -0002f230: 6f66 2066 665f 690a 6f6e 204d 2f28 6666 of ff_i.on M/(ff │ │ │ │ -0002f240: 5f30 2e2e 6666 5f7b 2869 2d31 2929 7d4d _0..ff_{(i-1))}M │ │ │ │ -0002f250: 2068 6173 2066 696e 6974 6520 6c65 6e67 has finite leng │ │ │ │ -0002f260: 7468 2066 6f72 2061 6c6c 2069 3d30 2e2e th for all i=0.. │ │ │ │ -0002f270: 286c 656e 6774 6820 6666 292d 312e 0a0a (length ff)-1... │ │ │ │ -0002f280: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ -0002f2b0: 3a20 6b6b 3d5a 5a2f 3130 313b 2020 2020 : kk=ZZ/101; │ │ │ │ -0002f2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f2d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f070: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0002f080: 2020 2020 2020 2074 203d 2069 7351 7561 t = isQua │ │ │ │ +0002f090: 7369 5265 6775 6c61 7228 6666 2c4d 290a siRegular(ff,M). │ │ │ │ +0002f0a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0002f0b0: 2020 2a20 6666 2c20 6120 2a6e 6f74 6520 * ff, a *note │ │ │ │ +0002f0c0: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +0002f0d0: 7932 446f 6329 4d61 7472 6978 2c2c 200a y2Doc)Matrix,, . │ │ │ │ +0002f0e0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0002f0f0: 6f74 6520 6c69 7374 3a20 284d 6163 6175 ote list: (Macau │ │ │ │ +0002f100: 6c61 7932 446f 6329 4c69 7374 2c2c 200a lay2Doc)List,, . │ │ │ │ +0002f110: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0002f120: 6f74 6520 7365 7175 656e 6365 3a20 284d ote sequence: (M │ │ │ │ +0002f130: 6163 6175 6c61 7932 446f 6329 5365 7175 acaulay2Doc)Sequ │ │ │ │ +0002f140: 656e 6365 2c2c 200a 2020 2020 2020 2a20 ence,, . * │ │ │ │ +0002f150: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ +0002f160: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +0002f170: 294d 6f64 756c 652c 2c20 0a20 202a 204f )Module,, . * O │ │ │ │ +0002f180: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +0002f190: 742c 2061 202a 6e6f 7465 2042 6f6f 6c65 t, a *note Boole │ │ │ │ +0002f1a0: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ +0002f1b0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ +0002f1c0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ +0002f1d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a66 6620 ===========..ff │ │ │ │ +0002f1e0: 6973 2071 7561 7369 2d72 6567 756c 6172 is quasi-regular │ │ │ │ +0002f1f0: 2069 6620 7468 6520 6c65 6e67 7468 206f if the length o │ │ │ │ +0002f200: 6620 6666 2069 7320 3c3d 2064 696d 204d f ff is <= dim M │ │ │ │ +0002f210: 2061 6e64 2074 6865 2061 6e6e 6968 696c and the annihil │ │ │ │ +0002f220: 6174 6f72 206f 6620 6666 5f69 0a6f 6e20 ator of ff_i.on │ │ │ │ +0002f230: 4d2f 2866 665f 302e 2e66 665f 7b28 692d M/(ff_0..ff_{(i- │ │ │ │ +0002f240: 3129 297d 4d20 6861 7320 6669 6e69 7465 1))}M has finite │ │ │ │ +0002f250: 206c 656e 6774 6820 666f 7220 616c 6c20 length for all │ │ │ │ +0002f260: 693d 302e 2e28 6c65 6e67 7468 2066 6629 i=0..(length ff) │ │ │ │ +0002f270: 2d31 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d -1...+---------- │ │ │ │ +0002f280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002f2a0: 0a7c 6931 203a 206b 6b3d 5a5a 2f31 3031 .|i1 : kk=ZZ/101 │ │ │ │ +0002f2b0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +0002f2c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f300: 2d2d 2b0a 7c69 3220 3a20 5320 3d20 6b6b --+.|i2 : S = kk │ │ │ │ -0002f310: 5b61 2c62 2c63 5d3b 2020 2020 2020 2020 [a,b,c]; │ │ │ │ -0002f320: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f330: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f350: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 ----------+.|i3 │ │ │ │ -0002f360: 3a20 4520 3d20 535e 312f 6964 6561 6c22 : E = S^1/ideal" │ │ │ │ -0002f370: 6162 222b 2b53 5e31 2f69 6465 616c 2076 ab"++S^1/ideal v │ │ │ │ -0002f380: 6172 7320 533b 7c0a 2b2d 2d2d 2d2d 2d2d ars S;|.+------- │ │ │ │ +0002f2f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ +0002f300: 203d 206b 6b5b 612c 622c 635d 3b20 2020 = kk[a,b,c]; │ │ │ │ +0002f310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f320: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002f330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002f350: 0a7c 6933 203a 2045 203d 2053 5e31 2f69 .|i3 : E = S^1/i │ │ │ │ +0002f360: 6465 616c 2261 6222 2b2b 535e 312f 6964 deal"ab"++S^1/id │ │ │ │ +0002f370: 6561 6c20 7661 7273 2053 3b7c 0a2b 2d2d eal vars S;|.+-- │ │ │ │ +0002f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f3a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f3b0: 2d2d 2b0a 7c69 3420 3a20 6631 203d 6d61 --+.|i4 : f1 =ma │ │ │ │ -0002f3c0: 7472 6978 2261 223b 2020 2020 2020 2020 trix"a"; │ │ │ │ -0002f3d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f3e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f400: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0002f410: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ -0002f420: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -0002f430: 2020 2020 2020 7c0a 7c6f 3420 3a20 4d61 |.|o4 : Ma │ │ │ │ -0002f440: 7472 6978 2053 2020 3c2d 2d20 5320 2020 trix S <-- S │ │ │ │ -0002f450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f460: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0002f3a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ +0002f3b0: 3120 3d6d 6174 7269 7822 6122 3b20 2020 1 =matrix"a"; │ │ │ │ +0002f3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f3f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f400: 0a7c 2020 2020 2020 2020 2020 2020 2031 .| 1 │ │ │ │ +0002f410: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +0002f420: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0002f430: 203a 204d 6174 7269 7820 5320 203c 2d2d : Matrix S <-- │ │ │ │ +0002f440: 2053 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0002f450: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0002f460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0002f490: 7c69 3520 3a20 6632 203d 6d61 7472 6978 |i5 : f2 =matrix │ │ │ │ -0002f4a0: 2261 2b62 2c63 223b 2020 2020 2020 2020 "a+b,c"; │ │ │ │ -0002f4b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0002f480: 2d2d 2d2b 0a7c 6935 203a 2066 3220 3d6d ---+.|i5 : f2 =m │ │ │ │ +0002f490: 6174 7269 7822 612b 622c 6322 3b20 2020 atrix"a+b,c"; │ │ │ │ +0002f4a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f4b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0002f4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f4e0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0002f4f0: 2020 2020 2020 3120 2020 2020 2032 2020 1 2 │ │ │ │ -0002f500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f510: 2020 7c0a 7c6f 3520 3a20 4d61 7472 6978 |.|o5 : Matrix │ │ │ │ -0002f520: 2053 2020 3c2d 2d20 5320 2020 2020 2020 S <-- S │ │ │ │ -0002f530: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f540: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f560: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0002f570: 3a20 6633 203d 206d 6174 7269 7822 612b : f3 = matrix"a+ │ │ │ │ -0002f580: 6222 3b20 2020 2020 2020 2020 2020 2020 b"; │ │ │ │ -0002f590: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0002f4d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002f4e0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +0002f4f0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0002f500: 2020 2020 2020 207c 0a7c 6f35 203a 204d |.|o5 : M │ │ │ │ +0002f510: 6174 7269 7820 5320 203c 2d2d 2053 2020 atrix S <-- S │ │ │ │ +0002f520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f530: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0002f540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0002f550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0002f560: 0a7c 6936 203a 2066 3320 3d20 6d61 7472 .|i6 : f3 = matr │ │ │ │ +0002f570: 6978 2261 2b62 223b 2020 2020 2020 2020 ix"a+b"; │ │ │ │ +0002f580: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0002f590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f5c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0002f5d0: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ -0002f5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f5f0: 7c6f 3620 3a20 4d61 7472 6978 2053 2020 |o6 : Matrix S │ │ │ │ -0002f600: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ -0002f610: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0002f5b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002f5c0: 2020 2020 2020 2031 2020 2020 2020 3120 1 1 │ │ │ │ +0002f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f5e0: 2020 207c 0a7c 6f36 203a 204d 6174 7269 |.|o6 : Matri │ │ │ │ +0002f5f0: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +0002f600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f610: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0002f620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f640: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6634 ------+.|i7 : f4 │ │ │ │ -0002f650: 203d 206d 6174 7269 7822 612b 622c 2061 = matrix"a+b, a │ │ │ │ -0002f660: 322b 6222 3b20 2020 2020 2020 2020 2020 2+b"; │ │ │ │ -0002f670: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ +0002f640: 203a 2066 3420 3d20 6d61 7472 6978 2261 : f4 = matrix"a │ │ │ │ +0002f650: 2b62 2c20 6132 2b62 223b 2020 2020 2020 +b, a2+b"; │ │ │ │ +0002f660: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0002f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002f680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f690: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f6a0: 7c20 2020 2020 2020 2020 2020 2020 3120 | 1 │ │ │ │ -0002f6b0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -0002f6c0: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -0002f6d0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ -0002f6e0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ -0002f6f0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f690: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f6a0: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +0002f6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f6c0: 0a7c 6f37 203a 204d 6174 7269 7820 5320 .|o7 : Matrix S │ │ │ │ +0002f6d0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0002f6e0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f6f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f720: 2d2d 2b0a 7c69 3820 3a20 6973 5175 6173 --+.|i8 : isQuas │ │ │ │ -0002f730: 6952 6567 756c 6172 2866 312c 4529 2020 iRegular(f1,E) │ │ │ │ -0002f740: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f750: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f770: 2020 2020 2020 2020 2020 7c0a 7c6f 3820 |.|o8 │ │ │ │ -0002f780: 3d20 6661 6c73 6520 2020 2020 2020 2020 = false │ │ │ │ -0002f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f710: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2069 -------+.|i8 : i │ │ │ │ +0002f720: 7351 7561 7369 5265 6775 6c61 7228 6631 sQuasiRegular(f1 │ │ │ │ +0002f730: 2c45 2920 2020 2020 2020 2020 2020 2020 ,E) │ │ │ │ +0002f740: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f760: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f770: 0a7c 6f38 203d 2066 616c 7365 2020 2020 .|o8 = false │ │ │ │ +0002f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f7d0: 2d2d 2b0a 7c69 3920 3a20 6973 5175 6173 --+.|i9 : isQuas │ │ │ │ -0002f7e0: 6952 6567 756c 6172 2866 322c 4529 2020 iRegular(f2,E) │ │ │ │ -0002f7f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f800: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f820: 2020 2020 2020 2020 2020 7c0a 7c6f 3920 |.|o9 │ │ │ │ -0002f830: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ -0002f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f850: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f7c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2069 -------+.|i9 : i │ │ │ │ +0002f7d0: 7351 7561 7369 5265 6775 6c61 7228 6632 sQuasiRegular(f2 │ │ │ │ +0002f7e0: 2c45 2920 2020 2020 2020 2020 2020 2020 ,E) │ │ │ │ +0002f7f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f820: 0a7c 6f39 203d 2074 7275 6520 2020 2020 .|o9 = true │ │ │ │ +0002f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f840: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f880: 2d2d 2b0a 7c69 3130 203a 2069 7351 7561 --+.|i10 : isQua │ │ │ │ -0002f890: 7369 5265 6775 6c61 7228 6633 2c45 2920 siRegular(f3,E) │ │ │ │ -0002f8a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f8b0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f8d0: 2020 2020 2020 2020 2020 7c0a 7c6f 3130 |.|o10 │ │ │ │ -0002f8e0: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ -0002f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f900: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f870: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ +0002f880: 6973 5175 6173 6952 6567 756c 6172 2866 isQuasiRegular(f │ │ │ │ +0002f890: 332c 4529 2020 2020 2020 2020 2020 2020 3,E) │ │ │ │ +0002f8a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f8d0: 0a7c 6f31 3020 3d20 7472 7565 2020 2020 .|o10 = true │ │ │ │ +0002f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f8f0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f930: 2d2d 2b0a 7c69 3131 203a 2069 7351 7561 --+.|i11 : isQua │ │ │ │ -0002f940: 7369 5265 6775 6c61 7228 6634 2c45 2920 siRegular(f4,E) │ │ │ │ -0002f950: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0002f960: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0002f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f980: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ -0002f990: 203d 2066 616c 7365 2020 2020 2020 2020 = false │ │ │ │ -0002f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002f9b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0002f920: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ +0002f930: 6973 5175 6173 6952 6567 756c 6172 2866 isQuasiRegular(f │ │ │ │ +0002f940: 342c 4529 2020 2020 2020 2020 2020 2020 4,E) │ │ │ │ +0002f950: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0002f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f970: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0002f980: 0a7c 6f31 3120 3d20 6661 6c73 6520 2020 .|o11 = false │ │ │ │ +0002f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002f9a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0002f9b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002f9c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002f9e0: 2d2d 2b0a 0a57 6179 7320 746f 2075 7365 --+..Ways to use │ │ │ │ -0002f9f0: 2069 7351 7561 7369 5265 6775 6c61 723a isQuasiRegular: │ │ │ │ -0002fa00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0002fa10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0002fa20: 2a20 2269 7351 7561 7369 5265 6775 6c61 * "isQuasiRegula │ │ │ │ -0002fa30: 7228 4c69 7374 2c4d 6f64 756c 6529 220a r(List,Module)". │ │ │ │ -0002fa40: 2020 2a20 2269 7351 7561 7369 5265 6775 * "isQuasiRegu │ │ │ │ -0002fa50: 6c61 7228 4d61 7472 6978 2c4d 6f64 756c lar(Matrix,Modul │ │ │ │ -0002fa60: 6529 220a 2020 2a20 2269 7351 7561 7369 e)". * "isQuasi │ │ │ │ -0002fa70: 5265 6775 6c61 7228 5365 7175 656e 6365 Regular(Sequence │ │ │ │ -0002fa80: 2c4d 6f64 756c 6529 220a 0a46 6f72 2074 ,Module)"..For t │ │ │ │ -0002fa90: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -0002faa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0002fab0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -0002fac0: 7465 2069 7351 7561 7369 5265 6775 6c61 te isQuasiRegula │ │ │ │ -0002fad0: 723a 2069 7351 7561 7369 5265 6775 6c61 r: isQuasiRegula │ │ │ │ -0002fae0: 722c 2069 7320 6120 2a6e 6f74 6520 6d65 r, is a *note me │ │ │ │ -0002faf0: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -0002fb00: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -0002fb10: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +0002f9d0: 2d2d 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 -------+..Ways t │ │ │ │ +0002f9e0: 6f20 7573 6520 6973 5175 6173 6952 6567 o use isQuasiReg │ │ │ │ +0002f9f0: 756c 6172 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ular:.========== │ │ │ │ +0002fa00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0002fa10: 3d0a 0a20 202a 2022 6973 5175 6173 6952 =.. * "isQuasiR │ │ │ │ +0002fa20: 6567 756c 6172 284c 6973 742c 4d6f 6475 egular(List,Modu │ │ │ │ +0002fa30: 6c65 2922 0a20 202a 2022 6973 5175 6173 le)". * "isQuas │ │ │ │ +0002fa40: 6952 6567 756c 6172 284d 6174 7269 782c iRegular(Matrix, │ │ │ │ +0002fa50: 4d6f 6475 6c65 2922 0a20 202a 2022 6973 Module)". * "is │ │ │ │ +0002fa60: 5175 6173 6952 6567 756c 6172 2853 6571 QuasiRegular(Seq │ │ │ │ +0002fa70: 7565 6e63 652c 4d6f 6475 6c65 2922 0a0a uence,Module)".. │ │ │ │ +0002fa80: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +0002fa90: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +0002faa0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +0002fab0: 7420 2a6e 6f74 6520 6973 5175 6173 6952 t *note isQuasiR │ │ │ │ +0002fac0: 6567 756c 6172 3a20 6973 5175 6173 6952 egular: isQuasiR │ │ │ │ +0002fad0: 6567 756c 6172 2c20 6973 2061 202a 6e6f egular, is a *no │ │ │ │ +0002fae0: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +0002faf0: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +0002fb00: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +0002fb10: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ 0002fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0002fb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0002fb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -0002fb70: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -0002fb80: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -0002fb90: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -0002fba0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -0002fbb0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -0002fbc0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -0002fbd0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ -0002fbe0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0002fbf0: 7574 696f 6e73 2e6d 323a 3436 3237 3a30 utions.m2:4627:0 │ │ │ │ -0002fc00: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ -0002fc10: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -0002fc20: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -0002fc30: 4e6f 6465 3a20 6973 5374 6162 6c79 5472 Node: isStablyTr │ │ │ │ -0002fc40: 6976 6961 6c2c 204e 6578 743a 206b 6f73 ivial, Next: kos │ │ │ │ -0002fc50: 7a75 6c45 7874 656e 7369 6f6e 2c20 5072 zulExtension, Pr │ │ │ │ -0002fc60: 6576 3a20 6973 5175 6173 6952 6567 756c ev: isQuasiRegul │ │ │ │ -0002fc70: 6172 2c20 5570 3a20 546f 700a 0a69 7353 ar, Up: Top..isS │ │ │ │ -0002fc80: 7461 626c 7954 7269 7669 616c 202d 2d20 tablyTrivial -- │ │ │ │ -0002fc90: 7265 7475 726e 7320 7472 7565 2069 6620 returns true if │ │ │ │ -0002fca0: 7468 6520 6d61 7020 676f 6573 2074 6f20 the map goes to │ │ │ │ -0002fcb0: 3020 756e 6465 7220 7374 6162 6c65 486f 0 under stableHo │ │ │ │ -0002fcc0: 6d0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a m.************** │ │ │ │ +0002fb60: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +0002fb70: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +0002fb80: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +0002fb90: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +0002fba0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +0002fbb0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +0002fbc0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +0002fbd0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0002fbe0: 5265 736f 6c75 7469 6f6e 732e 6d32 3a34 Resolutions.m2:4 │ │ │ │ +0002fbf0: 3632 373a 302e 0a1f 0a46 696c 653a 2043 627:0....File: C │ │ │ │ +0002fc00: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +0002fc10: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +0002fc20: 6e66 6f2c 204e 6f64 653a 2069 7353 7461 nfo, Node: isSta │ │ │ │ +0002fc30: 626c 7954 7269 7669 616c 2c20 4e65 7874 blyTrivial, Next │ │ │ │ +0002fc40: 3a20 6b6f 737a 756c 4578 7465 6e73 696f : koszulExtensio │ │ │ │ +0002fc50: 6e2c 2050 7265 763a 2069 7351 7561 7369 n, Prev: isQuasi │ │ │ │ +0002fc60: 5265 6775 6c61 722c 2055 703a 2054 6f70 Regular, Up: Top │ │ │ │ +0002fc70: 0a0a 6973 5374 6162 6c79 5472 6976 6961 ..isStablyTrivia │ │ │ │ +0002fc80: 6c20 2d2d 2072 6574 7572 6e73 2074 7275 l -- returns tru │ │ │ │ +0002fc90: 6520 6966 2074 6865 206d 6170 2067 6f65 e if the map goe │ │ │ │ +0002fca0: 7320 746f 2030 2075 6e64 6572 2073 7461 s to 0 under sta │ │ │ │ +0002fcb0: 626c 6548 6f6d 0a2a 2a2a 2a2a 2a2a 2a2a bleHom.********* │ │ │ │ +0002fcc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002fcd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0002fce0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0002fd00: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0002fd10: 653a 200a 2020 2020 2020 2020 6220 3d20 e: . b = │ │ │ │ -0002fd20: 6973 5374 6162 6c79 5472 6976 6961 6c20 isStablyTrivial │ │ │ │ -0002fd30: 660a 2020 2a20 496e 7075 7473 3a0a 2020 f. * Inputs:. │ │ │ │ -0002fd40: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ -0002fd50: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ -0002fd60: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ -0002fd70: 6d61 7020 4d20 746f 204e 0a20 202a 204f map M to N. * O │ │ │ │ -0002fd80: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ -0002fd90: 622c 2061 202a 6e6f 7465 2042 6f6f 6c65 b, a *note Boole │ │ │ │ -0002fda0: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ -0002fdb0: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ -0002fdc0: 2c20 7472 7565 2069 6666 2066 2066 6163 , true iff f fac │ │ │ │ -0002fdd0: 746f 7273 0a20 2020 2020 2020 2074 6872 tors. thr │ │ │ │ -0002fde0: 6f75 6768 2061 2070 726f 6a65 6374 6976 ough a projectiv │ │ │ │ -0002fdf0: 650a 0a44 6573 6372 6970 7469 6f6e 0a3d e..Description.= │ │ │ │ -0002fe00: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4120 706f ==========..A po │ │ │ │ -0002fe10: 7373 6962 6c65 206f 6273 7472 7563 7469 ssible obstructi │ │ │ │ -0002fe20: 6f6e 2074 6f20 7468 6520 636f 6d6d 7574 on to the commut │ │ │ │ -0002fe30: 6174 6976 6974 7920 6f66 2074 6865 2043 ativity of the C │ │ │ │ -0002fe40: 4920 6f70 6572 6174 6f72 7320 696e 2063 I operators in c │ │ │ │ -0002fe50: 6f64 696d 2063 2c0a 6576 656e 2061 7379 odim c,.even asy │ │ │ │ -0002fe60: 6d70 746f 7469 6361 6c6c 792c 2077 6f75 mptotically, wou │ │ │ │ -0002fe70: 6c64 2062 6520 7468 6520 6e6f 6e2d 7472 ld be the non-tr │ │ │ │ -0002fe80: 6976 6961 6c69 7479 206f 6620 7468 6520 iviality of the │ │ │ │ -0002fe90: 6d61 7020 4d5f 7b28 6b2b 3429 7d20 2d2d map M_{(k+4)} -- │ │ │ │ -0002fea0: 3e20 4d5f 6b0a 5c6f 7469 6d65 7320 5c77 > M_k.\otimes \w │ │ │ │ -0002feb0: 6564 6765 5e32 2853 5e63 2920 696e 2074 edge^2(S^c) in t │ │ │ │ -0002fec0: 6865 2073 7461 626c 6520 6361 7465 676f he stable catego │ │ │ │ -0002fed0: 7279 206f 6620 6d61 7869 6d61 6c20 436f ry of maximal Co │ │ │ │ -0002fee0: 6865 6e2d 4d61 6361 756c 6179 206d 6f64 hen-Macaulay mod │ │ │ │ -0002fef0: 756c 6573 2e0a 0a49 6e20 7468 6520 666f ules...In the fo │ │ │ │ -0002ff00: 6c6c 6f77 696e 6720 6578 616d 706c 652c llowing example, │ │ │ │ -0002ff10: 2073 7475 6469 6564 2069 6e20 7468 6520 studied in the │ │ │ │ -0002ff20: 7061 7065 7220 2254 6f72 2061 7320 6120 paper "Tor as a │ │ │ │ -0002ff30: 6d6f 6475 6c65 206f 7665 7220 616e 0a65 module over an.e │ │ │ │ -0002ff40: 7874 6572 696f 7220 616c 6765 6272 6122 xterior algebra" │ │ │ │ -0002ff50: 206f 6620 4569 7365 6e62 7564 2c20 5065 of Eisenbud, Pe │ │ │ │ -0002ff60: 6576 6120 616e 6420 5363 6872 6579 6572 eva and Schreyer │ │ │ │ -0002ff70: 2c20 7468 6520 6d61 7020 6973 206e 6f6e , the map is non │ │ │ │ -0002ff80: 2d74 7269 7669 616c 2e2e 2e62 7574 0a69 -trivial...but.i │ │ │ │ -0002ff90: 7420 6973 2073 7461 626c 7920 7472 6976 t is stably triv │ │ │ │ -0002ffa0: 6961 6c2e 2054 6865 2073 616d 6520 676f ial. The same go │ │ │ │ -0002ffb0: 6573 2066 6f72 2068 6967 6865 7220 7661 es for higher va │ │ │ │ -0002ffc0: 6c75 6573 206f 6620 6b20 2877 6869 6368 lues of k (which │ │ │ │ -0002ffd0: 2074 616b 6520 6c6f 6e67 6572 0a74 6f20 take longer.to │ │ │ │ -0002ffe0: 636f 6d70 7574 6529 2e20 286e 6f74 6520 compute). (note │ │ │ │ -0002fff0: 7468 6174 2069 6e20 7468 6973 2063 6173 that in this cas │ │ │ │ -00030000: 652c 2077 6974 6820 6320 3d20 332c 2074 e, with c = 3, t │ │ │ │ -00030010: 776f 206f 6620 7468 6520 7468 7265 6520 wo of the three │ │ │ │ -00030020: 616c 7465 726e 6174 696e 670a 7072 6f64 alternating.prod │ │ │ │ -00030030: 7563 7473 2061 7265 2061 6374 7561 6c6c ucts are actuall │ │ │ │ -00030040: 7920 6571 7561 6c20 746f 2030 2c20 736f y equal to 0, so │ │ │ │ -00030050: 2077 6520 7465 7374 206f 6e6c 7920 7468 we test only th │ │ │ │ -00030060: 6520 7468 6972 642e 290a 0a4e 6f74 6520 e third.)..Note │ │ │ │ -00030070: 7468 6174 2054 2069 7320 7765 6c6c 2d64 that T is well-d │ │ │ │ -00030080: 6566 696e 6564 2075 7020 746f 2068 6f6d efined up to hom │ │ │ │ -00030090: 6f74 6f70 793b 2073 6f20 545e 3220 6973 otopy; so T^2 is │ │ │ │ -000300a0: 2077 656c 6c2d 6465 6669 6e65 6420 6d6f well-defined mo │ │ │ │ -000300b0: 6420 6d6d 5e32 2e0a 0a2b 2d2d 2d2d 2d2d d mm^2...+------ │ │ │ │ +0002fcf0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0002fd00: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0002fd10: 2062 203d 2069 7353 7461 626c 7954 7269 b = isStablyTri │ │ │ │ +0002fd20: 7669 616c 2066 0a20 202a 2049 6e70 7574 vial f. * Input │ │ │ │ +0002fd30: 733a 0a20 2020 2020 202a 2066 2c20 6120 s:. * f, a │ │ │ │ +0002fd40: 2a6e 6f74 6520 6d61 7472 6978 3a20 284d *note matrix: (M │ │ │ │ +0002fd50: 6163 6175 6c61 7932 446f 6329 4d61 7472 acaulay2Doc)Matr │ │ │ │ +0002fd60: 6978 2c2c 206d 6170 204d 2074 6f20 4e0a ix,, map M to N. │ │ │ │ +0002fd70: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ +0002fd80: 2020 202a 2062 2c20 6120 2a6e 6f74 6520 * b, a *note │ │ │ │ +0002fd90: 426f 6f6c 6561 6e20 7661 6c75 653a 2028 Boolean value: ( │ │ │ │ +0002fda0: 4d61 6361 756c 6179 3244 6f63 2942 6f6f Macaulay2Doc)Boo │ │ │ │ +0002fdb0: 6c65 616e 2c2c 2074 7275 6520 6966 6620 lean,, true iff │ │ │ │ +0002fdc0: 6620 6661 6374 6f72 730a 2020 2020 2020 f factors. │ │ │ │ +0002fdd0: 2020 7468 726f 7567 6820 6120 7072 6f6a through a proj │ │ │ │ +0002fde0: 6563 7469 7665 0a0a 4465 7363 7269 7074 ective..Descript │ │ │ │ +0002fdf0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ +0002fe00: 0a41 2070 6f73 7369 626c 6520 6f62 7374 .A possible obst │ │ │ │ +0002fe10: 7275 6374 696f 6e20 746f 2074 6865 2063 ruction to the c │ │ │ │ +0002fe20: 6f6d 6d75 7461 7469 7669 7479 206f 6620 ommutativity of │ │ │ │ +0002fe30: 7468 6520 4349 206f 7065 7261 746f 7273 the CI operators │ │ │ │ +0002fe40: 2069 6e20 636f 6469 6d20 632c 0a65 7665 in codim c,.eve │ │ │ │ +0002fe50: 6e20 6173 796d 7074 6f74 6963 616c 6c79 n asymptotically │ │ │ │ +0002fe60: 2c20 776f 756c 6420 6265 2074 6865 206e , would be the n │ │ │ │ +0002fe70: 6f6e 2d74 7269 7669 616c 6974 7920 6f66 on-triviality of │ │ │ │ +0002fe80: 2074 6865 206d 6170 204d 5f7b 286b 2b34 the map M_{(k+4 │ │ │ │ +0002fe90: 297d 202d 2d3e 204d 5f6b 0a5c 6f74 696d )} --> M_k.\otim │ │ │ │ +0002fea0: 6573 205c 7765 6467 655e 3228 535e 6329 es \wedge^2(S^c) │ │ │ │ +0002feb0: 2069 6e20 7468 6520 7374 6162 6c65 2063 in the stable c │ │ │ │ +0002fec0: 6174 6567 6f72 7920 6f66 206d 6178 696d ategory of maxim │ │ │ │ +0002fed0: 616c 2043 6f68 656e 2d4d 6163 6175 6c61 al Cohen-Macaula │ │ │ │ +0002fee0: 7920 6d6f 6475 6c65 732e 0a0a 496e 2074 y modules...In t │ │ │ │ +0002fef0: 6865 2066 6f6c 6c6f 7769 6e67 2065 7861 he following exa │ │ │ │ +0002ff00: 6d70 6c65 2c20 7374 7564 6965 6420 696e mple, studied in │ │ │ │ +0002ff10: 2074 6865 2070 6170 6572 2022 546f 7220 the paper "Tor │ │ │ │ +0002ff20: 6173 2061 206d 6f64 756c 6520 6f76 6572 as a module over │ │ │ │ +0002ff30: 2061 6e0a 6578 7465 7269 6f72 2061 6c67 an.exterior alg │ │ │ │ +0002ff40: 6562 7261 2220 6f66 2045 6973 656e 6275 ebra" of Eisenbu │ │ │ │ +0002ff50: 642c 2050 6565 7661 2061 6e64 2053 6368 d, Peeva and Sch │ │ │ │ +0002ff60: 7265 7965 722c 2074 6865 206d 6170 2069 reyer, the map i │ │ │ │ +0002ff70: 7320 6e6f 6e2d 7472 6976 6961 6c2e 2e2e s non-trivial... │ │ │ │ +0002ff80: 6275 740a 6974 2069 7320 7374 6162 6c79 but.it is stably │ │ │ │ +0002ff90: 2074 7269 7669 616c 2e20 5468 6520 7361 trivial. The sa │ │ │ │ +0002ffa0: 6d65 2067 6f65 7320 666f 7220 6869 6768 me goes for high │ │ │ │ +0002ffb0: 6572 2076 616c 7565 7320 6f66 206b 2028 er values of k ( │ │ │ │ +0002ffc0: 7768 6963 6820 7461 6b65 206c 6f6e 6765 which take longe │ │ │ │ +0002ffd0: 720a 746f 2063 6f6d 7075 7465 292e 2028 r.to compute). ( │ │ │ │ +0002ffe0: 6e6f 7465 2074 6861 7420 696e 2074 6869 note that in thi │ │ │ │ +0002fff0: 7320 6361 7365 2c20 7769 7468 2063 203d s case, with c = │ │ │ │ +00030000: 2033 2c20 7477 6f20 6f66 2074 6865 2074 3, two of the t │ │ │ │ +00030010: 6872 6565 2061 6c74 6572 6e61 7469 6e67 hree alternating │ │ │ │ +00030020: 0a70 726f 6475 6374 7320 6172 6520 6163 .products are ac │ │ │ │ +00030030: 7475 616c 6c79 2065 7175 616c 2074 6f20 tually equal to │ │ │ │ +00030040: 302c 2073 6f20 7765 2074 6573 7420 6f6e 0, so we test on │ │ │ │ +00030050: 6c79 2074 6865 2074 6869 7264 2e29 0a0a ly the third.).. │ │ │ │ +00030060: 4e6f 7465 2074 6861 7420 5420 6973 2077 Note that T is w │ │ │ │ +00030070: 656c 6c2d 6465 6669 6e65 6420 7570 2074 ell-defined up t │ │ │ │ +00030080: 6f20 686f 6d6f 746f 7079 3b20 736f 2054 o homotopy; so T │ │ │ │ +00030090: 5e32 2069 7320 7765 6c6c 2d64 6566 696e ^2 is well-defin │ │ │ │ +000300a0: 6564 206d 6f64 206d 6d5e 322e 0a0a 2b2d ed mod mm^2...+- │ │ │ │ +000300b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000300c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000300d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000300e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030100: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -00030110: 6b20 3d20 5a5a 2f31 3031 2020 2020 2020 k = ZZ/101 │ │ │ │ +000300f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030100: 3120 3a20 6b6b 203d 205a 5a2f 3130 3120 1 : kk = ZZ/101 │ │ │ │ +00030110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030150: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030140: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000301a0: 2020 2020 2020 207c 0a7c 6f31 203d 206b |.|o1 = k │ │ │ │ -000301b0: 6b20 2020 2020 2020 2020 2020 2020 2020 k │ │ │ │ +00030190: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000301a0: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +000301b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000301c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000301d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000301e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000301f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000301e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000301f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030240: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -00030250: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00030230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030240: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +00030250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030290: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030280: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000302a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000302b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000302c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000302e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ -000302f0: 203d 206b 6b5b 612c 622c 635d 2020 2020 = kk[a,b,c] │ │ │ │ +000302d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000302e0: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63 2 : S = kk[a,b,c │ │ │ │ +000302f0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 00030300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030330: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030320: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030380: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +00030370: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030380: 3220 3d20 5320 2020 2020 2020 2020 2020 2 = S │ │ │ │ 00030390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000303c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000303d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000303c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000303d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000303f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030420: 2020 2020 2020 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -00030430: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00030410: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030420: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ +00030430: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 00030440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030470: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030460: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000304a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000304c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 -------+.|i3 : f │ │ │ │ -000304d0: 6620 3d20 6d61 7472 6978 2261 322c 6232 f = matrix"a2,b2 │ │ │ │ -000304e0: 2c63 3222 2020 2020 2020 2020 2020 2020 ,c2" │ │ │ │ +000304b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000304c0: 3320 3a20 6666 203d 206d 6174 7269 7822 3 : ff = matrix" │ │ │ │ +000304d0: 6132 2c62 322c 6332 2220 2020 2020 2020 a2,b2,c2" │ │ │ │ +000304e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000304f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030560: 2020 2020 2020 207c 0a7c 6f33 203d 207c |.|o3 = | │ │ │ │ -00030570: 2061 3220 6232 2063 3220 7c20 2020 2020 a2 b2 c2 | │ │ │ │ +00030550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030560: 3320 3d20 7c20 6132 2062 3220 6332 207c 3 = | a2 b2 c2 | │ │ │ │ +00030570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000305a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000305b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000305a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000305b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000305c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000305d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000305e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000305f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030600: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00030610: 2020 2020 2020 2031 2020 2020 2020 3320 1 3 │ │ │ │ +000305f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030600: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00030610: 2020 2033 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00030620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030650: 2020 2020 2020 207c 0a7c 6f33 203a 204d |.|o3 : M │ │ │ │ -00030660: 6174 7269 7820 5320 203c 2d2d 2053 2020 atrix S <-- S │ │ │ │ +00030640: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030650: 3320 3a20 4d61 7472 6978 2053 2020 3c2d 3 : Matrix S <- │ │ │ │ +00030660: 2d20 5320 2020 2020 2020 2020 2020 2020 - S │ │ │ │ 00030670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000306a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030690: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000306a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000306b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000306c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000306d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000306f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ -00030700: 203d 2053 2f69 6465 616c 2066 6620 2020 = S/ideal ff │ │ │ │ +000306e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000306f0: 3420 3a20 5220 3d20 532f 6964 6561 6c20 4 : R = S/ideal │ │ │ │ +00030700: 6666 2020 2020 2020 2020 2020 2020 2020 ff │ │ │ │ 00030710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030740: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030790: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ +00030780: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030790: 3420 3d20 5220 2020 2020 2020 2020 2020 4 = R │ │ │ │ 000307a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000307b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000307c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000307d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000307e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000307d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000307e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000307f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030830: 2020 2020 2020 207c 0a7c 6f34 203a 2051 |.|o4 : Q │ │ │ │ -00030840: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00030820: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030830: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +00030840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030880: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030870: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000308a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000308b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000308d0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204d -------+.|i5 : M │ │ │ │ -000308e0: 203d 2052 5e31 2f69 6465 616c 2261 2c62 = R^1/ideal"a,b │ │ │ │ -000308f0: 6322 2020 2020 2020 2020 2020 2020 2020 c" │ │ │ │ +000308c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000308d0: 3520 3a20 4d20 3d20 525e 312f 6964 6561 5 : M = R^1/idea │ │ │ │ +000308e0: 6c22 612c 6263 2220 2020 2020 2020 2020 l"a,bc" │ │ │ │ +000308f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030920: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030910: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030970: 2020 2020 2020 207c 0a7c 6f35 203d 2063 |.|o5 = c │ │ │ │ -00030980: 6f6b 6572 6e65 6c20 7c20 6120 6263 207c okernel | a bc | │ │ │ │ +00030960: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030970: 3520 3d20 636f 6b65 726e 656c 207c 2061 5 = cokernel | a │ │ │ │ +00030980: 2062 6320 7c20 2020 2020 2020 2020 2020 bc | │ │ │ │ 00030990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000309c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000309b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000309c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000309f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00030a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a30: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00030a00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030a20: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +00030a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030a60: 2020 2020 2020 207c 0a7c 6f35 203a 2052 |.|o5 : R │ │ │ │ -00030a70: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00030a80: 7420 6f66 2052 2020 2020 2020 2020 2020 t of R │ │ │ │ +00030a50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030a60: 3520 3a20 522d 6d6f 6475 6c65 2c20 7175 5 : R-module, qu │ │ │ │ +00030a70: 6f74 6965 6e74 206f 6620 5220 2020 2020 otient of R │ │ │ │ +00030a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ab0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030aa0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030b00: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 206b -------+.|i6 : k │ │ │ │ -00030b10: 203d 2031 2020 2020 2020 2020 2020 2020 = 1 │ │ │ │ +00030af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030b00: 3620 3a20 6b20 3d20 3120 2020 2020 2020 6 : k = 1 │ │ │ │ +00030b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030b40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ba0: 2020 2020 2020 207c 0a7c 6f36 203d 2031 |.|o6 = 1 │ │ │ │ +00030b90: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030ba0: 3620 3d20 3120 2020 2020 2020 2020 2020 6 = 1 │ │ │ │ 00030bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030bf0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030be0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030c40: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 206d -------+.|i7 : m │ │ │ │ -00030c50: 203d 206b 2b35 2020 2020 2020 2020 2020 = k+5 │ │ │ │ +00030c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030c40: 3720 3a20 6d20 3d20 6b2b 3520 2020 2020 7 : m = k+5 │ │ │ │ +00030c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030c90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030c80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ce0: 2020 2020 2020 207c 0a7c 6f37 203d 2036 |.|o7 = 6 │ │ │ │ +00030cd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030ce0: 3720 3d20 3620 2020 2020 2020 2020 2020 7 = 6 │ │ │ │ 00030cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030d30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030d20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00030d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00030d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00030d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 -------+.|i8 : F │ │ │ │ -00030d90: 203d 2066 7265 6552 6573 6f6c 7574 696f = freeResolutio │ │ │ │ -00030da0: 6e28 4d2c 204c 656e 6774 684c 696d 6974 n(M, LengthLimit │ │ │ │ -00030db0: 203d 3e20 6d29 2020 2020 2020 2020 2020 => m) │ │ │ │ -00030dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030dd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00030d80: 3820 3a20 4620 3d20 6672 6565 5265 736f 8 : F = freeReso │ │ │ │ +00030d90: 6c75 7469 6f6e 284d 2c20 4c65 6e67 7468 lution(M, Length │ │ │ │ +00030da0: 4c69 6d69 7420 3d3e 206d 2920 2020 2020 Limit => m) │ │ │ │ +00030db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00030dc0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00030e30: 3120 2020 2020 2032 2020 2020 2020 3420 1 2 4 │ │ │ │ -00030e40: 2020 2020 2037 2020 2020 2020 3131 2020 7 11 │ │ │ │ -00030e50: 2020 2020 3136 2020 2020 2020 3232 2020 16 22 │ │ │ │ -00030e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030e70: 2020 2020 2020 207c 0a7c 6f38 203d 2052 |.|o8 = R │ │ │ │ -00030e80: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020 <-- R <-- R │ │ │ │ -00030e90: 3c2d 2d20 5220 203c 2d2d 2052 2020 203c <-- R <-- R < │ │ │ │ -00030ea0: 2d2d 2052 2020 203c 2d2d 2052 2020 2020 -- R <-- R │ │ │ │ -00030eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030e10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030e20: 2020 2020 2031 2020 2020 2020 3220 2020 1 2 │ │ │ │ +00030e30: 2020 2034 2020 2020 2020 3720 2020 2020 4 7 │ │ │ │ +00030e40: 2031 3120 2020 2020 2031 3620 2020 2020 11 16 │ │ │ │ +00030e50: 2032 3220 2020 2020 2020 2020 2020 2020 22 │ │ │ │ +00030e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030e70: 3820 3d20 5220 203c 2d2d 2052 2020 3c2d 8 = R <-- R <- │ │ │ │ +00030e80: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20 - R <-- R <-- │ │ │ │ +00030e90: 5220 2020 3c2d 2d20 5220 2020 3c2d 2d20 R <-- R <-- │ │ │ │ +00030ea0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00030eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f10: 2020 2020 2020 207c 0a7c 2020 2020 2030 |.| 0 │ │ │ │ -00030f20: 2020 2020 2020 3120 2020 2020 2032 2020 1 2 │ │ │ │ -00030f30: 2020 2020 3320 2020 2020 2034 2020 2020 3 4 │ │ │ │ -00030f40: 2020 2035 2020 2020 2020 2036 2020 2020 5 6 │ │ │ │ -00030f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030f60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00030f00: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030f10: 2020 2020 3020 2020 2020 2031 2020 2020 0 1 │ │ │ │ +00030f20: 2020 3220 2020 2020 2033 2020 2020 2020 2 3 │ │ │ │ +00030f30: 3420 2020 2020 2020 3520 2020 2020 2020 4 5 │ │ │ │ +00030f40: 3620 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ +00030f50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00030f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030fb0: 2020 2020 2020 207c 0a7c 6f38 203a 2043 |.|o8 : C │ │ │ │ -00030fc0: 6f6d 706c 6578 2020 2020 2020 2020 2020 omplex │ │ │ │ +00030fa0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00030fb0: 3820 3a20 436f 6d70 6c65 7820 2020 2020 8 : Complex │ │ │ │ +00030fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00030fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00030ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031000: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00030ff0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031050: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2073 -------+.|i9 : s │ │ │ │ -00031060: 797a 7967 6965 7320 3d20 6170 706c 7928 yzygies = apply( │ │ │ │ -00031070: 312e 2e6d 2c20 692d 3e63 6f6b 6572 2046 1..m, i->coker F │ │ │ │ -00031080: 2e64 645f 6929 3b20 2020 2020 2020 2020 .dd_i); │ │ │ │ -00031090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000310a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031050: 3920 3a20 7379 7a79 6769 6573 203d 2061 9 : syzygies = a │ │ │ │ +00031060: 7070 6c79 2831 2e2e 6d2c 2069 2d3e 636f pply(1..m, i->co │ │ │ │ +00031070: 6b65 7220 462e 6464 5f69 293b 2020 2020 ker F.dd_i); │ │ │ │ +00031080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031090: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000310a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000310b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000310c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000310d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000310f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 3a20 -------+.|i10 : │ │ │ │ -00031100: 7431 203d 206d 616b 6554 2866 662c 462c t1 = makeT(ff,F, │ │ │ │ -00031110: 6b2b 3429 3b20 2020 2020 2020 2020 2020 k+4); │ │ │ │ +000310e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000310f0: 3130 203a 2074 3120 3d20 6d61 6b65 5428 10 : t1 = makeT( │ │ │ │ +00031100: 6666 2c46 2c6b 2b34 293b 2020 2020 2020 ff,F,k+4); │ │ │ │ +00031110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031140: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031130: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031190: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -000311a0: 7432 203d 206d 616b 6554 2866 662c 462c t2 = makeT(ff,F, │ │ │ │ -000311b0: 6b2b 3229 3b20 2020 2020 2020 2020 2020 k+2); │ │ │ │ +00031180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031190: 3131 203a 2074 3220 3d20 6d61 6b65 5428 11 : t2 = makeT( │ │ │ │ +000311a0: 6666 2c46 2c6b 2b32 293b 2020 2020 2020 ff,F,k+2); │ │ │ │ +000311b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000311c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000311e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000311d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000311e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000311f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031230: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00031240: 5432 436f 6d70 6f6e 656e 7473 203d 2066 T2Components = f │ │ │ │ -00031250: 6c61 7474 656e 2066 6f72 2069 2066 726f latten for i fro │ │ │ │ -00031260: 6d20 3020 746f 2031 206c 6973 7428 666f m 0 to 1 list(fo │ │ │ │ -00031270: 7220 6a20 6672 6f6d 2069 2b31 2074 6f20 r j from i+1 to │ │ │ │ -00031280: 3220 6c69 7374 207c 0a7c 2d2d 2d2d 2d2d 2 list |.|------ │ │ │ │ +00031220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031230: 3132 203a 2054 3243 6f6d 706f 6e65 6e74 12 : T2Component │ │ │ │ +00031240: 7320 3d20 666c 6174 7465 6e20 666f 7220 s = flatten for │ │ │ │ +00031250: 6920 6672 6f6d 2030 2074 6f20 3120 6c69 i from 0 to 1 li │ │ │ │ +00031260: 7374 2866 6f72 206a 2066 726f 6d20 692b st(for j from i+ │ │ │ │ +00031270: 3120 746f 2032 206c 6973 7420 7c0a 7c2d 1 to 2 list |.|- │ │ │ │ +00031280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000312a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000312b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000312d0: 2d2d 2d2d 2d2d 2d7c 0a7c 6d61 7028 465f -------|.|map(F_ │ │ │ │ -000312e0: 6b2c 2046 5f28 6b2b 3429 2c20 7432 5f69 k, F_(k+4), t2_i │ │ │ │ -000312f0: 2a74 315f 6a2d 7432 5f6a 2a74 315f 6929 *t1_j-t2_j*t1_i) │ │ │ │ -00031300: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -00031310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031320: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000312c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6d ------------|.|m │ │ │ │ +000312d0: 6170 2846 5f6b 2c20 465f 286b 2b34 292c ap(F_k, F_(k+4), │ │ │ │ +000312e0: 2074 325f 692a 7431 5f6a 2d74 325f 6a2a t2_i*t1_j-t2_j* │ │ │ │ +000312f0: 7431 5f69 2929 3b20 2020 2020 2020 2020 t1_i)); │ │ │ │ +00031300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031310: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031370: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 3a20 -------+.|i13 : │ │ │ │ -00031380: 6720 3d20 6d61 7028 7379 7a79 6769 6573 g = map(syzygies │ │ │ │ -00031390: 5f6b 2c20 7379 7a79 6769 6573 5f28 6b2b _k, syzygies_(k+ │ │ │ │ -000313a0: 3429 2c20 5432 436f 6d70 6f6e 656e 7473 4), T2Components │ │ │ │ -000313b0: 5f32 2920 2020 2020 2020 2020 2020 2020 _2) │ │ │ │ -000313c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00031360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00031370: 3133 203a 2067 203d 206d 6170 2873 797a 13 : g = map(syz │ │ │ │ +00031380: 7967 6965 735f 6b2c 2073 797a 7967 6965 ygies_k, syzygie │ │ │ │ +00031390: 735f 286b 2b34 292c 2054 3243 6f6d 706f s_(k+4), T2Compo │ │ │ │ +000313a0: 6e65 6e74 735f 3229 2020 2020 2020 2020 nents_2) │ │ │ │ +000313b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000313c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000313f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031410: 2020 2020 2020 207c 0a7c 6f31 3320 3d20 |.|o13 = │ │ │ │ -00031420: 7b31 7d20 7c20 3020 3020 3020 3020 3020 {1} | 0 0 0 0 0 │ │ │ │ -00031430: 2d63 2030 2030 2062 2030 2030 2030 2030 -c 0 0 b 0 0 0 0 │ │ │ │ -00031440: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00031450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031460: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00031470: 7b32 7d20 7c20 3020 3020 3020 3020 3020 {2} | 0 0 0 0 0 │ │ │ │ -00031480: 3020 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ -00031490: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -000314a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00031400: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031410: 3133 203d 207b 317d 207c 2030 2030 2030 13 = {1} | 0 0 0 │ │ │ │ +00031420: 2030 2030 202d 6320 3020 3020 6220 3020 0 0 -c 0 0 b 0 │ │ │ │ +00031430: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00031440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031450: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00031460: 2020 2020 207b 327d 207c 2030 2030 2030 {2} | 0 0 0 │ │ │ │ +00031470: 2030 2030 2030 2020 3020 3020 3020 3020 0 0 0 0 0 0 0 │ │ │ │ +00031480: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00031490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000314a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000314b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000314e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000314f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031500: 2020 2020 2020 207c 0a7c 6f31 3320 3a20 |.|o13 : │ │ │ │ -00031510: 4d61 7472 6978 2020 2020 2020 2020 2020 Matrix │ │ │ │ +000314f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031500: 3133 203a 204d 6174 7269 7820 2020 2020 13 : Matrix │ │ │ │ +00031510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031550: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031540: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000315a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 3a20 -------+.|i14 : │ │ │ │ -000315b0: 6973 5374 6162 6c79 5472 6976 6961 6c20 isStablyTrivial │ │ │ │ -000315c0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +00031590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000315a0: 3134 203a 2069 7353 7461 626c 7954 7269 14 : isStablyTri │ │ │ │ +000315b0: 7669 616c 2067 2020 2020 2020 2020 2020 vial g │ │ │ │ +000315c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000315d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000315e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000315f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000315e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000315f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031640: 2020 2020 2020 207c 0a7c 6f31 3420 3d20 |.|o14 = │ │ │ │ -00031650: 7472 7565 2020 2020 2020 2020 2020 2020 true │ │ │ │ +00031630: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00031640: 3134 203d 2074 7275 6520 2020 2020 2020 14 = true │ │ │ │ +00031650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031690: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00031680: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00031690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000316a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000316b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000316c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000316e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -000316f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00031700: 202a 6e6f 7465 2073 7461 626c 6548 6f6d *note stableHom │ │ │ │ -00031710: 3a20 7374 6162 6c65 486f 6d2c 202d 2d20 : stableHom, -- │ │ │ │ -00031720: 6d61 7020 6672 6f6d 2048 6f6d 284d 2c4e map from Hom(M,N │ │ │ │ -00031730: 2920 746f 2074 6865 2073 7461 626c 6520 ) to the stable │ │ │ │ -00031740: 486f 6d20 6d6f 6475 6c65 0a0a 5761 7973 Hom module..Ways │ │ │ │ -00031750: 2074 6f20 7573 6520 6973 5374 6162 6c79 to use isStably │ │ │ │ -00031760: 5472 6976 6961 6c3a 0a3d 3d3d 3d3d 3d3d Trivial:.======= │ │ │ │ -00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00031780: 3d3d 3d3d 3d0a 0a20 202a 2022 6973 5374 =====.. * "isSt │ │ │ │ -00031790: 6162 6c79 5472 6976 6961 6c28 4d61 7472 ablyTrivial(Matr │ │ │ │ -000317a0: 6978 2922 0a0a 466f 7220 7468 6520 7072 ix)"..For the pr │ │ │ │ -000317b0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -000317c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -000317d0: 206f 626a 6563 7420 2a6e 6f74 6520 6973 object *note is │ │ │ │ -000317e0: 5374 6162 6c79 5472 6976 6961 6c3a 2069 StablyTrivial: i │ │ │ │ -000317f0: 7353 7461 626c 7954 7269 7669 616c 2c20 sStablyTrivial, │ │ │ │ -00031800: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00031810: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00031820: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00031830: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +000316d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +000316e0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +000316f0: 0a0a 2020 2a20 2a6e 6f74 6520 7374 6162 .. * *note stab │ │ │ │ +00031700: 6c65 486f 6d3a 2073 7461 626c 6548 6f6d leHom: stableHom │ │ │ │ +00031710: 2c20 2d2d 206d 6170 2066 726f 6d20 486f , -- map from Ho │ │ │ │ +00031720: 6d28 4d2c 4e29 2074 6f20 7468 6520 7374 m(M,N) to the st │ │ │ │ +00031730: 6162 6c65 2048 6f6d 206d 6f64 756c 650a able Hom module. │ │ │ │ +00031740: 0a57 6179 7320 746f 2075 7365 2069 7353 .Ways to use isS │ │ │ │ +00031750: 7461 626c 7954 7269 7669 616c 3a0a 3d3d tablyTrivial:.== │ │ │ │ +00031760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031770: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00031780: 2269 7353 7461 626c 7954 7269 7669 616c "isStablyTrivial │ │ │ │ +00031790: 284d 6174 7269 7829 220a 0a46 6f72 2074 (Matrix)"..For t │ │ │ │ +000317a0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ +000317b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000317c0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ +000317d0: 7465 2069 7353 7461 626c 7954 7269 7669 te isStablyTrivi │ │ │ │ +000317e0: 616c 3a20 6973 5374 6162 6c79 5472 6976 al: isStablyTriv │ │ │ │ +000317f0: 6961 6c2c 2069 7320 6120 2a6e 6f74 6520 ial, is a *note │ │ │ │ +00031800: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00031810: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00031820: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00031830: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 00031840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00031870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00031880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -00031890: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -000318a0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -000318b0: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -000318c0: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -000318d0: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -000318e0: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -000318f0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -00031900: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00031910: 6f6e 732e 6d32 3a34 3639 393a 302e 0a1f ons.m2:4699:0... │ │ │ │ -00031920: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00031930: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00031940: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -00031950: 653a 206b 6f73 7a75 6c45 7874 656e 7369 e: koszulExtensi │ │ │ │ -00031960: 6f6e 2c20 4e65 7874 3a20 4c61 7965 7265 on, Next: Layere │ │ │ │ -00031970: 642c 2050 7265 763a 2069 7353 7461 626c d, Prev: isStabl │ │ │ │ -00031980: 7954 7269 7669 616c 2c20 5570 3a20 546f yTrivial, Up: To │ │ │ │ -00031990: 700a 0a6b 6f73 7a75 6c45 7874 656e 7369 p..koszulExtensi │ │ │ │ -000319a0: 6f6e 202d 2d20 6372 6561 7465 7320 7468 on -- creates th │ │ │ │ -000319b0: 6520 4b6f 737a 756c 2065 7874 656e 7369 e Koszul extensi │ │ │ │ -000319c0: 6f6e 2063 6f6d 706c 6578 206f 6620 6120 on complex of a │ │ │ │ -000319d0: 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a map.************ │ │ │ │ +00031880: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00031890: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000318a0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000318b0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +000318c0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +000318d0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +000318e0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +000318f0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00031900: 6f6c 7574 696f 6e73 2e6d 323a 3436 3939 olutions.m2:4699 │ │ │ │ +00031910: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +00031920: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00031930: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +00031940: 2c20 4e6f 6465 3a20 6b6f 737a 756c 4578 , Node: koszulEx │ │ │ │ +00031950: 7465 6e73 696f 6e2c 204e 6578 743a 204c tension, Next: L │ │ │ │ +00031960: 6179 6572 6564 2c20 5072 6576 3a20 6973 ayered, Prev: is │ │ │ │ +00031970: 5374 6162 6c79 5472 6976 6961 6c2c 2055 StablyTrivial, U │ │ │ │ +00031980: 703a 2054 6f70 0a0a 6b6f 737a 756c 4578 p: Top..koszulEx │ │ │ │ +00031990: 7465 6e73 696f 6e20 2d2d 2063 7265 6174 tension -- creat │ │ │ │ +000319a0: 6573 2074 6865 204b 6f73 7a75 6c20 6578 es the Koszul ex │ │ │ │ +000319b0: 7465 6e73 696f 6e20 636f 6d70 6c65 7820 tension complex │ │ │ │ +000319c0: 6f66 2061 206d 6170 0a2a 2a2a 2a2a 2a2a of a map.******* │ │ │ │ +000319d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000319e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000319f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031a10: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00031a20: 200a 2020 2020 2020 2020 4d4d 203d 206b . MM = k │ │ │ │ -00031a30: 6f73 7a75 6c45 7874 656e 7369 6f6e 2846 oszulExtension(F │ │ │ │ -00031a40: 462c 4242 2c70 7369 312c 6666 290a 2020 F,BB,psi1,ff). │ │ │ │ -00031a50: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -00031a60: 2a20 4646 2c20 6120 2a6e 6f74 6520 636f * FF, a *note co │ │ │ │ -00031a70: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ -00031a80: 7329 436f 6d70 6c65 782c 2c20 7265 736f s)Complex,, reso │ │ │ │ -00031a90: 6c75 7469 6f6e 206f 7665 7220 530a 2020 lution over S. │ │ │ │ -00031aa0: 2020 2020 2a20 4242 2c20 6120 2a6e 6f74 * BB, a *not │ │ │ │ -00031ab0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ -00031ac0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ -00031ad0: 7477 6f2d 7465 726d 2063 6f6d 706c 6578 two-term complex │ │ │ │ -00031ae0: 2042 425f 312d 2d3e 4242 5f30 0a20 2020 BB_1-->BB_0. │ │ │ │ -00031af0: 2020 202a 2070 7369 312c 2061 202a 6e6f * psi1, a *no │ │ │ │ -00031b00: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -00031b10: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -00031b20: 2c20 6672 6f6d 2042 425f 3120 746f 2046 , from BB_1 to F │ │ │ │ -00031b30: 465f 300a 2020 2020 2020 2a20 6666 2c20 F_0. * ff, │ │ │ │ -00031b40: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -00031b50: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -00031b60: 7472 6978 2c2c 2072 6567 756c 6172 2073 trix,, regular s │ │ │ │ -00031b70: 6571 7565 6e63 650a 2020 2020 2020 2020 equence. │ │ │ │ -00031b80: 616e 6e69 6869 6c61 7469 6e67 2074 6865 annihilating the │ │ │ │ -00031b90: 206d 6f64 756c 6520 7265 736f 6c76 6564 module resolved │ │ │ │ -00031ba0: 2062 7920 4646 0a20 202a 204f 7574 7075 by FF. * Outpu │ │ │ │ -00031bb0: 7473 3a0a 2020 2020 2020 2a20 4d4d 2c20 ts:. * MM, │ │ │ │ -00031bc0: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a a *note complex: │ │ │ │ -00031bd0: 2028 436f 6d70 6c65 7865 7329 436f 6d70 (Complexes)Comp │ │ │ │ -00031be0: 6c65 782c 2c20 7468 6520 6d61 7070 696e lex,, the mappin │ │ │ │ -00031bf0: 6720 636f 6e65 206f 6620 7468 650a 2020 g cone of the. │ │ │ │ -00031c00: 2020 2020 2020 696e 6475 6365 6420 6d61 induced ma │ │ │ │ -00031c10: 7020 425b 2d31 5d5c 6f74 696d 6573 204b p B[-1]\otimes K │ │ │ │ -00031c20: 4b28 6666 2920 746f 2057 2065 7874 656e K(ff) to W exten │ │ │ │ -00031c30: 6469 6e67 2070 7369 0a0a 4465 7363 7269 ding psi..Descri │ │ │ │ -00031c40: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00031c50: 3d0a 0a49 6d70 6c65 6d65 6e74 7320 7468 =..Implements th │ │ │ │ -00031c60: 6520 636f 6e73 7472 7563 7469 6f6e 2069 e construction i │ │ │ │ -00031c70: 6e20 7468 6520 7061 7065 7220 224d 6174 n the paper "Mat │ │ │ │ -00031c80: 7269 7820 4661 6374 6f72 697a 6174 696f rix Factorizatio │ │ │ │ -00031c90: 6e73 2069 6e20 4869 6768 6572 0a43 6f64 ns in Higher.Cod │ │ │ │ -00031ca0: 696d 656e 7369 6f6e 2220 6279 2045 6973 imension" by Eis │ │ │ │ -00031cb0: 656e 6275 6420 616e 6420 5065 6576 612e enbud and Peeva. │ │ │ │ -00031cc0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00031cd0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -00031ce0: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ -00031cf0: 696f 6e3a 206d 616b 6546 696e 6974 6552 ion: makeFiniteR │ │ │ │ -00031d00: 6573 6f6c 7574 696f 6e2c 202d 2d20 6669 esolution, -- fi │ │ │ │ -00031d10: 6e69 7465 2072 6573 6f6c 7574 696f 6e20 nite resolution │ │ │ │ -00031d20: 6f66 2061 0a20 2020 206d 6174 7269 7820 of a. matrix │ │ │ │ -00031d30: 6661 6374 6f72 697a 6174 696f 6e20 6d6f factorization mo │ │ │ │ -00031d40: 6475 6c65 204d 0a0a 5761 7973 2074 6f20 dule M..Ways to │ │ │ │ -00031d50: 7573 6520 6b6f 737a 756c 4578 7465 6e73 use koszulExtens │ │ │ │ -00031d60: 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d ion:.=========== │ │ │ │ -00031d70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00031d80: 3d0a 0a20 202a 2022 6b6f 737a 756c 4578 =.. * "koszulEx │ │ │ │ -00031d90: 7465 6e73 696f 6e28 436f 6d70 6c65 782c tension(Complex, │ │ │ │ -00031da0: 436f 6d70 6c65 782c 4d61 7472 6978 2c4d Complex,Matrix,M │ │ │ │ -00031db0: 6174 7269 7829 220a 0a46 6f72 2074 6865 atrix)"..For the │ │ │ │ -00031dc0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00031dd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00031de0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00031df0: 206b 6f73 7a75 6c45 7874 656e 7369 6f6e koszulExtension │ │ │ │ -00031e00: 3a20 6b6f 737a 756c 4578 7465 6e73 696f : koszulExtensio │ │ │ │ -00031e10: 6e2c 2069 7320 6120 2a6e 6f74 6520 6d65 n, is a *note me │ │ │ │ -00031e20: 7468 6f64 2066 756e 6374 696f 6e3a 0a28 thod function:.( │ │ │ │ -00031e30: 4d61 6361 756c 6179 3244 6f63 294d 6574 Macaulay2Doc)Met │ │ │ │ -00031e40: 686f 6446 756e 6374 696f 6e2c 2e0a 0a2d hodFunction,...- │ │ │ │ +00031a00: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00031a10: 7361 6765 3a20 0a20 2020 2020 2020 204d sage: . M │ │ │ │ +00031a20: 4d20 3d20 6b6f 737a 756c 4578 7465 6e73 M = koszulExtens │ │ │ │ +00031a30: 696f 6e28 4646 2c42 422c 7073 6931 2c66 ion(FF,BB,psi1,f │ │ │ │ +00031a40: 6629 0a20 202a 2049 6e70 7574 733a 0a20 f). * Inputs:. │ │ │ │ +00031a50: 2020 2020 202a 2046 462c 2061 202a 6e6f * FF, a *no │ │ │ │ +00031a60: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ +00031a70: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ +00031a80: 2072 6573 6f6c 7574 696f 6e20 6f76 6572 resolution over │ │ │ │ +00031a90: 2053 0a20 2020 2020 202a 2042 422c 2061 S. * BB, a │ │ │ │ +00031aa0: 202a 6e6f 7465 2063 6f6d 706c 6578 3a20 *note complex: │ │ │ │ +00031ab0: 2843 6f6d 706c 6578 6573 2943 6f6d 706c (Complexes)Compl │ │ │ │ +00031ac0: 6578 2c2c 2074 776f 2d74 6572 6d20 636f ex,, two-term co │ │ │ │ +00031ad0: 6d70 6c65 7820 4242 5f31 2d2d 3e42 425f mplex BB_1-->BB_ │ │ │ │ +00031ae0: 300a 2020 2020 2020 2a20 7073 6931 2c20 0. * psi1, │ │ │ │ +00031af0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +00031b00: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +00031b10: 7472 6978 2c2c 2066 726f 6d20 4242 5f31 trix,, from BB_1 │ │ │ │ +00031b20: 2074 6f20 4646 5f30 0a20 2020 2020 202a to FF_0. * │ │ │ │ +00031b30: 2066 662c 2061 202a 6e6f 7465 206d 6174 ff, a *note mat │ │ │ │ +00031b40: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +00031b50: 6f63 294d 6174 7269 782c 2c20 7265 6775 oc)Matrix,, regu │ │ │ │ +00031b60: 6c61 7220 7365 7175 656e 6365 0a20 2020 lar sequence. │ │ │ │ +00031b70: 2020 2020 2061 6e6e 6968 696c 6174 696e annihilatin │ │ │ │ +00031b80: 6720 7468 6520 6d6f 6475 6c65 2072 6573 g the module res │ │ │ │ +00031b90: 6f6c 7665 6420 6279 2046 460a 2020 2a20 olved by FF. * │ │ │ │ +00031ba0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00031bb0: 204d 4d2c 2061 202a 6e6f 7465 2063 6f6d MM, a *note com │ │ │ │ +00031bc0: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ +00031bd0: 2943 6f6d 706c 6578 2c2c 2074 6865 206d )Complex,, the m │ │ │ │ +00031be0: 6170 7069 6e67 2063 6f6e 6520 6f66 2074 apping cone of t │ │ │ │ +00031bf0: 6865 0a20 2020 2020 2020 2069 6e64 7563 he. induc │ │ │ │ +00031c00: 6564 206d 6170 2042 5b2d 315d 5c6f 7469 ed map B[-1]\oti │ │ │ │ +00031c10: 6d65 7320 4b4b 2866 6629 2074 6f20 5720 mes KK(ff) to W │ │ │ │ +00031c20: 6578 7465 6e64 696e 6720 7073 690a 0a44 extending psi..D │ │ │ │ +00031c30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00031c40: 3d3d 3d3d 3d3d 0a0a 496d 706c 656d 656e ======..Implemen │ │ │ │ +00031c50: 7473 2074 6865 2063 6f6e 7374 7275 6374 ts the construct │ │ │ │ +00031c60: 696f 6e20 696e 2074 6865 2070 6170 6572 ion in the paper │ │ │ │ +00031c70: 2022 4d61 7472 6978 2046 6163 746f 7269 "Matrix Factori │ │ │ │ +00031c80: 7a61 7469 6f6e 7320 696e 2048 6967 6865 zations in Highe │ │ │ │ +00031c90: 720a 436f 6469 6d65 6e73 696f 6e22 2062 r.Codimension" b │ │ │ │ +00031ca0: 7920 4569 7365 6e62 7564 2061 6e64 2050 y Eisenbud and P │ │ │ │ +00031cb0: 6565 7661 2e0a 0a53 6565 2061 6c73 6f0a eeva...See also. │ │ │ │ +00031cc0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00031cd0: 6f74 6520 6d61 6b65 4669 6e69 7465 5265 ote makeFiniteRe │ │ │ │ +00031ce0: 736f 6c75 7469 6f6e 3a20 6d61 6b65 4669 solution: makeFi │ │ │ │ +00031cf0: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20 niteResolution, │ │ │ │ +00031d00: 2d2d 2066 696e 6974 6520 7265 736f 6c75 -- finite resolu │ │ │ │ +00031d10: 7469 6f6e 206f 6620 610a 2020 2020 6d61 tion of a. ma │ │ │ │ +00031d20: 7472 6978 2066 6163 746f 7269 7a61 7469 trix factorizati │ │ │ │ +00031d30: 6f6e 206d 6f64 756c 6520 4d0a 0a57 6179 on module M..Way │ │ │ │ +00031d40: 7320 746f 2075 7365 206b 6f73 7a75 6c45 s to use koszulE │ │ │ │ +00031d50: 7874 656e 7369 6f6e 3a0a 3d3d 3d3d 3d3d xtension:.====== │ │ │ │ +00031d60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00031d70: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226b 6f73 ======.. * "kos │ │ │ │ +00031d80: 7a75 6c45 7874 656e 7369 6f6e 2843 6f6d zulExtension(Com │ │ │ │ +00031d90: 706c 6578 2c43 6f6d 706c 6578 2c4d 6174 plex,Complex,Mat │ │ │ │ +00031da0: 7269 782c 4d61 7472 6978 2922 0a0a 466f rix,Matrix)"..Fo │ │ │ │ +00031db0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00031dc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00031dd0: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00031de0: 2a6e 6f74 6520 6b6f 737a 756c 4578 7465 *note koszulExte │ │ │ │ +00031df0: 6e73 696f 6e3a 206b 6f73 7a75 6c45 7874 nsion: koszulExt │ │ │ │ +00031e00: 656e 7369 6f6e 2c20 6973 2061 202a 6e6f ension, is a *no │ │ │ │ +00031e10: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ +00031e20: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ +00031e30: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +00031e40: 2c2e 0a0a 2d2d 2d2d 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6f6c ntersectionResol │ │ │ │ -00031f20: 7574 696f 6e73 2e6d 323a 3330 3038 3a30 utions.m2:3008:0 │ │ │ │ -00031f30: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ -00031f40: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ -00031f50: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ -00031f60: 4e6f 6465 3a20 4c61 7965 7265 642c 204e Node: Layered, N │ │ │ │ -00031f70: 6578 743a 206c 6179 6572 6564 5265 736f ext: layeredReso │ │ │ │ -00031f80: 6c75 7469 6f6e 2c20 5072 6576 3a20 6b6f lution, Prev: ko │ │ │ │ -00031f90: 737a 756c 4578 7465 6e73 696f 6e2c 2055 szulExtension, U │ │ │ │ -00031fa0: 703a 2054 6f70 0a0a 4c61 7965 7265 6420 p: Top..Layered │ │ │ │ -00031fb0: 2d2d 204f 7074 696f 6e20 666f 7220 6d61 -- Option for ma │ │ │ │ -00031fc0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -00031fd0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ +00031e90: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ +00031ea0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ +00031eb0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ +00031ec0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ +00031ed0: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ +00031ee0: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ +00031ef0: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ +00031f00: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00031f10: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ +00031f20: 3030 383a 302e 0a1f 0a46 696c 653a 2043 008:0....File: C │ │ │ │ +00031f30: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ +00031f40: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ +00031f50: 6e66 6f2c 204e 6f64 653a 204c 6179 6572 nfo, Node: Layer │ │ │ │ +00031f60: 6564 2c20 4e65 7874 3a20 6c61 7965 7265 ed, Next: layere │ │ │ │ +00031f70: 6452 6573 6f6c 7574 696f 6e2c 2050 7265 dResolution, Pre │ │ │ │ +00031f80: 763a 206b 6f73 7a75 6c45 7874 656e 7369 v: koszulExtensi │ │ │ │ +00031f90: 6f6e 2c20 5570 3a20 546f 700a 0a4c 6179 on, Up: Top..Lay │ │ │ │ +00031fa0: 6572 6564 202d 2d20 4f70 7469 6f6e 2066 ered -- Option f │ │ │ │ +00031fb0: 6f72 206d 6174 7269 7846 6163 746f 7269 or matrixFactori │ │ │ │ +00031fc0: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a zation.********* │ │ │ │ +00031fd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00031fe0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00031ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00032000: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -00032010: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -00032020: 7469 6f6e 2866 662c 6d2c 4c61 7965 7265 tion(ff,m,Layere │ │ │ │ -00032030: 6420 3d3e 2074 7275 6529 0a20 202a 2049 d => true). * I │ │ │ │ -00032040: 6e70 7574 733a 0a20 2020 2020 202a 2043 nputs:. * C │ │ │ │ -00032050: 6865 636b 2c20 6120 2a6e 6f74 6520 426f heck, a *note Bo │ │ │ │ -00032060: 6f6c 6561 6e20 7661 6c75 653a 2028 4d61 olean value: (Ma │ │ │ │ -00032070: 6361 756c 6179 3244 6f63 2942 6f6f 6c65 caulay2Doc)Boole │ │ │ │ -00032080: 616e 2c2c 200a 0a44 6573 6372 6970 7469 an,, ..Descripti │ │ │ │ -00032090: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -000320a0: 4d61 6b65 7320 6d61 7472 6978 4661 6374 Makes matrixFact │ │ │ │ -000320b0: 6f72 697a 6174 696f 6e20 7573 6520 7468 orization use th │ │ │ │ -000320c0: 6520 226c 6179 6572 6564 2220 616c 676f e "layered" algo │ │ │ │ -000320d0: 7269 7468 6d2c 2077 6869 6368 2077 6f72 rithm, which wor │ │ │ │ -000320e0: 6b73 2066 6f72 2061 6e79 204d 434d 0a6d ks for any MCM.m │ │ │ │ -000320f0: 6f64 756c 652c 2062 7574 2072 6574 7572 odule, but retur │ │ │ │ -00032100: 6e73 2073 6f6d 6574 6869 6e67 206e 6f6e ns something non │ │ │ │ -00032110: 2d6d 696e 696d 616c 2069 6620 7468 6520 -minimal if the │ │ │ │ -00032120: 6d6f 6475 6c65 2069 7320 6e6f 7420 6120 module is not a │ │ │ │ -00032130: 2268 6967 6820 7379 7a79 6779 220a 696e "high syzygy".in │ │ │ │ -00032140: 2061 2073 7569 7461 626c 6520 7365 6e73 a suitable sens │ │ │ │ -00032150: 652e 2044 6566 6175 6c74 2069 7320 2274 e. Default is "t │ │ │ │ -00032160: 7275 6522 2e20 4e6f 7465 2074 6861 7420 rue". Note that │ │ │ │ -00032170: 7768 656e 2074 6865 206d 6f64 756c 6520 when the module │ │ │ │ -00032180: 6973 2061 2068 6967 680a 7379 7a79 6779 is a high.syzygy │ │ │ │ -00032190: 2c20 4c61 7965 7265 643d 3e20 6661 6c73 , Layered=> fals │ │ │ │ -000321a0: 6520 6973 206d 7563 6820 6661 7374 6572 e is much faster │ │ │ │ -000321b0: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -000321c0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -000321d0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ -000321e0: 696f 6e3a 206d 6174 7269 7846 6163 746f ion: matrixFacto │ │ │ │ -000321f0: 7269 7a61 7469 6f6e 2c20 2d2d 204d 6170 rization, -- Map │ │ │ │ -00032200: 7320 696e 2061 2068 6967 6865 720a 2020 s in a higher. │ │ │ │ -00032210: 2020 636f 6469 6d65 6e73 696f 6e20 6d61 codimension ma │ │ │ │ -00032220: 7472 6978 2066 6163 746f 7269 7a61 7469 trix factorizati │ │ │ │ -00032230: 6f6e 0a0a 4675 6e63 7469 6f6e 7320 7769 on..Functions wi │ │ │ │ -00032240: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -00032250: 6d65 6e74 206e 616d 6564 204c 6179 6572 ment named Layer │ │ │ │ -00032260: 6564 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ed:.============ │ │ │ │ +00031ff0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +00032000: 2020 2020 2020 6d61 7472 6978 4661 6374 matrixFact │ │ │ │ +00032010: 6f72 697a 6174 696f 6e28 6666 2c6d 2c4c orization(ff,m,L │ │ │ │ +00032020: 6179 6572 6564 203d 3e20 7472 7565 290a ayered => true). │ │ │ │ +00032030: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00032040: 2020 2a20 4368 6563 6b2c 2061 202a 6e6f * Check, a *no │ │ │ │ +00032050: 7465 2042 6f6f 6c65 616e 2076 616c 7565 te Boolean value │ │ │ │ +00032060: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00032070: 426f 6f6c 6561 6e2c 2c20 0a0a 4465 7363 Boolean,, ..Desc │ │ │ │ +00032080: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00032090: 3d3d 3d0a 0a4d 616b 6573 206d 6174 7269 ===..Makes matri │ │ │ │ +000320a0: 7846 6163 746f 7269 7a61 7469 6f6e 2075 xFactorization u │ │ │ │ +000320b0: 7365 2074 6865 2022 6c61 7965 7265 6422 se the "layered" │ │ │ │ +000320c0: 2061 6c67 6f72 6974 686d 2c20 7768 6963 algorithm, whic │ │ │ │ +000320d0: 6820 776f 726b 7320 666f 7220 616e 7920 h works for any │ │ │ │ +000320e0: 4d43 4d0a 6d6f 6475 6c65 2c20 6275 7420 MCM.module, but │ │ │ │ +000320f0: 7265 7475 726e 7320 736f 6d65 7468 696e returns somethin │ │ │ │ +00032100: 6720 6e6f 6e2d 6d69 6e69 6d61 6c20 6966 g non-minimal if │ │ │ │ +00032110: 2074 6865 206d 6f64 756c 6520 6973 206e the module is n │ │ │ │ +00032120: 6f74 2061 2022 6869 6768 2073 797a 7967 ot a "high syzyg │ │ │ │ +00032130: 7922 0a69 6e20 6120 7375 6974 6162 6c65 y".in a suitable │ │ │ │ +00032140: 2073 656e 7365 2e20 4465 6661 756c 7420 sense. Default │ │ │ │ +00032150: 6973 2022 7472 7565 222e 204e 6f74 6520 is "true". Note │ │ │ │ +00032160: 7468 6174 2077 6865 6e20 7468 6520 6d6f that when the mo │ │ │ │ +00032170: 6475 6c65 2069 7320 6120 6869 6768 0a73 dule is a high.s │ │ │ │ +00032180: 797a 7967 792c 204c 6179 6572 6564 3d3e yzygy, Layered=> │ │ │ │ +00032190: 2066 616c 7365 2069 7320 6d75 6368 2066 false is much f │ │ │ │ +000321a0: 6173 7465 722e 0a0a 5365 6520 616c 736f aster...See also │ │ │ │ +000321b0: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +000321c0: 6e6f 7465 206d 6174 7269 7846 6163 746f note matrixFacto │ │ │ │ +000321d0: 7269 7a61 7469 6f6e 3a20 6d61 7472 6978 rization: matrix │ │ │ │ +000321e0: 4661 6374 6f72 697a 6174 696f 6e2c 202d Factorization, - │ │ │ │ +000321f0: 2d20 4d61 7073 2069 6e20 6120 6869 6768 - Maps in a high │ │ │ │ +00032200: 6572 0a20 2020 2063 6f64 696d 656e 7369 er. codimensi │ │ │ │ +00032210: 6f6e 206d 6174 7269 7820 6661 6374 6f72 on matrix factor │ │ │ │ +00032220: 697a 6174 696f 6e0a 0a46 756e 6374 696f ization..Functio │ │ │ │ +00032230: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00032240: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ +00032250: 4c61 7965 7265 643a 0a3d 3d3d 3d3d 3d3d Layered:.======= │ │ │ │ +00032260: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00032270: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032290: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978 ===.. * "matrix │ │ │ │ -000322a0: 4661 6374 6f72 697a 6174 696f 6e28 2e2e Factorization(.. │ │ │ │ -000322b0: 2e2c 4c61 7965 7265 643d 3e2e 2e2e 2922 .,Layered=>...)" │ │ │ │ -000322c0: 202d 2d20 7365 6520 2a6e 6f74 6520 6d61 -- see *note ma │ │ │ │ -000322d0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -000322e0: 6e3a 0a20 2020 206d 6174 7269 7846 6163 n:. matrixFac │ │ │ │ -000322f0: 746f 7269 7a61 7469 6f6e 2c20 2d2d 204d torization, -- M │ │ │ │ -00032300: 6170 7320 696e 2061 2068 6967 6865 7220 aps in a higher │ │ │ │ -00032310: 636f 6469 6d65 6e73 696f 6e20 6d61 7472 codimension matr │ │ │ │ -00032320: 6978 2066 6163 746f 7269 7a61 7469 6f6e ix factorization │ │ │ │ -00032330: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ -00032340: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ -00032350: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ -00032360: 6563 7420 2a6e 6f74 6520 4c61 7965 7265 ect *note Layere │ │ │ │ -00032370: 643a 204c 6179 6572 6564 2c20 6973 2061 d: Layered, is a │ │ │ │ -00032380: 202a 6e6f 7465 2073 796d 626f 6c3a 2028 *note symbol: ( │ │ │ │ -00032390: 4d61 6361 756c 6179 3244 6f63 2953 796d Macaulay2Doc)Sym │ │ │ │ -000323a0: 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d bol,...--------- │ │ │ │ +00032280: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ +00032290: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +000322a0: 6f6e 282e 2e2e 2c4c 6179 6572 6564 3d3e on(...,Layered=> │ │ │ │ +000322b0: 2e2e 2e29 2220 2d2d 2073 6565 202a 6e6f ...)" -- see *no │ │ │ │ +000322c0: 7465 206d 6174 7269 7846 6163 746f 7269 te matrixFactori │ │ │ │ +000322d0: 7a61 7469 6f6e 3a0a 2020 2020 6d61 7472 zation:. matr │ │ │ │ +000322e0: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ +000322f0: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ +00032300: 6768 6572 2063 6f64 696d 656e 7369 6f6e gher codimension │ │ │ │ +00032310: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ +00032320: 6174 696f 6e0a 0a46 6f72 2074 6865 2070 ation..For the p │ │ │ │ +00032330: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00032340: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00032350: 6520 6f62 6a65 6374 202a 6e6f 7465 204c e object *note L │ │ │ │ +00032360: 6179 6572 6564 3a20 4c61 7965 7265 642c ayered: Layered, │ │ │ │ +00032370: 2069 7320 6120 2a6e 6f74 6520 7379 6d62 is a *note symb │ │ │ │ +00032380: 6f6c 3a20 284d 6163 6175 6c61 7932 446f ol: (Macaulay2Do │ │ │ │ +00032390: 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d c)Symbol,...---- │ │ │ │ +000323a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000323d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000323f0: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ -00032400: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ -00032410: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ -00032420: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ -00032430: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ -00032440: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ -00032450: 6c61 7932 2f70 6163 6b61 6765 732f 0a43 lay2/packages/.C │ │ │ │ -00032460: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -00032470: 696f 6e52 6573 6f6c 7574 696f 6e73 2e6d ionResolutions.m │ │ │ │ -00032480: 323a 3332 3531 3a30 2e0a 1f0a 4669 6c65 2:3251:0....File │ │ │ │ -00032490: 3a20 436f 6d70 6c65 7465 496e 7465 7273 : CompleteInters │ │ │ │ -000324a0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -000324b0: 732e 696e 666f 2c20 4e6f 6465 3a20 6c61 s.info, Node: la │ │ │ │ -000324c0: 7965 7265 6452 6573 6f6c 7574 696f 6e2c yeredResolution, │ │ │ │ -000324d0: 204e 6578 743a 204c 6966 742c 2050 7265 Next: Lift, Pre │ │ │ │ -000324e0: 763a 204c 6179 6572 6564 2c20 5570 3a20 v: Layered, Up: │ │ │ │ -000324f0: 546f 700a 0a6c 6179 6572 6564 5265 736f Top..layeredReso │ │ │ │ -00032500: 6c75 7469 6f6e 202d 2d20 6c61 7965 7265 lution -- layere │ │ │ │ -00032510: 6420 6669 6e69 7465 2061 6e64 2069 6e66 d finite and inf │ │ │ │ -00032520: 696e 6974 6520 6c61 7965 7265 6420 7265 inite layered re │ │ │ │ -00032530: 736f 6c75 7469 6f6e 7320 6f66 2043 4d20 solutions of CM │ │ │ │ -00032540: 6d6f 6475 6c65 730a 2a2a 2a2a 2a2a 2a2a modules.******** │ │ │ │ +000323e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +000323f0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00032400: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00032410: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00032420: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00032430: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +00032440: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00032450: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +00032460: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +00032470: 6f6e 732e 6d32 3a33 3235 313a 302e 0a1f ons.m2:3251:0... │ │ │ │ +00032480: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +00032490: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000324a0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +000324b0: 653a 206c 6179 6572 6564 5265 736f 6c75 e: layeredResolu │ │ │ │ +000324c0: 7469 6f6e 2c20 4e65 7874 3a20 4c69 6674 tion, Next: Lift │ │ │ │ +000324d0: 2c20 5072 6576 3a20 4c61 7965 7265 642c , Prev: Layered, │ │ │ │ +000324e0: 2055 703a 2054 6f70 0a0a 6c61 7965 7265 Up: Top..layere │ │ │ │ +000324f0: 6452 6573 6f6c 7574 696f 6e20 2d2d 206c dResolution -- l │ │ │ │ +00032500: 6179 6572 6564 2066 696e 6974 6520 616e ayered finite an │ │ │ │ +00032510: 6420 696e 6669 6e69 7465 206c 6179 6572 d infinite layer │ │ │ │ +00032520: 6564 2072 6573 6f6c 7574 696f 6e73 206f ed resolutions o │ │ │ │ +00032530: 6620 434d 206d 6f64 756c 6573 0a2a 2a2a f CM modules.*** │ │ │ │ +00032540: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032550: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032560: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00032570: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00032590: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -000325a0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -000325b0: 2846 462c 2061 7567 2920 3d20 6c61 7965 (FF, aug) = laye │ │ │ │ -000325c0: 7265 6452 6573 6f6c 7574 696f 6e28 6666 redResolution(ff │ │ │ │ -000325d0: 2c4d 290a 2020 2020 2020 2020 2846 462c ,M). (FF, │ │ │ │ -000325e0: 2061 7567 2920 3d20 6c61 7965 7265 6452 aug) = layeredR │ │ │ │ -000325f0: 6573 6f6c 7574 696f 6e28 6666 2c4d 2c6c esolution(ff,M,l │ │ │ │ -00032600: 656e 290a 2020 2a20 496e 7075 7473 3a0a en). * Inputs:. │ │ │ │ -00032610: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ -00032620: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ -00032630: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ -00032640: 2c2c 2031 2078 2063 206d 6174 7269 7820 ,, 1 x c matrix │ │ │ │ -00032650: 7768 6f73 6520 656e 7472 6965 730a 2020 whose entries. │ │ │ │ -00032660: 2020 2020 2020 6172 6520 6120 7265 6775 are a regu │ │ │ │ -00032670: 6c61 7220 7365 7175 656e 6365 2069 6e20 lar sequence in │ │ │ │ -00032680: 7468 6520 476f 7265 6e73 7465 696e 2072 the Gorenstein r │ │ │ │ -00032690: 696e 6720 530a 2020 2020 2020 2a20 4d2c ing S. * M, │ │ │ │ -000326a0: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ -000326b0: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ -000326c0: 6f64 756c 652c 2c20 4d43 4d20 6d6f 6475 odule,, MCM modu │ │ │ │ -000326d0: 6c65 206f 7665 7220 522c 0a20 2020 2020 le over R,. │ │ │ │ -000326e0: 2020 2072 6570 7265 7365 6e74 6564 2061 represented a │ │ │ │ -000326f0: 7320 616e 2053 2d6d 6f64 756c 6520 696e s an S-module in │ │ │ │ -00032700: 2074 6865 2066 6972 7374 2063 6173 6520 the first case │ │ │ │ -00032710: 616e 6420 6173 2061 6e20 522d 6d6f 6475 and as an R-modu │ │ │ │ -00032720: 6c65 2069 6e20 7468 650a 2020 2020 2020 le in the. │ │ │ │ -00032730: 2020 7365 636f 6e64 0a20 2020 2020 202a second. * │ │ │ │ -00032740: 206c 656e 2c20 616e 202a 6e6f 7465 2069 len, an *note i │ │ │ │ -00032750: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ -00032760: 7932 446f 6329 5a5a 2c2c 206c 656e 6774 y2Doc)ZZ,, lengt │ │ │ │ -00032770: 6820 6f66 2074 6865 2073 6567 6d65 6e74 h of the segment │ │ │ │ -00032780: 206f 6620 7468 650a 2020 2020 2020 2020 of the. │ │ │ │ -00032790: 7265 736f 6c75 7469 6f6e 2074 6f20 6265 resolution to be │ │ │ │ -000327a0: 2063 6f6d 7075 7465 6420 6f76 6572 2052 computed over R │ │ │ │ -000327b0: 2c20 696e 2074 6865 2073 6563 6f6e 6420 , in the second │ │ │ │ -000327c0: 666f 726d 2e0a 2020 2a20 2a6e 6f74 6520 form.. * *note │ │ │ │ -000327d0: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ -000327e0: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ -000327f0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ -00032800: 6974 6820 6f70 7469 6f6e 616c 2069 6e70 ith optional inp │ │ │ │ -00032810: 7574 732c 3a0a 2020 2020 2020 2a20 4368 uts,:. * Ch │ │ │ │ -00032820: 6563 6b20 3d3e 202e 2e2e 2c20 6465 6661 eck => ..., defa │ │ │ │ -00032830: 756c 7420 7661 6c75 6520 6661 6c73 650a ult value false. │ │ │ │ -00032840: 2020 2020 2020 2a20 5665 7262 6f73 6520 * Verbose │ │ │ │ -00032850: 3d3e 202e 2e2e 2c20 6465 6661 756c 7420 => ..., default │ │ │ │ -00032860: 7661 6c75 6520 6661 6c73 650a 2020 2a20 value false. * │ │ │ │ -00032870: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00032880: 2046 462c 2061 202a 6e6f 7465 2063 6f6d FF, a *note com │ │ │ │ -00032890: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -000328a0: 2943 6f6d 706c 6578 2c2c 2072 6573 6f6c )Complex,, resol │ │ │ │ -000328b0: 7574 696f 6e20 6f66 204d 206f 7665 7220 ution of M over │ │ │ │ -000328c0: 5320 696e 2074 6865 0a20 2020 2020 2020 S in the. │ │ │ │ -000328d0: 2066 6972 7374 2063 6173 653b 206c 656e first case; len │ │ │ │ -000328e0: 6774 6820 6c65 6e20 7365 676d 656e 7420 gth len segment │ │ │ │ -000328f0: 6f66 2074 6865 2072 6573 6f6c 7574 696f of the resolutio │ │ │ │ -00032900: 6e20 6f76 6572 2052 2069 6e20 7468 6520 n over R in the │ │ │ │ -00032910: 7365 636f 6e64 2e0a 0a44 6573 6372 6970 second...Descrip │ │ │ │ -00032920: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00032930: 0a0a 5468 6520 7265 736f 6c75 7469 6f6e ..The resolution │ │ │ │ -00032940: 7320 636f 6d70 7574 6564 2061 7265 2074 s computed are t │ │ │ │ -00032950: 686f 7365 2064 6573 6372 6962 6564 2069 hose described i │ │ │ │ -00032960: 6e20 7468 6520 7061 7065 7220 224c 6179 n the paper "Lay │ │ │ │ -00032970: 6572 6564 2052 6573 6f6c 7574 696f 6e73 ered Resolutions │ │ │ │ -00032980: 0a6f 6620 436f 6865 6e2d 4d61 6361 756c .of Cohen-Macaul │ │ │ │ -00032990: 6179 206d 6f64 756c 6573 2220 6279 2045 ay modules" by E │ │ │ │ -000329a0: 6973 656e 6275 6420 616e 6420 5065 6576 isenbud and Peev │ │ │ │ -000329b0: 612e 2054 6865 7920 6172 6520 626f 7468 a. They are both │ │ │ │ -000329c0: 206d 696e 696d 616c 2077 6865 6e20 4d0a minimal when M. │ │ │ │ -000329d0: 6973 2061 2073 7566 6669 6369 656e 746c is a sufficientl │ │ │ │ -000329e0: 7920 6869 6768 2073 797a 7967 7920 6f66 y high syzygy of │ │ │ │ -000329f0: 2061 206d 6f64 756c 6520 4e2e 2049 6620 a module N. If │ │ │ │ -00032a00: 7468 6520 6f70 7469 6f6e 2056 6572 626f the option Verbo │ │ │ │ -00032a10: 7365 3d3e 7472 7565 2069 730a 7365 742c se=>true is.set, │ │ │ │ -00032a20: 2074 6865 6e20 2869 6e20 7468 6520 6361 then (in the ca │ │ │ │ -00032a30: 7365 206f 6620 7468 6520 7265 736f 6c75 se of the resolu │ │ │ │ -00032a40: 7469 6f6e 206f 7665 7220 5329 2074 6865 tion over S) the │ │ │ │ -00032a50: 2072 616e 6b73 206f 6620 7468 6520 6d6f ranks of the mo │ │ │ │ -00032a60: 6475 6c65 7320 425f 730a 696e 2074 6865 dules B_s.in the │ │ │ │ -00032a70: 2072 6573 6f6c 7574 696f 6e20 6172 6520 resolution are │ │ │ │ -00032a80: 6f75 7470 7574 2e0a 0a48 6572 6520 6973 output...Here is │ │ │ │ -00032a90: 2061 6e20 6578 616d 706c 6520 636f 6d70 an example comp │ │ │ │ -00032aa0: 7574 696e 6720 3520 7465 726d 7320 6f66 uting 5 terms of │ │ │ │ -00032ab0: 2061 6e20 696e 6669 6e69 7465 2072 6573 an infinite res │ │ │ │ -00032ac0: 6f6c 7574 696f 6e3a 0a0a 2b2d 2d2d 2d2d olution:..+----- │ │ │ │ +00032580: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +00032590: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +000325a0: 2020 2020 2028 4646 2c20 6175 6729 203d (FF, aug) = │ │ │ │ +000325b0: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ +000325c0: 6f6e 2866 662c 4d29 0a20 2020 2020 2020 on(ff,M). │ │ │ │ +000325d0: 2028 4646 2c20 6175 6729 203d 206c 6179 (FF, aug) = lay │ │ │ │ +000325e0: 6572 6564 5265 736f 6c75 7469 6f6e 2866 eredResolution(f │ │ │ │ +000325f0: 662c 4d2c 6c65 6e29 0a20 202a 2049 6e70 f,M,len). * Inp │ │ │ │ +00032600: 7574 733a 0a20 2020 2020 202a 2066 662c uts:. * ff, │ │ │ │ +00032610: 2061 202a 6e6f 7465 206d 6174 7269 783a a *note matrix: │ │ │ │ +00032620: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +00032630: 6174 7269 782c 2c20 3120 7820 6320 6d61 atrix,, 1 x c ma │ │ │ │ +00032640: 7472 6978 2077 686f 7365 2065 6e74 7269 trix whose entri │ │ │ │ +00032650: 6573 0a20 2020 2020 2020 2061 7265 2061 es. are a │ │ │ │ +00032660: 2072 6567 756c 6172 2073 6571 7565 6e63 regular sequenc │ │ │ │ +00032670: 6520 696e 2074 6865 2047 6f72 656e 7374 e in the Gorenst │ │ │ │ +00032680: 6569 6e20 7269 6e67 2053 0a20 2020 2020 ein ring S. │ │ │ │ +00032690: 202a 204d 2c20 6120 2a6e 6f74 6520 6d6f * M, a *note mo │ │ │ │ +000326a0: 6475 6c65 3a20 284d 6163 6175 6c61 7932 dule: (Macaulay2 │ │ │ │ +000326b0: 446f 6329 4d6f 6475 6c65 2c2c 204d 434d Doc)Module,, MCM │ │ │ │ +000326c0: 206d 6f64 756c 6520 6f76 6572 2052 2c0a module over R,. │ │ │ │ +000326d0: 2020 2020 2020 2020 7265 7072 6573 656e represen │ │ │ │ +000326e0: 7465 6420 6173 2061 6e20 532d 6d6f 6475 ted as an S-modu │ │ │ │ +000326f0: 6c65 2069 6e20 7468 6520 6669 7273 7420 le in the first │ │ │ │ +00032700: 6361 7365 2061 6e64 2061 7320 616e 2052 case and as an R │ │ │ │ +00032710: 2d6d 6f64 756c 6520 696e 2074 6865 0a20 -module in the. │ │ │ │ +00032720: 2020 2020 2020 2073 6563 6f6e 640a 2020 second. │ │ │ │ +00032730: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e * len, an *n │ │ │ │ +00032740: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +00032750: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +00032760: 6c65 6e67 7468 206f 6620 7468 6520 7365 length of the se │ │ │ │ +00032770: 676d 656e 7420 6f66 2074 6865 0a20 2020 gment of the. │ │ │ │ +00032780: 2020 2020 2072 6573 6f6c 7574 696f 6e20 resolution │ │ │ │ +00032790: 746f 2062 6520 636f 6d70 7574 6564 206f to be computed o │ │ │ │ +000327a0: 7665 7220 522c 2069 6e20 7468 6520 7365 ver R, in the se │ │ │ │ +000327b0: 636f 6e64 2066 6f72 6d2e 0a20 202a 202a cond form.. * * │ │ │ │ +000327c0: 6e6f 7465 204f 7074 696f 6e61 6c20 696e note Optional in │ │ │ │ +000327d0: 7075 7473 3a20 284d 6163 6175 6c61 7932 puts: (Macaulay2 │ │ │ │ +000327e0: 446f 6329 7573 696e 6720 6675 6e63 7469 Doc)using functi │ │ │ │ +000327f0: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ +00032800: 6c20 696e 7075 7473 2c3a 0a20 2020 2020 l inputs,:. │ │ │ │ +00032810: 202a 2043 6865 636b 203d 3e20 2e2e 2e2c * Check => ..., │ │ │ │ +00032820: 2064 6566 6175 6c74 2076 616c 7565 2066 default value f │ │ │ │ +00032830: 616c 7365 0a20 2020 2020 202a 2056 6572 alse. * Ver │ │ │ │ +00032840: 626f 7365 203d 3e20 2e2e 2e2c 2064 6566 bose => ..., def │ │ │ │ +00032850: 6175 6c74 2076 616c 7565 2066 616c 7365 ault value false │ │ │ │ +00032860: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00032870: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74 * FF, a *not │ │ │ │ +00032880: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +00032890: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +000328a0: 7265 736f 6c75 7469 6f6e 206f 6620 4d20 resolution of M │ │ │ │ +000328b0: 6f76 6572 2053 2069 6e20 7468 650a 2020 over S in the. │ │ │ │ +000328c0: 2020 2020 2020 6669 7273 7420 6361 7365 first case │ │ │ │ +000328d0: 3b20 6c65 6e67 7468 206c 656e 2073 6567 ; length len seg │ │ │ │ +000328e0: 6d65 6e74 206f 6620 7468 6520 7265 736f ment of the reso │ │ │ │ +000328f0: 6c75 7469 6f6e 206f 7665 7220 5220 696e lution over R in │ │ │ │ +00032900: 2074 6865 2073 6563 6f6e 642e 0a0a 4465 the second...De │ │ │ │ +00032910: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00032920: 3d3d 3d3d 3d0a 0a54 6865 2072 6573 6f6c =====..The resol │ │ │ │ +00032930: 7574 696f 6e73 2063 6f6d 7075 7465 6420 utions computed │ │ │ │ +00032940: 6172 6520 7468 6f73 6520 6465 7363 7269 are those descri │ │ │ │ +00032950: 6265 6420 696e 2074 6865 2070 6170 6572 bed in the paper │ │ │ │ +00032960: 2022 4c61 7965 7265 6420 5265 736f 6c75 "Layered Resolu │ │ │ │ +00032970: 7469 6f6e 730a 6f66 2043 6f68 656e 2d4d tions.of Cohen-M │ │ │ │ +00032980: 6163 6175 6c61 7920 6d6f 6475 6c65 7322 acaulay modules" │ │ │ │ +00032990: 2062 7920 4569 7365 6e62 7564 2061 6e64 by Eisenbud and │ │ │ │ +000329a0: 2050 6565 7661 2e20 5468 6579 2061 7265 Peeva. They are │ │ │ │ +000329b0: 2062 6f74 6820 6d69 6e69 6d61 6c20 7768 both minimal wh │ │ │ │ +000329c0: 656e 204d 0a69 7320 6120 7375 6666 6963 en M.is a suffic │ │ │ │ +000329d0: 6965 6e74 6c79 2068 6967 6820 7379 7a79 iently high syzy │ │ │ │ +000329e0: 6779 206f 6620 6120 6d6f 6475 6c65 204e gy of a module N │ │ │ │ +000329f0: 2e20 4966 2074 6865 206f 7074 696f 6e20 . If the option │ │ │ │ +00032a00: 5665 7262 6f73 653d 3e74 7275 6520 6973 Verbose=>true is │ │ │ │ +00032a10: 0a73 6574 2c20 7468 656e 2028 696e 2074 .set, then (in t │ │ │ │ +00032a20: 6865 2063 6173 6520 6f66 2074 6865 2072 he case of the r │ │ │ │ +00032a30: 6573 6f6c 7574 696f 6e20 6f76 6572 2053 esolution over S │ │ │ │ +00032a40: 2920 7468 6520 7261 6e6b 7320 6f66 2074 ) the ranks of t │ │ │ │ +00032a50: 6865 206d 6f64 756c 6573 2042 5f73 0a69 he modules B_s.i │ │ │ │ +00032a60: 6e20 7468 6520 7265 736f 6c75 7469 6f6e n the resolution │ │ │ │ +00032a70: 2061 7265 206f 7574 7075 742e 0a0a 4865 are output...He │ │ │ │ +00032a80: 7265 2069 7320 616e 2065 7861 6d70 6c65 re is an example │ │ │ │ +00032a90: 2063 6f6d 7075 7469 6e67 2035 2074 6572 computing 5 ter │ │ │ │ +00032aa0: 6d73 206f 6620 616e 2069 6e66 696e 6974 ms of an infinit │ │ │ │ +00032ab0: 6520 7265 736f 6c75 7469 6f6e 3a0a 0a2b e resolution:..+ │ │ │ │ +00032ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032b10: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ -00032b20: 5320 3d20 5a5a 2f31 3031 5b61 2c62 2c63 S = ZZ/101[a,b,c │ │ │ │ -00032b30: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00032b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00032b10: 6931 203a 2053 203d 205a 5a2f 3130 315b i1 : S = ZZ/101[ │ │ │ │ +00032b20: 612c 622c 635d 2020 2020 2020 2020 2020 a,b,c] │ │ │ │ +00032b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032b60: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032b50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032bb0: 2020 2020 2020 2020 7c0a 7c6f 3120 3d20 |.|o1 = │ │ │ │ -00032bc0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00032ba0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032bb0: 6f31 203d 2053 2020 2020 2020 2020 2020 o1 = S │ │ │ │ +00032bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032bf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c50: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ -00032c60: 506f 6c79 6e6f 6d69 616c 5269 6e67 2020 PolynomialRing │ │ │ │ +00032c40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032c50: 6f31 203a 2050 6f6c 796e 6f6d 6961 6c52 o1 : PolynomialR │ │ │ │ +00032c60: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ 00032c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ca0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00032c90: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00032ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032cb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032cf0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -00032d00: 6666 203d 206d 6174 7269 7822 6133 2c20 ff = matrix"a3, │ │ │ │ -00032d10: 6233 2c20 6333 2220 2020 2020 2020 2020 b3, c3" │ │ │ │ +00032ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00032cf0: 6932 203a 2066 6620 3d20 6d61 7472 6978 i2 : ff = matrix │ │ │ │ +00032d00: 2261 332c 2062 332c 2063 3322 2020 2020 "a3, b3, c3" │ │ │ │ +00032d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032d30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032d90: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ -00032da0: 7c20 6133 2062 3320 6333 207c 2020 2020 | a3 b3 c3 | │ │ │ │ +00032d80: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032d90: 6f32 203d 207c 2061 3320 6233 2063 3320 o2 = | a3 b3 c3 │ │ │ │ +00032da0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00032db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032de0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032dd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00032e40: 2020 2020 2020 2020 3120 2020 2020 2033 1 3 │ │ │ │ +00032e20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032e30: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +00032e40: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ 00032e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032e80: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ -00032e90: 4d61 7472 6978 2053 2020 3c2d 2d20 5320 Matrix S <-- S │ │ │ │ +00032e70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032e80: 6f32 203a 204d 6174 7269 7820 5320 203c o2 : Matrix S < │ │ │ │ +00032e90: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ 00032ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032ed0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00032ec0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00032ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00032f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00032f20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ -00032f30: 5220 3d20 532f 6964 6561 6c20 6666 2020 R = S/ideal ff │ │ │ │ +00032f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00032f20: 6933 203a 2052 203d 2053 2f69 6465 616c i3 : R = S/ideal │ │ │ │ +00032f30: 2066 6620 2020 2020 2020 2020 2020 2020 ff │ │ │ │ 00032f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032f70: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00032f60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032fc0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ -00032fd0: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +00032fb0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00032fc0: 6f33 203d 2052 2020 2020 2020 2020 2020 o3 = R │ │ │ │ +00032fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033010: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00033000: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033060: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -00033070: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +00033050: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033060: 6f33 203a 2051 756f 7469 656e 7452 696e o3 : QuotientRin │ │ │ │ +00033070: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ 00033080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000330a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000330b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000330a0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000330b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000330c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000330d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000330e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00033100: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 --------+.|i4 : │ │ │ │ -00033110: 4d20 3d20 7379 7a79 6779 4d6f 6475 6c65 M = syzygyModule │ │ │ │ -00033120: 2832 2c63 6f6b 6572 2076 6172 7320 5229 (2,coker vars R) │ │ │ │ +000330f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00033100: 6934 203a 204d 203d 2073 797a 7967 794d i4 : M = syzygyM │ │ │ │ +00033110: 6f64 756c 6528 322c 636f 6b65 7220 7661 odule(2,coker va │ │ │ │ +00033120: 7273 2052 2920 2020 2020 2020 2020 2020 rs R) │ │ │ │ 00033130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033150: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00033140: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000331a0: 2020 2020 2020 2020 7c0a 7c6f 3420 3d20 |.|o4 = │ │ │ │ -000331b0: 636f 6b65 726e 656c 207b 327d 207c 2061 cokernel {2} | a │ │ │ │ -000331c0: 2020 3020 2d63 3220 3020 2020 6232 2030 0 -c2 0 b2 0 │ │ │ │ -000331d0: 2030 2020 2030 2020 3020 2030 207c 2020 0 0 0 0 | │ │ │ │ -000331e0: 2020 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7c0a 7c74 6f74 616c |.|total │ │ │ │ -00033570: 3a20 3420 3420 3420 3420 3420 3420 2020 : 4 4 4 4 4 4 │ │ │ │ +00033550: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033560: 746f 7461 6c3a 2034 2034 2034 2034 2034 total: 4 4 4 4 4 │ │ │ │ +00033570: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ 00033580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000335a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000335b0: 2020 2020 2020 2020 7c0a 7c20 2020 2032 |.| 2 │ │ │ │ -000335c0: 3a20 3320 3120 2e20 2e20 2e20 2e20 2020 : 3 1 . . . . │ │ │ │ +000335a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000335b0: 2020 2020 323a 2033 2031 202e 202e 202e 2: 3 1 . . . │ │ │ │ +000335c0: 202e 2020 2020 2020 2020 2020 2020 2020 . │ │ │ │ 000335d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000335e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000335f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 │ │ │ │ -000336a0: 2020 2020 2020 2020 7c0a 7c20 2020 2035 |.| 5 │ │ │ │ -000336b0: 3a20 2e20 2e20 2e20 2e20 3120 3320 2020 : . . . . 1 3 │ │ │ │ +00033690: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000336a0: 2020 2020 353a 202e 202e 202e 202e 2031 5: . . . . 1 │ │ │ │ +000336b0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000336c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000336d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000336e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000336f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00033700: 2020 3020 3120 3220 2033 2020 3420 2035 0 1 2 3 4 5 │ │ │ │ +000336e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000336f0: 2020 2020 2020 2030 2031 2032 2020 3320 0 1 2 3 │ │ │ │ +00033700: 2034 2020 3520 2020 2020 2020 2020 2020 4 5 │ │ │ │ 00033710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033740: 2020 2020 2020 2020 7c0a 7c74 6f74 616c |.|total │ │ │ │ -00033750: 3a20 3520 3720 3920 3131 2031 3320 3135 : 5 7 9 11 13 15 │ │ │ │ +00033730: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033740: 746f 7461 6c3a 2035 2037 2039 2031 3120 total: 5 7 9 11 │ │ │ │ +00033750: 3133 2031 3520 2020 2020 2020 2020 2020 13 15 │ │ │ │ 00033760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00033770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00033790: 2020 2020 2020 2020 7c0a 7c20 2020 2032 |.| 2 │ │ │ │ -000337a0: 3a20 3320 3120 2e20 202e 2020 2e20 202e : 3 1 . . . . │ │ │ │ +00033780: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00033790: 2020 2020 323a 2033 2031 202e 2020 2e20 2: 3 1 . . │ │ │ │ +000337a0: 202e 2020 2e20 2020 2020 2020 2020 2020 . . │ │ │ │ 000337b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000337c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 │ │ │ │ 00036410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036440: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00036430: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00036440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036490: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3130 203a --------+.|i10 : │ │ │ │ -000364a0: 2061 7070 6c79 2835 2c20 6a2d 3e20 7072 apply(5, j-> pr │ │ │ │ -000364b0: 756e 6520 4848 5f6a 2043 203d 3d20 3029 une HH_j C == 0) │ │ │ │ +00036480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00036490: 6931 3020 3a20 6170 706c 7928 352c 206a i10 : apply(5, j │ │ │ │ +000364a0: 2d3e 2070 7275 6e65 2048 485f 6a20 4320 -> prune HH_j C │ │ │ │ +000364b0: 3d3d 2030 2920 2020 2020 2020 2020 2020 == 0) │ │ │ │ 000364c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000364d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000364e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000364d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000364e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000364f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036530: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ -00036540: 207b 7472 7565 2c20 6661 6c73 652c 2066 {true, false, f │ │ │ │ -00036550: 616c 7365 2c20 6661 6c73 652c 2066 616c alse, false, fal │ │ │ │ -00036560: 7365 7d20 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+000365c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000365d0: 6f31 3020 3a20 4c69 7374 2020 2020 2020 o10 : List │ │ │ │ +000365e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000365f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036620: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00036610: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00036620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036670: 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 6e64 206f --------+..And o │ │ │ │ -00036680: 6e65 2063 6f6d 7075 7469 6e67 2074 6865 ne computing the │ │ │ │ -00036690: 2077 686f 6c65 2066 696e 6974 6520 7265 whole finite re │ │ │ │ -000366a0: 736f 6c75 7469 6f6e 3a0a 0a2b 2d2d 2d2d solution:..+---- │ │ │ │ +00036660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00036670: 416e 6420 6f6e 6520 636f 6d70 7574 696e And one computin │ │ │ │ +00036680: 6720 7468 6520 7768 6f6c 6520 6669 6e69 g the whole fini │ │ │ │ +00036690: 7465 2072 6573 6f6c 7574 696f 6e3a 0a0a te resolution:.. │ │ │ │ +000366a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000366b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000366c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000366d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000366e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000366f0: 2d2d 2d2b 0a7c 6931 3120 3a20 4d53 203d ---+.|i11 : MS = │ │ │ │ -00036700: 2070 7573 6846 6f72 7761 7264 286d 6170 pushForward(map │ │ │ │ -00036710: 2852 2c53 292c 204d 293b 2020 2020 2020 (R,S), M); │ │ │ │ +000366e0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ +000366f0: 204d 5320 3d20 7075 7368 466f 7277 6172 MS = pushForwar │ │ │ │ +00036700: 6428 6d61 7028 522c 5329 2c20 4d29 3b20 d(map(R,S), M); │ │ │ │ +00036710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036730: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00036730: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00036740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036780: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 3a20 -------+.|i12 : │ │ │ │ -00036790: 2847 472c 2061 7567 2920 3d20 6c61 7965 (GG, aug) = laye │ │ │ │ -000367a0: 7265 6452 6573 6f6c 7574 696f 6e28 6666 redResolution(ff │ │ │ │ -000367b0: 2c4d 5329 2020 2020 2020 2020 2020 2020 ,MS) │ │ │ │ -000367c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000367d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00036780: 3132 203a 2028 4747 2c20 6175 6729 203d 12 : (GG, aug) = │ │ │ │ +00036790: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ +000367a0: 6f6e 2866 662c 4d53 2920 2020 2020 2020 on(ff,MS) │ │ │ │ +000367b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000367c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000367d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000367e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000367f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036810: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00036820: 2020 2020 2020 3620 2020 2020 2031 3320 6 13 │ │ │ │ -00036830: 2020 2020 2031 3020 2020 2020 2033 2020 10 3 │ │ │ │ +00036810: 7c0a 7c20 2020 2020 2020 2036 2020 2020 |.| 6 │ │ │ │ +00036820: 2020 3133 2020 2020 2020 3130 2020 2020 13 10 │ │ │ │ +00036830: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00036840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036860: 2020 2020 207c 0a7c 6f31 3220 3d20 2853 |.|o12 = (S │ │ │ │ -00036870: 2020 3c2d 2d20 5320 2020 3c2d 2d20 5320 <-- S <-- S │ │ │ │ -00036880: 2020 3c2d 2d20 5320 2c20 7b32 7d20 7c20 <-- S , {2} | │ │ │ │ -00036890: 3020 3020 3020 3020 2030 2020 3120 7c29 0 0 0 0 0 1 |) │ │ │ │ -000368a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000368b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -000368c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000368d0: 2020 2020 7b32 7d20 7c20 3020 3020 3020 {2} | 0 0 0 │ │ │ │ -000368e0: 2d31 2030 2020 3020 7c20 2020 2020 2020 -1 0 0 | │ │ │ │ -000368f0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00036900: 2020 2030 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ -00036910: 2020 3220 2020 2020 2020 3320 2020 7b32 2 3 {2 │ │ │ │ -00036920: 7d20 7c20 3020 3020 3020 3020 202d 3120 } | 0 0 0 0 -1 │ │ │ │ -00036930: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ -00036940: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00036950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036960: 2020 2020 2020 2020 7b33 7d20 7c20 3120 {3} | 1 │ │ │ │ -00036970: 3020 3020 3020 2030 2020 3020 7c20 2020 0 0 0 0 0 | │ │ │ │ -00036980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00036850: 2020 2020 2020 2020 2020 7c0a 7c6f 3132 |.|o12 │ │ │ │ +00036860: 203d 2028 5320 203c 2d2d 2053 2020 203c = (S <-- S < │ │ │ │ +00036870: 2d2d 2053 2020 203c 2d2d 2053 202c 207b -- S <-- S , { │ │ │ │ +00036880: 327d 207c 2030 2030 2030 2030 2020 3020 2} | 0 0 0 0 0 │ │ │ │ +00036890: 2031 207c 2920 2020 2020 2020 2020 2020 1 |) │ │ │ │ +000368a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000368b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000368c0: 2020 2020 2020 2020 207b 327d 207c 2030 {2} | 0 │ │ │ │ +000368d0: 2030 2030 202d 3120 3020 2030 207c 2020 0 0 -1 0 0 | │ │ │ │ +000368e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000368f0: 7c20 2020 2020 2020 3020 2020 2020 2031 | 0 1 │ │ │ │ +00036900: 2020 2020 2020 2032 2020 2020 2020 2033 2 3 │ │ │ │ +00036910: 2020 207b 327d 207c 2030 2030 2030 2030 {2} | 0 0 0 0 │ │ │ │ +00036920: 2020 2d31 2030 207c 2020 2020 2020 2020 -1 0 | │ │ │ │ +00036930: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00036940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036950: 2020 2020 2020 2020 2020 2020 207b 337d {3} │ │ │ │ +00036960: 207c 2031 2030 2030 2030 2020 3020 2030 | 1 0 0 0 0 0 │ │ │ │ +00036970: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00036980: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00036990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369b0: 2020 7b33 7d20 7c20 3020 3120 3020 3020 {3} | 0 1 0 0 │ │ │ │ -000369c0: 2030 2020 3020 7c20 2020 2020 2020 2020 0 0 | │ │ │ │ -000369d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000369a0: 2020 2020 2020 207b 337d 207c 2030 2031 {3} | 0 1 │ │ │ │ +000369b0: 2030 2030 2020 3020 2030 207c 2020 2020 0 0 0 0 | │ │ │ │ +000369c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000369d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000369e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000369f0: 2020 2020 2020 2020 2020 2020 7b33 7d20 {3} │ │ │ │ -00036a00: 7c20 3020 3020 3120 3020 2030 2020 3020 | 0 0 1 0 0 0 │ │ │ │ -00036a10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00036a20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000369f0: 207b 337d 207c 2030 2030 2031 2030 2020 {3} | 0 0 1 0 │ │ │ │ +00036a00: 3020 2030 207c 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +00036a10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00036a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036a60: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00036a70: 3220 3a20 5365 7175 656e 6365 2020 2020 2 : Sequence │ │ │ │ +00036a60: 7c0a 7c6f 3132 203a 2053 6571 7565 6e63 |.|o12 : Sequenc │ │ │ │ +00036a70: 6520 2020 2020 2020 2020 2020 2020 2020 e │ │ │ │ 00036a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ab0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00036aa0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00036ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00036b00: 0a7c 6931 3320 3a20 2847 472c 2061 7567 .|i13 : (GG, aug │ │ │ │ -00036b10: 2920 3d20 6c61 7965 7265 6452 6573 6f6c ) = layeredResol │ │ │ │ -00036b20: 7574 696f 6e28 6666 2c4d 532c 2056 6572 ution(ff,MS, Ver │ │ │ │ -00036b30: 626f 7365 203d 3e74 7275 6529 2020 2020 bose =>true) │ │ │ │ -00036b40: 2020 2020 2020 2020 207c 0a7c 7b33 2c20 |.|{3, │ │ │ │ -00036b50: 317d 2069 6e20 636f 6469 6d65 6e73 696f 1} in codimensio │ │ │ │ -00036b60: 6e20 3320 2020 2020 2020 2020 2020 2020 n 3 │ │ │ │ +00036af0: 2d2d 2d2d 2b0a 7c69 3133 203a 2028 4747 ----+.|i13 : (GG │ │ │ │ +00036b00: 2c20 6175 6729 203d 206c 6179 6572 6564 , aug) = layered │ │ │ │ +00036b10: 5265 736f 6c75 7469 6f6e 2866 662c 4d53 Resolution(ff,MS │ │ │ │ +00036b20: 2c20 5665 7262 6f73 6520 3d3e 7472 7565 , Verbose =>true │ │ │ │ +00036b30: 2920 2020 2020 2020 2020 2020 2020 7c0a ) |. │ │ │ │ +00036b40: 7c7b 332c 2031 7d20 696e 2063 6f64 696d |{3, 1} in codim │ │ │ │ +00036b50: 656e 7369 6f6e 2033 2020 2020 2020 2020 ension 3 │ │ │ │ +00036b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036b90: 2020 207c 0a7c 7b33 2c20 317d 2069 6e20 |.|{3, 1} in │ │ │ │ -00036ba0: 636f 6469 6d65 6e73 696f 6e20 3220 2020 codimension 2 │ │ │ │ +00036b80: 2020 2020 2020 2020 7c0a 7c7b 332c 2031 |.|{3, 1 │ │ │ │ +00036b90: 7d20 696e 2063 6f64 696d 656e 7369 6f6e } in codimension │ │ │ │ +00036ba0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 00036bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036bd0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00036bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00036be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c20: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00036c30: 2020 3620 2020 2020 2031 3320 2020 2020 6 13 │ │ │ │ -00036c40: 2031 3020 2020 2020 2033 2020 2020 2020 10 3 │ │ │ │ +00036c10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00036c20: 2020 2020 2020 2036 2020 2020 2020 3133 6 13 │ │ │ │ +00036c30: 2020 2020 2020 3130 2020 2020 2020 3320 10 3 │ │ │ │ +00036c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036c70: 207c 0a7c 6f31 3320 3d20 2853 2020 3c2d |.|o13 = (S <- │ │ │ │ -00036c80: 2d20 5320 2020 3c2d 2d20 5320 2020 3c2d - S <-- S <- │ │ │ │ -00036c90: 2d20 5320 2c20 7b32 7d20 7c20 3020 3020 - S , {2} | 0 0 │ │ │ │ -00036ca0: 3020 3020 2030 2020 3120 7c29 2020 2020 0 0 0 1 |) │ │ │ │ -00036cb0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00036c60: 2020 2020 2020 7c0a 7c6f 3133 203d 2028 |.|o13 = ( │ │ │ │ +00036c70: 5320 203c 2d2d 2053 2020 203c 2d2d 2053 S <-- S <-- S │ │ │ │ +00036c80: 2020 203c 2d2d 2053 202c 207b 327d 207c <-- S , {2} | │ │ │ │ +00036c90: 2030 2030 2030 2030 2020 3020 2031 207c 0 0 0 0 0 1 | │ │ │ │ +00036ca0: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00036cb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00036cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036ce0: 7b32 7d20 7c20 3020 3020 3020 2d31 2030 {2} | 0 0 0 -1 0 │ │ │ │ -00036cf0: 2020 3020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -00036d00: 2020 2020 207c 0a7c 2020 2020 2020 2030 |.| 0 │ │ │ │ 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2020 S │ │ │ │ 00037790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000377a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000377b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000377a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000377b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000377c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000377d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000377e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000377f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037800: 207c 0a7c 2020 2020 2020 3020 2020 2020 |.| 0 │ │ │ │ -00037810: 2031 2020 2020 2020 3220 2020 2020 2020 1 2 │ │ │ │ -00037820: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +000377f0: 2020 2020 2020 7c0a 7c20 2020 2020 2030 |.| 0 │ │ │ │ +00037800: 2020 2020 2020 3120 2020 2020 2032 2020 1 2 │ │ │ │ +00037810: 2020 2020 2033 2020 2020 2020 2020 2020 3 │ │ │ │ +00037820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037830: 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j-> │ │ │ │ +00037940: 7072 756e 6520 4848 5f6a 2043 203d 3d20 prune HH_j C == │ │ │ │ +00037950: 3029 2020 2020 2020 2020 2020 2020 2020 0) │ │ │ │ +00037960: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00037970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000379a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000379b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000379c0: 6f31 3720 3d20 7b74 7275 652c 2074 7275 o17 = {true, tru │ │ │ │ -000379d0: 652c 2074 7275 652c 2066 616c 7365 7d20 e, true, false} │ │ │ │ +000379b0: 2020 7c0a 7c6f 3137 203d 207b 7472 7565 |.|o17 = {true │ │ │ │ +000379c0: 2c20 7472 7565 2c20 7472 7565 2c20 6661 , true, true, fa │ │ │ │ +000379d0: 6c73 657d 2020 2020 2020 2020 2020 2020 lse} │ │ │ │ 000379e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000379f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000379f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00037a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a50: 207c 0a7c 6f31 3720 3a20 4c69 7374 2020 |.|o17 : List │ │ │ │ +00037a40: 2020 2020 2020 7c0a 7c6f 3137 203a 204c |.|o17 : L │ │ │ │ +00037a50: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 00037a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a90: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00037a90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00037aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037ae0: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ -00037af0: 7573 6520 6c61 7965 7265 6452 6573 6f6c use layeredResol │ │ │ │ -00037b00: 7574 696f 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d ution:.========= │ │ │ │ -00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037b20: 3d3d 3d3d 3d0a 0a20 202a 2022 6c61 7965 =====.. * "laye │ │ │ │ -00037b30: 7265 6452 6573 6f6c 7574 696f 6e28 4d61 redResolution(Ma │ │ │ │ -00037b40: 7472 6978 2c4d 6f64 756c 6529 220a 2020 trix,Module)". │ │ │ │ -00037b50: 2a20 226c 6179 6572 6564 5265 736f 6c75 * "layeredResolu │ │ │ │ -00037b60: 7469 6f6e 284d 6174 7269 782c 4d6f 6475 tion(Matrix,Modu │ │ │ │ -00037b70: 6c65 2c5a 5a29 220a 0a46 6f72 2074 6865 le,ZZ)"..For the │ │ │ │ -00037b80: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00037b90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00037ba0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00037bb0: 206c 6179 6572 6564 5265 736f 6c75 7469 layeredResoluti │ │ │ │ -00037bc0: 6f6e 3a20 6c61 7965 7265 6452 6573 6f6c on: layeredResol │ │ │ │ -00037bd0: 7574 696f 6e2c 2069 7320 6120 2a6e 6f74 ution, is a *not │ │ │ │ -00037be0: 6520 6d65 7468 6f64 0a66 756e 6374 696f e method.functio │ │ │ │ -00037bf0: 6e20 7769 7468 206f 7074 696f 6e73 3a20 n with options: │ │ │ │ -00037c00: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -00037c10: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ -00037c20: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00037ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a57 6179 ----------+..Way │ │ │ │ +00037ae0: 7320 746f 2075 7365 206c 6179 6572 6564 s to use layered │ │ │ │ +00037af0: 5265 736f 6c75 7469 6f6e 3a0a 3d3d 3d3d Resolution:.==== │ │ │ │ +00037b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00037b10: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00037b20: 226c 6179 6572 6564 5265 736f 6c75 7469 "layeredResoluti │ │ │ │ +00037b30: 6f6e 284d 6174 7269 782c 4d6f 6475 6c65 on(Matrix,Module │ │ │ │ +00037b40: 2922 0a20 202a 2022 6c61 7965 7265 6452 )". * "layeredR │ │ │ │ +00037b50: 6573 6f6c 7574 696f 6e28 4d61 7472 6978 esolution(Matrix │ │ │ │ +00037b60: 2c4d 6f64 756c 652c 5a5a 2922 0a0a 466f ,Module,ZZ)"..Fo │ │ │ │ +00037b70: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ +00037b80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00037b90: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ +00037ba0: 2a6e 6f74 6520 6c61 7965 7265 6452 6573 *note layeredRes │ │ │ │ +00037bb0: 6f6c 7574 696f 6e3a 206c 6179 6572 6564 olution: layered │ │ │ │ +00037bc0: 5265 736f 6c75 7469 6f6e 2c20 6973 2061 Resolution, is a │ │ │ │ +00037bd0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ +00037be0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ +00037bf0: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ +00037c00: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ +00037c10: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +00037c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00037c70: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -00037c80: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00037c90: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00037ca0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00037cb0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00037cc0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -00037cd0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00037ce0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00037cf0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00037d00: 6e73 2e6d 323a 3438 3935 3a30 2e0a 1f0a ns.m2:4895:0.... │ │ │ │ -00037d10: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00037d20: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -00037d30: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -00037d40: 3a20 4c69 6674 2c20 4e65 7874 3a20 6d61 : Lift, Next: ma │ │ │ │ -00037d50: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ -00037d60: 6f6e 2c20 5072 6576 3a20 6c61 7965 7265 on, Prev: layere │ │ │ │ -00037d70: 6452 6573 6f6c 7574 696f 6e2c 2055 703a dResolution, Up: │ │ │ │ -00037d80: 2054 6f70 0a0a 4c69 6674 202d 2d20 4f70 Top..Lift -- Op │ │ │ │ -00037d90: 7469 6f6e 2066 6f72 206e 6577 4578 740a tion for newExt. │ │ │ │ -00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00037db0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -00037dc0: 7361 6765 3a20 0a20 2020 2020 2020 206e sage: . n │ │ │ │ -00037dd0: 6577 4578 7428 4d2c 4e2c 4368 6563 6b20 ewExt(M,N,Check │ │ │ │ -00037de0: 3d3e 7472 7565 290a 2020 2a20 496e 7075 =>true). * Inpu │ │ │ │ -00037df0: 7473 3a0a 2020 2020 2020 2a20 4368 6563 ts:. * Chec │ │ │ │ -00037e00: 6b2c 2061 202a 6e6f 7465 2042 6f6f 6c65 k, a *note Boole │ │ │ │ -00037e10: 616e 2076 616c 7565 3a20 284d 6163 6175 an value: (Macau │ │ │ │ -00037e20: 6c61 7932 446f 6329 426f 6f6c 6561 6e2c lay2Doc)Boolean, │ │ │ │ -00037e30: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ -00037e40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a4d 616b ===========..Mak │ │ │ │ -00037e50: 6573 206e 6577 4578 7420 7065 7266 6f72 es newExt perfor │ │ │ │ -00037e60: 6d20 7661 7269 6f75 7320 6368 6563 6b73 m various checks │ │ │ │ -00037e70: 2061 7320 6974 2063 6f6d 7075 7465 732e as it computes. │ │ │ │ -00037e80: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00037e90: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206e ===.. * *note n │ │ │ │ -00037ea0: 6577 4578 743a 206e 6577 4578 742c 202d ewExt: newExt, - │ │ │ │ -00037eb0: 2d20 476c 6f62 616c 2045 7874 2066 6f72 - Global Ext for │ │ │ │ -00037ec0: 206d 6f64 756c 6573 206f 7665 7220 6120 modules over a │ │ │ │ -00037ed0: 636f 6d70 6c65 7465 0a20 2020 2049 6e74 complete. Int │ │ │ │ -00037ee0: 6572 7365 6374 696f 6e0a 0a46 756e 6374 ersection..Funct │ │ │ │ -00037ef0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ -00037f00: 616c 2061 7267 756d 656e 7420 6e61 6d65 al argument name │ │ │ │ -00037f10: 6420 4c69 6674 3a0a 3d3d 3d3d 3d3d 3d3d d Lift:.======== │ │ │ │ +00037c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00037c70: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +00037c80: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +00037c90: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +00037ca0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +00037cb0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +00037cc0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +00037cd0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00037ce0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00037cf0: 6c75 7469 6f6e 732e 6d32 3a34 3839 353a lutions.m2:4895: │ │ │ │ +00037d00: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00037d10: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00037d20: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +00037d30: 204e 6f64 653a 204c 6966 742c 204e 6578 Node: Lift, Nex │ │ │ │ +00037d40: 743a 206d 616b 6546 696e 6974 6552 6573 t: makeFiniteRes │ │ │ │ +00037d50: 6f6c 7574 696f 6e2c 2050 7265 763a 206c olution, Prev: l │ │ │ │ +00037d60: 6179 6572 6564 5265 736f 6c75 7469 6f6e ayeredResolution │ │ │ │ +00037d70: 2c20 5570 3a20 546f 700a 0a4c 6966 7420 , Up: Top..Lift │ │ │ │ +00037d80: 2d2d 204f 7074 696f 6e20 666f 7220 6e65 -- Option for ne │ │ │ │ +00037d90: 7745 7874 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a wExt.*********** │ │ │ │ +00037da0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00037db0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00037dc0: 2020 2020 6e65 7745 7874 284d 2c4e 2c43 newExt(M,N,C │ │ │ │ +00037dd0: 6865 636b 203d 3e74 7275 6529 0a20 202a heck =>true). * │ │ │ │ +00037de0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00037df0: 2043 6865 636b 2c20 6120 2a6e 6f74 6520 Check, a *note │ │ │ │ +00037e00: 426f 6f6c 6561 6e20 7661 6c75 653a 2028 Boolean value: ( │ │ │ │ +00037e10: 4d61 6361 756c 6179 3244 6f63 2942 6f6f Macaulay2Doc)Boo │ │ │ │ +00037e20: 6c65 616e 2c2c 200a 0a44 6573 6372 6970 lean,, ..Descrip │ │ │ │ +00037e30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +00037e40: 0a0a 4d61 6b65 7320 6e65 7745 7874 2070 ..Makes newExt p │ │ │ │ +00037e50: 6572 666f 726d 2076 6172 696f 7573 2063 erform various c │ │ │ │ +00037e60: 6865 636b 7320 6173 2069 7420 636f 6d70 hecks as it comp │ │ │ │ +00037e70: 7574 6573 2e0a 0a53 6565 2061 6c73 6f0a utes...See also. │ │ │ │ +00037e80: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00037e90: 6f74 6520 6e65 7745 7874 3a20 6e65 7745 ote newExt: newE │ │ │ │ +00037ea0: 7874 2c20 2d2d 2047 6c6f 6261 6c20 4578 xt, -- Global Ex │ │ │ │ +00037eb0: 7420 666f 7220 6d6f 6475 6c65 7320 6f76 t for modules ov │ │ │ │ +00037ec0: 6572 2061 2063 6f6d 706c 6574 650a 2020 er a complete. │ │ │ │ +00037ed0: 2020 496e 7465 7273 6563 7469 6f6e 0a0a Intersection.. │ │ │ │ +00037ee0: 4675 6e63 7469 6f6e 7320 7769 7468 206f Functions with o │ │ │ │ +00037ef0: 7074 696f 6e61 6c20 6172 6775 6d65 6e74 ptional argument │ │ │ │ +00037f00: 206e 616d 6564 204c 6966 743a 0a3d 3d3d named Lift:.=== │ │ │ │ +00037f10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00037f20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037f30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037f40: 3d3d 3d3d 0a0a 2020 2a20 226e 6577 4578 ====.. * "newEx │ │ │ │ -00037f50: 7428 2e2e 2e2c 4c69 6674 3d3e 2e2e 2e29 t(...,Lift=>...) │ │ │ │ -00037f60: 2220 2d2d 2073 6565 202a 6e6f 7465 206e " -- see *note n │ │ │ │ -00037f70: 6577 4578 743a 206e 6577 4578 742c 202d ewExt: newExt, - │ │ │ │ -00037f80: 2d20 476c 6f62 616c 2045 7874 2066 6f72 - Global Ext for │ │ │ │ -00037f90: 0a20 2020 206d 6f64 756c 6573 206f 7665 . modules ove │ │ │ │ -00037fa0: 7220 6120 636f 6d70 6c65 7465 2049 6e74 r a complete Int │ │ │ │ -00037fb0: 6572 7365 6374 696f 6e0a 0a46 6f72 2074 ersection..For t │ │ │ │ -00037fc0: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -00037fd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00037fe0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -00037ff0: 7465 204c 6966 743a 204c 6966 742c 2069 te Lift: Lift, i │ │ │ │ -00038000: 7320 6120 2a6e 6f74 6520 7379 6d62 6f6c s a *note symbol │ │ │ │ -00038010: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00038020: 5379 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d Symbol,...------ │ │ │ │ +00037f30: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00037f40: 6e65 7745 7874 282e 2e2e 2c4c 6966 743d newExt(...,Lift= │ │ │ │ +00037f50: 3e2e 2e2e 2922 202d 2d20 7365 6520 2a6e >...)" -- see *n │ │ │ │ +00037f60: 6f74 6520 6e65 7745 7874 3a20 6e65 7745 ote newExt: newE │ │ │ │ +00037f70: 7874 2c20 2d2d 2047 6c6f 6261 6c20 4578 xt, -- Global Ex │ │ │ │ +00037f80: 7420 666f 720a 2020 2020 6d6f 6475 6c65 t for. module │ │ │ │ +00037f90: 7320 6f76 6572 2061 2063 6f6d 706c 6574 s over a complet │ │ │ │ +00037fa0: 6520 496e 7465 7273 6563 7469 6f6e 0a0a e Intersection.. │ │ │ │ +00037fb0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +00037fc0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +00037fd0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +00037fe0: 7420 2a6e 6f74 6520 4c69 6674 3a20 4c69 t *note Lift: Li │ │ │ │ +00037ff0: 6674 2c20 6973 2061 202a 6e6f 7465 2073 ft, is a *note s │ │ │ │ +00038000: 796d 626f 6c3a 2028 4d61 6361 756c 6179 ymbol: (Macaulay │ │ │ │ +00038010: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ +00038020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038070: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00038080: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00038090: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -000380a0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -000380b0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -000380c0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -000380d0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -000380e0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -000380f0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -00038100: 732e 6d32 3a33 3139 383a 302e 0a1f 0a46 s.m2:3198:0....F │ │ │ │ -00038110: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -00038120: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00038130: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -00038140: 206d 616b 6546 696e 6974 6552 6573 6f6c makeFiniteResol │ │ │ │ -00038150: 7574 696f 6e2c 204e 6578 743a 206d 616b ution, Next: mak │ │ │ │ -00038160: 6546 696e 6974 6552 6573 6f6c 7574 696f eFiniteResolutio │ │ │ │ -00038170: 6e43 6f64 696d 322c 2050 7265 763a 204c nCodim2, Prev: L │ │ │ │ -00038180: 6966 742c 2055 703a 2054 6f70 0a0a 6d61 ift, Up: Top..ma │ │ │ │ -00038190: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ -000381a0: 6f6e 202d 2d20 6669 6e69 7465 2072 6573 on -- finite res │ │ │ │ -000381b0: 6f6c 7574 696f 6e20 6f66 2061 206d 6174 olution of a mat │ │ │ │ -000381c0: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -000381d0: 6e20 6d6f 6475 6c65 204d 0a2a 2a2a 2a2a n module M.***** │ │ │ │ +00038060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00038070: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00038080: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00038090: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +000380a0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +000380b0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +000380c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +000380d0: 6b61 6765 732f 0a43 6f6d 706c 6574 6549 kages/.CompleteI │ │ │ │ +000380e0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000380f0: 7574 696f 6e73 2e6d 323a 3331 3938 3a30 utions.m2:3198:0 │ │ │ │ +00038100: 2e0a 1f0a 4669 6c65 3a20 436f 6d70 6c65 ....File: Comple │ │ │ │ +00038110: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +00038120: 736f 6c75 7469 6f6e 732e 696e 666f 2c20 solutions.info, │ │ │ │ +00038130: 4e6f 6465 3a20 6d61 6b65 4669 6e69 7465 Node: makeFinite │ │ │ │ +00038140: 5265 736f 6c75 7469 6f6e 2c20 4e65 7874 Resolution, Next │ │ │ │ +00038150: 3a20 6d61 6b65 4669 6e69 7465 5265 736f : makeFiniteReso │ │ │ │ +00038160: 6c75 7469 6f6e 436f 6469 6d32 2c20 5072 lutionCodim2, Pr │ │ │ │ +00038170: 6576 3a20 4c69 6674 2c20 5570 3a20 546f ev: Lift, Up: To │ │ │ │ +00038180: 700a 0a6d 616b 6546 696e 6974 6552 6573 p..makeFiniteRes │ │ │ │ +00038190: 6f6c 7574 696f 6e20 2d2d 2066 696e 6974 olution -- finit │ │ │ │ +000381a0: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +000381b0: 6120 6d61 7472 6978 2066 6163 746f 7269 a matrix factori │ │ │ │ +000381c0: 7a61 7469 6f6e 206d 6f64 756c 6520 4d0a zation module M. │ │ │ │ +000381d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000381e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000381f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00038200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00038220: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -00038230: 6765 3a20 0a20 2020 2020 2020 2041 203d ge: . A = │ │ │ │ -00038240: 206d 616b 6546 696e 6974 6552 6573 6f6c makeFiniteResol │ │ │ │ -00038250: 7574 696f 6e28 6666 2c6d 6629 0a20 202a ution(ff,mf). * │ │ │ │ -00038260: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -00038270: 206d 662c 2061 202a 6e6f 7465 206c 6973 mf, a *note lis │ │ │ │ -00038280: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -00038290: 294c 6973 742c 2c20 6f75 7470 7574 206f )List,, output o │ │ │ │ -000382a0: 6620 6d61 7472 6978 4661 6374 6f72 697a f matrixFactoriz │ │ │ │ -000382b0: 6174 696f 6e0a 2020 2020 2020 2a20 6666 ation. * ff │ │ │ │ -000382c0: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ -000382d0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -000382e0: 4d61 7472 6978 2c2c 2074 6865 2072 6567 Matrix,, the reg │ │ │ │ -000382f0: 756c 6172 2073 6571 7565 6e63 6520 7573 ular sequence us │ │ │ │ -00038300: 6564 0a20 2020 2020 2020 2066 6f72 2074 ed. for t │ │ │ │ -00038310: 6865 206d 6174 7269 7846 6163 746f 7269 he matrixFactori │ │ │ │ -00038320: 7a61 7469 6f6e 2063 6f6d 7075 7461 7469 zation computati │ │ │ │ -00038330: 6f6e 0a20 202a 204f 7574 7075 7473 3a0a on. * Outputs:. │ │ │ │ -00038340: 2020 2020 2020 2a20 412c 2061 202a 6e6f * A, a *no │ │ │ │ -00038350: 7465 2063 6f6d 706c 6578 3a20 2843 6f6d te complex: (Com │ │ │ │ -00038360: 706c 6578 6573 2943 6f6d 706c 6578 2c2c plexes)Complex,, │ │ │ │ -00038370: 2041 2069 7320 7468 6520 6d69 6e69 6d61 A is the minima │ │ │ │ -00038380: 6c20 6669 6e69 7465 0a20 2020 2020 2020 l finite. │ │ │ │ -00038390: 2072 6573 6f6c 7574 696f 6e20 6f66 204d resolution of M │ │ │ │ -000383a0: 206f 7665 7220 522e 0a0a 4465 7363 7269 over R...Descri │ │ │ │ -000383b0: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -000383c0: 3d0a 0a53 7570 706f 7365 2074 6861 7420 =..Suppose that │ │ │ │ -000383d0: 665f 312e 2e66 5f63 2069 7320 6120 686f f_1..f_c is a ho │ │ │ │ -000383e0: 6d6f 6765 6e65 6f75 7320 7265 6775 6c61 mogeneous regula │ │ │ │ -000383f0: 7220 7365 7175 656e 6365 206f 6620 666f r sequence of fo │ │ │ │ -00038400: 726d 7320 6f66 2074 6865 2073 616d 650a rms of the same. │ │ │ │ -00038410: 6465 6772 6565 2069 6e20 6120 706f 6c79 degree in a poly │ │ │ │ -00038420: 6e6f 6d69 616c 2072 696e 6720 5320 616e nomial ring S an │ │ │ │ -00038430: 6420 4d20 6973 2061 2068 6967 6820 7379 d M is a high sy │ │ │ │ -00038440: 7a79 6779 206d 6f64 756c 6520 6f76 6572 zygy module over │ │ │ │ -00038450: 2053 2f28 665f 312c 2e2e 2c66 5f63 290a S/(f_1,..,f_c). │ │ │ │ -00038460: 3d20 5228 6329 2c20 616e 6420 6d66 203d = R(c), and mf = │ │ │ │ -00038470: 2028 642c 6829 2069 7320 7468 6520 6f75 (d,h) is the ou │ │ │ │ -00038480: 7470 7574 206f 6620 6d61 7472 6978 4661 tput of matrixFa │ │ │ │ -00038490: 6374 6f72 697a 6174 696f 6e28 4d2c 6666 ctorization(M,ff │ │ │ │ -000384a0: 292e 2049 6620 7468 650a 636f 6d70 6c65 ). If the.comple │ │ │ │ -000384b0: 7869 7479 206f 6620 4d20 6973 2063 272c xity of M is c', │ │ │ │ -000384c0: 2074 6865 6e20 4d20 6861 7320 6120 6669 then M has a fi │ │ │ │ -000384d0: 6e69 7465 2066 7265 6520 7265 736f 6c75 nite free resolu │ │ │ │ -000384e0: 7469 6f6e 206f 7665 7220 5220 3d0a 532f tion over R =.S/ │ │ │ │ -000384f0: 2866 5f31 2c2e 2e2c 665f 7b28 632d 6327 (f_1,..,f_{(c-c' │ │ │ │ -00038500: 297d 2920 2861 6e64 2c20 6d6f 7265 2067 )}) (and, more g │ │ │ │ -00038510: 656e 6572 616c 6c79 2c20 6861 7320 636f enerally, has co │ │ │ │ -00038520: 6d70 6c65 7869 7479 2063 2d64 206f 7665 mplexity c-d ove │ │ │ │ -00038530: 720a 532f 2866 5f31 2c2e 2e2c 665f 7b28 r.S/(f_1,..,f_{( │ │ │ │ -00038540: 632d 6429 7d29 2066 6f72 2064 3e3d 6327 c-d)}) for d>=c' │ │ │ │ -00038550: 292e 0a0a 5468 6520 636f 6d70 6c65 7820 )...The complex │ │ │ │ -00038560: 4120 6973 2074 6865 206d 696e 696d 616c A is the minimal │ │ │ │ -00038570: 2066 696e 6974 6520 6672 6565 2072 6573 finite free res │ │ │ │ -00038580: 6f6c 7574 696f 6e20 6f66 204d 206f 7665 olution of M ove │ │ │ │ -00038590: 7220 412c 2063 6f6e 7374 7275 6374 6564 r A, constructed │ │ │ │ -000385a0: 2061 730a 616e 2069 7465 7261 7465 6420 as.an iterated │ │ │ │ -000385b0: 4b6f 737a 756c 2065 7874 656e 7369 6f6e Koszul extension │ │ │ │ -000385c0: 2c20 6d61 6465 2066 726f 6d20 7468 6520 , made from the │ │ │ │ -000385d0: 6d61 7073 2069 6e20 624d 6170 7320 6d66 maps in bMaps mf │ │ │ │ -000385e0: 2061 6e64 2070 7369 4d61 7073 206d 662c and psiMaps mf, │ │ │ │ -000385f0: 2061 730a 6465 7363 7269 6265 6420 696e as.described in │ │ │ │ -00038600: 2045 6973 656e 6275 642d 5065 6576 612e Eisenbud-Peeva. │ │ │ │ -00038610: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +00038210: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +00038220: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +00038230: 2020 4120 3d20 6d61 6b65 4669 6e69 7465 A = makeFinite │ │ │ │ +00038240: 5265 736f 6c75 7469 6f6e 2866 662c 6d66 Resolution(ff,mf │ │ │ │ +00038250: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +00038260: 2020 2020 2a20 6d66 2c20 6120 2a6e 6f74 * mf, a *not │ │ │ │ +00038270: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +00038280: 7932 446f 6329 4c69 7374 2c2c 206f 7574 y2Doc)List,, out │ │ │ │ +00038290: 7075 7420 6f66 206d 6174 7269 7846 6163 put of matrixFac │ │ │ │ +000382a0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020 torization. │ │ │ │ +000382b0: 202a 2066 662c 2061 202a 6e6f 7465 206d * ff, a *note m │ │ │ │ +000382c0: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ +000382d0: 3244 6f63 294d 6174 7269 782c 2c20 7468 2Doc)Matrix,, th │ │ │ │ +000382e0: 6520 7265 6775 6c61 7220 7365 7175 656e e regular sequen │ │ │ │ +000382f0: 6365 2075 7365 640a 2020 2020 2020 2020 ce used. │ │ │ │ +00038300: 666f 7220 7468 6520 6d61 7472 6978 4661 for the matrixFa │ │ │ │ +00038310: 6374 6f72 697a 6174 696f 6e20 636f 6d70 ctorization comp │ │ │ │ +00038320: 7574 6174 696f 6e0a 2020 2a20 4f75 7470 utation. * Outp │ │ │ │ +00038330: 7574 733a 0a20 2020 2020 202a 2041 2c20 uts:. * A, │ │ │ │ +00038340: 6120 2a6e 6f74 6520 636f 6d70 6c65 783a a *note complex: │ │ │ │ +00038350: 2028 436f 6d70 6c65 7865 7329 436f 6d70 (Complexes)Comp │ │ │ │ +00038360: 6c65 782c 2c20 4120 6973 2074 6865 206d lex,, A is the m │ │ │ │ +00038370: 696e 696d 616c 2066 696e 6974 650a 2020 inimal finite. │ │ │ │ +00038380: 2020 2020 2020 7265 736f 6c75 7469 6f6e resolution │ │ │ │ +00038390: 206f 6620 4d20 6f76 6572 2052 2e0a 0a44 of M over R...D │ │ │ │ +000383a0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +000383b0: 3d3d 3d3d 3d3d 0a0a 5375 7070 6f73 6520 ======..Suppose │ │ │ │ +000383c0: 7468 6174 2066 5f31 2e2e 665f 6320 6973 that f_1..f_c is │ │ │ │ +000383d0: 2061 2068 6f6d 6f67 656e 656f 7573 2072 a homogeneous r │ │ │ │ +000383e0: 6567 756c 6172 2073 6571 7565 6e63 6520 egular sequence │ │ │ │ +000383f0: 6f66 2066 6f72 6d73 206f 6620 7468 6520 of forms of the │ │ │ │ +00038400: 7361 6d65 0a64 6567 7265 6520 696e 2061 same.degree in a │ │ │ │ +00038410: 2070 6f6c 796e 6f6d 6961 6c20 7269 6e67 polynomial ring │ │ │ │ +00038420: 2053 2061 6e64 204d 2069 7320 6120 6869 S and M is a hi │ │ │ │ +00038430: 6768 2073 797a 7967 7920 6d6f 6475 6c65 gh syzygy module │ │ │ │ +00038440: 206f 7665 7220 532f 2866 5f31 2c2e 2e2c over S/(f_1,.., │ │ │ │ +00038450: 665f 6329 0a3d 2052 2863 292c 2061 6e64 f_c).= R(c), and │ │ │ │ +00038460: 206d 6620 3d20 2864 2c68 2920 6973 2074 mf = (d,h) is t │ │ │ │ +00038470: 6865 206f 7574 7075 7420 6f66 206d 6174 he output of mat │ │ │ │ +00038480: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +00038490: 284d 2c66 6629 2e20 4966 2074 6865 0a63 (M,ff). If the.c │ │ │ │ +000384a0: 6f6d 706c 6578 6974 7920 6f66 204d 2069 omplexity of M i │ │ │ │ +000384b0: 7320 6327 2c20 7468 656e 204d 2068 6173 s c', then M has │ │ │ │ +000384c0: 2061 2066 696e 6974 6520 6672 6565 2072 a finite free r │ │ │ │ +000384d0: 6573 6f6c 7574 696f 6e20 6f76 6572 2052 esolution over R │ │ │ │ +000384e0: 203d 0a53 2f28 665f 312c 2e2e 2c66 5f7b =.S/(f_1,..,f_{ │ │ │ │ +000384f0: 2863 2d63 2729 7d29 2028 616e 642c 206d (c-c')}) (and, m │ │ │ │ +00038500: 6f72 6520 6765 6e65 7261 6c6c 792c 2068 ore generally, h │ │ │ │ +00038510: 6173 2063 6f6d 706c 6578 6974 7920 632d as complexity c- │ │ │ │ +00038520: 6420 6f76 6572 0a53 2f28 665f 312c 2e2e d over.S/(f_1,.. │ │ │ │ +00038530: 2c66 5f7b 2863 2d64 297d 2920 666f 7220 ,f_{(c-d)}) for │ │ │ │ +00038540: 643e 3d63 2729 2e0a 0a54 6865 2063 6f6d d>=c')...The com │ │ │ │ +00038550: 706c 6578 2041 2069 7320 7468 6520 6d69 plex A is the mi │ │ │ │ +00038560: 6e69 6d61 6c20 6669 6e69 7465 2066 7265 nimal finite fre │ │ │ │ +00038570: 6520 7265 736f 6c75 7469 6f6e 206f 6620 e resolution of │ │ │ │ +00038580: 4d20 6f76 6572 2041 2c20 636f 6e73 7472 M over A, constr │ │ │ │ +00038590: 7563 7465 6420 6173 0a61 6e20 6974 6572 ucted as.an iter │ │ │ │ +000385a0: 6174 6564 204b 6f73 7a75 6c20 6578 7465 ated Koszul exte │ │ │ │ +000385b0: 6e73 696f 6e2c 206d 6164 6520 6672 6f6d nsion, made from │ │ │ │ +000385c0: 2074 6865 206d 6170 7320 696e 2062 4d61 the maps in bMa │ │ │ │ +000385d0: 7073 206d 6620 616e 6420 7073 694d 6170 ps mf and psiMap │ │ │ │ +000385e0: 7320 6d66 2c20 6173 0a64 6573 6372 6962 s mf, as.describ │ │ │ │ +000385f0: 6564 2069 6e20 4569 7365 6e62 7564 2d50 ed in Eisenbud-P │ │ │ │ +00038600: 6565 7661 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d eeva...+-------- │ │ │ │ +00038610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038660: 2b0a 7c69 3120 3a20 7365 7452 616e 646f +.|i1 : setRando │ │ │ │ -00038670: 6d53 6565 6420 3020 2020 2020 2020 2020 mSeed 0 │ │ │ │ +00038650: 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 6574 -----+.|i1 : set │ │ │ │ +00038660: 5261 6e64 6f6d 5365 6564 2030 2020 2020 RandomSeed 0 │ │ │ │ +00038670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386b0: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072 |.| -- setting r │ │ │ │ -000386c0: 616e 646f 6d20 7365 6564 2074 6f20 3020 andom seed to 0 │ │ │ │ +000386a0: 2020 2020 207c 0a7c 202d 2d20 7365 7474 |.| -- sett │ │ │ │ +000386b0: 696e 6720 7261 6e64 6f6d 2073 6565 6420 ing random seed │ │ │ │ +000386c0: 746f 2030 2020 2020 2020 2020 2020 2020 to 0 │ │ │ │ 000386d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000386e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000386f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038700: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000386f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038750: 7c0a 7c6f 3120 3d20 3020 2020 2020 2020 |.|o1 = 0 │ │ │ │ +00038740: 2020 2020 207c 0a7c 6f31 203d 2030 2020 |.|o1 = 0 │ │ │ │ +00038750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000387a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038790: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000387a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000387b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000387c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000387d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000387e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000387f0: 2b0a 7c69 3220 3a20 5320 3d20 5a5a 2f31 +.|i2 : S = ZZ/1 │ │ │ │ -00038800: 3031 5b61 2c62 2c63 5d3b 2020 2020 2020 01[a,b,c]; │ │ │ │ +000387e0: 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 203d -----+.|i2 : S = │ │ │ │ +000387f0: 205a 5a2f 3130 315b 612c 622c 635d 3b20 ZZ/101[a,b,c]; │ │ │ │ +00038800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038840: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038830: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038890: 2b0a 7c69 3320 3a20 6666 203d 206d 6174 +.|i3 : ff = mat │ │ │ │ -000388a0: 7269 7822 6133 2c62 3322 3b20 2020 2020 rix"a3,b3"; │ │ │ │ +00038880: 2d2d 2d2d 2d2b 0a7c 6933 203a 2066 6620 -----+.|i3 : ff │ │ │ │ +00038890: 3d20 6d61 7472 6978 2261 332c 6233 223b = matrix"a3,b3"; │ │ │ │ +000388a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000388d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000388e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000388d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000388e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038930: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00038940: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +00038920: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038930: 2020 2020 2031 2020 2020 2020 3220 2020 1 2 │ │ │ │ +00038940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038980: 7c0a 7c6f 3320 3a20 4d61 7472 6978 2053 |.|o3 : Matrix S │ │ │ │ -00038990: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ +00038970: 2020 2020 207c 0a7c 6f33 203a 204d 6174 |.|o3 : Mat │ │ │ │ +00038980: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +00038990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000389a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000389b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000389c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000389d0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000389c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000389d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000389e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000389f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038a20: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964 +.|i4 : R = S/id │ │ │ │ -00038a30: 6561 6c20 6666 3b20 2020 2020 2020 2020 eal ff; │ │ │ │ +00038a10: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d -----+.|i4 : R = │ │ │ │ +00038a20: 2053 2f69 6465 616c 2066 663b 2020 2020 S/ideal ff; │ │ │ │ +00038a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038a70: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038a60: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038ac0: 2b0a 7c69 3520 3a20 4d20 3d20 6869 6768 +.|i5 : M = high │ │ │ │ -00038ad0: 5379 7a79 6779 2028 525e 312f 6964 6561 Syzygy (R^1/idea │ │ │ │ -00038ae0: 6c20 7661 7273 2052 293b 2020 2020 2020 l vars R); │ │ │ │ +00038ab0: 2d2d 2d2d 2d2b 0a7c 6935 203a 204d 203d -----+.|i5 : M = │ │ │ │ +00038ac0: 2068 6967 6853 797a 7967 7920 2852 5e31 highSyzygy (R^1 │ │ │ │ +00038ad0: 2f69 6465 616c 2076 6172 7320 5229 3b20 /ideal vars R); │ │ │ │ +00038ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038b10: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00038b00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00038b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038b20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038b60: 2b0a 7c69 3620 3a20 6d66 203d 206d 6174 +.|i6 : mf = mat │ │ │ │ -00038b70: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -00038b80: 2028 6666 2c20 4d29 2020 2020 2020 2020 (ff, M) │ │ │ │ +00038b50: 2d2d 2d2d 2d2b 0a7c 6936 203a 206d 6620 -----+.|i6 : mf │ │ │ │ +00038b60: 3d20 6d61 7472 6978 4661 6374 6f72 697a = matrixFactoriz │ │ │ │ +00038b70: 6174 696f 6e20 2866 662c 204d 2920 2020 ation (ff, M) │ │ │ │ +00038b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038bb0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038ba0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00038bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c00: 7c0a 7c6f 3620 3d20 7b7b 347d 207c 202d |.|o6 = {{4} | - │ │ │ │ -00038c10: 6320 6220 3020 2061 3220 3020 2030 2020 c b 0 a2 0 0 │ │ │ │ -00038c20: 3020 2030 2020 3020 2020 7c2c 207b 357d 0 0 0 |, {5} │ │ │ │ -00038c30: 207c 2030 2061 3220 3020 202d 6220 3020 | 0 a2 0 -b 0 │ │ │ │ -00038c40: 2030 2020 2030 2020 2030 2030 2020 3020 0 0 0 0 0 │ │ │ │ -00038c50: 7c0a 7c20 2020 2020 207b 347d 207c 2061 |.| {4} | a │ │ │ │ -00038c60: 2020 3020 6220 2030 2020 3020 2030 2020 0 b 0 0 0 │ │ │ │ -00038c70: 3020 2030 2020 3020 2020 7c20 207b 357d 0 0 0 | {5} │ │ │ │ -00038c80: 207c 2030 2030 2020 6132 202d 6320 6232 | 0 0 a2 -c b2 │ │ │ │ -00038c90: 2030 2020 2030 2020 2030 2030 2020 3020 0 0 0 0 0 │ │ │ │ -00038ca0: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038cb0: 2020 6120 6320 2030 2020 3020 2030 2020 a c 0 0 0 │ │ │ │ -00038cc0: 3020 2030 2020 2d62 3220 7c20 207b 357d 0 0 -b2 | {5} │ │ │ │ -00038cd0: 207c 2030 2030 2020 3020 2061 2020 3020 | 0 0 0 a 0 │ │ │ │ -00038ce0: 2062 3220 2030 2020 2030 2030 2020 3020 b2 0 0 0 0 │ │ │ │ -00038cf0: 7c0a 7c20 2020 2020 207b 337d 207c 2030 |.| {3} | 0 │ │ │ │ -00038d00: 2020 3020 6132 2030 2020 3020 2062 3220 0 a2 0 0 b2 │ │ │ │ -00038d10: 3020 2030 2020 3020 2020 7c20 207b 367d 0 0 0 | {6} │ │ │ │ -00038d20: 207c 2061 2063 2020 2d62 2030 2020 3020 | a c -b 0 0 │ │ │ │ -00038d30: 2030 2020 2030 2020 2030 2030 2020 3020 0 0 0 0 0 │ │ │ │ -00038d40: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038d50: 2020 3020 3020 2030 2020 6220 202d 6120 0 0 0 b -a │ │ │ │ -00038d60: 3020 2030 2020 3020 2020 7c20 207b 357d 0 0 0 | {5} │ │ │ │ -00038d70: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038d80: 2030 2020 2030 2020 2061 2062 3220 3020 0 0 a b2 0 │ │ │ │ -00038d90: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038da0: 2020 3020 3020 2030 2020 2d63 2030 2020 0 0 0 -c 0 │ │ │ │ -00038db0: 6120 2062 3220 3020 2020 7c20 207b 357d a b2 0 | {5} │ │ │ │ -00038dc0: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038dd0: 202d 6132 2030 2020 2062 2030 2020 3020 -a2 0 b 0 0 │ │ │ │ -00038de0: 7c0a 7c20 2020 2020 207b 347d 207c 2030 |.| {4} | 0 │ │ │ │ -00038df0: 2020 3020 3020 2030 2020 3020 2063 2020 0 0 0 0 c │ │ │ │ -00038e00: 2d62 2030 2020 6132 2020 7c20 207b 357d -b 0 a2 | {5} │ │ │ │ -00038e10: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038e20: 2030 2020 202d 6132 2063 2030 2020 3020 0 -a2 c 0 0 │ │ │ │ -00038e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038bf0: 2020 2020 207c 0a7c 6f36 203d 207b 7b34 |.|o6 = {{4 │ │ │ │ +00038c00: 7d20 7c20 2d63 2062 2030 2020 6132 2030 } | -c b 0 a2 0 │ │ │ │ +00038c10: 2020 3020 2030 2020 3020 2030 2020 207c 0 0 0 0 | │ │ │ │ +00038c20: 2c20 7b35 7d20 7c20 3020 6132 2030 2020 , {5} | 0 a2 0 │ │ │ │ +00038c30: 2d62 2030 2020 3020 2020 3020 2020 3020 -b 0 0 0 0 │ │ │ │ +00038c40: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038c50: 7d20 7c20 6120 2030 2062 2020 3020 2030 } | a 0 b 0 0 │ │ │ │ +00038c60: 2020 3020 2030 2020 3020 2030 2020 207c 0 0 0 0 | │ │ │ │ +00038c70: 2020 7b35 7d20 7c20 3020 3020 2061 3220 {5} | 0 0 a2 │ │ │ │ +00038c80: 2d63 2062 3220 3020 2020 3020 2020 3020 -c b2 0 0 0 │ │ │ │ +00038c90: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038ca0: 7d20 7c20 3020 2061 2063 2020 3020 2030 } | 0 a c 0 0 │ │ │ │ +00038cb0: 2020 3020 2030 2020 3020 202d 6232 207c 0 0 0 -b2 | │ │ │ │ +00038cc0: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038cd0: 6120 2030 2020 6232 2020 3020 2020 3020 a 0 b2 0 0 │ │ │ │ +00038ce0: 3020 2030 207c 0a7c 2020 2020 2020 7b33 0 0 |.| {3 │ │ │ │ +00038cf0: 7d20 7c20 3020 2030 2061 3220 3020 2030 } | 0 0 a2 0 0 │ │ │ │ +00038d00: 2020 6232 2030 2020 3020 2030 2020 207c b2 0 0 0 | │ │ │ │ +00038d10: 2020 7b36 7d20 7c20 6120 6320 202d 6220 {6} | a c -b │ │ │ │ +00038d20: 3020 2030 2020 3020 2020 3020 2020 3020 0 0 0 0 0 │ │ │ │ +00038d30: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038d40: 7d20 7c20 3020 2030 2030 2020 3020 2062 } | 0 0 0 0 b │ │ │ │ +00038d50: 2020 2d61 2030 2020 3020 2030 2020 207c -a 0 0 0 | │ │ │ │ +00038d60: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038d70: 3020 2030 2020 3020 2020 3020 2020 6120 0 0 0 0 a │ │ │ │ +00038d80: 6232 2030 207c 0a7c 2020 2020 2020 7b34 b2 0 |.| {4 │ │ │ │ +00038d90: 7d20 7c20 3020 2030 2030 2020 3020 202d } | 0 0 0 0 - │ │ │ │ +00038da0: 6320 3020 2061 2020 6232 2030 2020 207c c 0 a b2 0 | │ │ │ │ +00038db0: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038dc0: 3020 2030 2020 2d61 3220 3020 2020 6220 0 0 -a2 0 b │ │ │ │ +00038dd0: 3020 2030 207c 0a7c 2020 2020 2020 7b34 0 0 |.| {4 │ │ │ │ +00038de0: 7d20 7c20 3020 2030 2030 2020 3020 2030 } | 0 0 0 0 0 │ │ │ │ +00038df0: 2020 6320 202d 6220 3020 2061 3220 207c c -b 0 a2 | │ │ │ │ +00038e00: 2020 7b35 7d20 7c20 3020 3020 2030 2020 {5} | 0 0 0 │ │ │ │ +00038e10: 3020 2030 2020 3020 2020 2d61 3220 6320 0 0 0 -a2 c │ │ │ │ +00038e20: 3020 2030 207c 0a7c 2020 2020 2020 2020 0 0 |.| │ │ │ │ +00038e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038e50: 2020 2020 2020 2020 2020 2020 207b 367d {6} │ │ │ │ -00038e60: 207c 2030 2030 2020 3020 2030 2020 3020 | 0 0 0 0 0 │ │ │ │ -00038e70: 2030 2020 2030 2020 2030 2063 2020 6220 0 0 0 c b │ │ │ │ -00038e80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038e50: 2020 7b36 7d20 7c20 3020 3020 2030 2020 {6} | 0 0 0 │ │ │ │ +00038e60: 3020 2030 2020 3020 2020 3020 2020 3020 0 0 0 0 0 │ │ │ │ +00038e70: 6320 2062 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7468 656e 2074 maximal, then t │ │ │ │ -0003ac10: 6865 2066 696e 6974 6520 7265 736f 6c75 he finite resolu │ │ │ │ -0003ac20: 7469 6f6e 2074 616b 6573 2070 6c61 6365 tion takes place │ │ │ │ -0003ac30: 0a6f 7665 7220 616e 2069 6e74 6572 6d65 .over an interme │ │ │ │ -0003ac40: 6469 6174 6520 636f 6d70 6c65 7465 2069 diate complete i │ │ │ │ -0003ac50: 6e74 6572 7365 6374 696f 6e3a 0a0a 2b2d ntersection:..+- │ │ │ │ +0003abd0: 2d2d 2d2d 2d2b 0a0a 4966 2074 6865 2063 -----+..If the c │ │ │ │ +0003abe0: 6f6d 706c 6578 6974 7920 6f66 204d 2069 omplexity of M i │ │ │ │ +0003abf0: 7320 6e6f 7420 6d61 7869 6d61 6c2c 2074 s not maximal, t │ │ │ │ +0003ac00: 6865 6e20 7468 6520 6669 6e69 7465 2072 hen the finite r │ │ │ │ +0003ac10: 6573 6f6c 7574 696f 6e20 7461 6b65 7320 esolution takes │ │ │ │ +0003ac20: 706c 6163 650a 6f76 6572 2061 6e20 696e place.over an in │ │ │ │ +0003ac30: 7465 726d 6564 6961 7465 2063 6f6d 706c termediate compl │ │ │ │ +0003ac40: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ +0003ac50: 3a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d :..+------------ │ │ │ │ 0003ac60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ac70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ac80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003ac90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003aca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003acb0: 3133 203a 2053 203d 205a 5a2f 3130 315b 13 : S = ZZ/101[ │ │ │ │ -0003acc0: 612c 622c 632c 645d 2020 2020 2020 2020 a,b,c,d] │ │ │ │ +0003aca0: 2d2b 0a7c 6931 3320 3a20 5320 3d20 5a5a -+.|i13 : S = ZZ │ │ │ │ +0003acb0: 2f31 3031 5b61 2c62 2c63 2c64 5d20 2020 /101[a,b,c,d] │ │ │ │ +0003acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003acf0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003acf0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ad40: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003ad50: 3133 203d 2053 2020 2020 2020 2020 2020 13 = S │ │ │ │ +0003ad40: 207c 0a7c 6f31 3320 3d20 5320 2020 2020 |.|o13 = S │ │ │ │ +0003ad50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ad80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ad90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003ad90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003adc0: 2020 2020 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2020 2020 2020 2020 2020 4 │ │ │ │ +0003afc0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003afd0: 2020 3120 2020 2020 2034 2020 2020 2020 1 4 │ │ │ │ +0003afe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003aff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b010: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b020: 3134 203a 204d 6174 7269 7820 5320 203c 14 : Matrix S < │ │ │ │ -0003b030: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ +0003b010: 207c 0a7c 6f31 3420 3a20 4d61 7472 6978 |.|o14 : Matrix │ │ │ │ +0003b020: 2053 2020 3c2d 2d20 5320 2020 2020 2020 S <-- S │ │ │ │ +0003b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b060: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b060: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 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│ │ +0003b100: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b150: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b160: 3135 203d 207c 2032 3461 332d 3336 6233 15 = | 24a3-36b3 │ │ │ │ -0003b170: 2d33 3063 332d 3239 6433 2031 3961 332b -30c3-29d3 19a3+ │ │ │ │ -0003b180: 3139 6233 2d31 3063 332d 3239 6433 202d 19b3-10c3-29d3 - │ │ │ │ -0003b190: 3861 332d 3232 6233 2d32 3963 332d 3234 8a3-22b3-29c3-24 │ │ │ │ -0003b1a0: 6433 2020 2020 2020 2020 2020 7c0a 7c20 d3 |.| │ │ │ │ -0003b1b0: 2020 2020 202d 2d2d 2d2d 2d2d 2d2d 2d2d ----------- │ │ │ │ +0003b150: 207c 0a7c 6f31 3520 3d20 7c20 3234 6133 |.|o15 = | 24a3 │ │ │ │ +0003b160: 2d33 3662 332d 3330 6333 2d32 3964 3320 -36b3-30c3-29d3 │ │ │ │ +0003b170: 3139 6133 2b31 3962 332d 3130 6333 2d32 19a3+19b3-10c3-2 │ │ │ │ +0003b180: 3964 3320 2d38 6133 2d32 3262 332d 3239 9d3 -8a3-22b3-29 │ │ │ │ +0003b190: 6333 2d32 3464 3320 2020 2020 2020 2020 c3-24d3 │ │ │ │ +0003b1a0: 207c 0a7c 2020 2020 2020 2d2d 2d2d 2d2d |.| ------ │ │ │ │ +0003b1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b1f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ -0003b200: 2020 2020 202d 3338 6133 2d31 3662 332b -38a3-16b3+ │ │ │ │ -0003b210: 3339 6333 2b32 3164 3320 7c20 2020 2020 39c3+21d3 | │ │ │ │ +0003b1f0: 2d7c 0a7c 2020 2020 2020 2d33 3861 332d -|.| -38a3- │ │ │ │ +0003b200: 3136 6233 2b33 3963 332b 3231 6433 207c 16b3+39c3+21d3 | │ │ │ │ +0003b210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b240: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b240: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b290: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003b2a0: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0003b2b0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ +0003b290: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b2a0: 2020 3120 2020 2020 2034 2020 2020 2020 1 4 │ │ │ │ +0003b2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b2e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b2f0: 3135 203a 204d 6174 7269 7820 5320 203c 15 : Matrix S < │ │ │ │ -0003b300: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ +0003b2e0: 207c 0a7c 6f31 3520 3a20 4d61 7472 6978 |.|o15 : Matrix │ │ │ │ +0003b2f0: 2053 2020 3c2d 2d20 5320 2020 2020 2020 S <-- S │ │ │ │ +0003b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b330: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b330: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b390: 3136 203a 2052 203d 2053 2f69 6465 616c 16 : R = S/ideal │ │ │ │ -0003b3a0: 2066 6620 2020 2020 2020 2020 2020 2020 ff │ │ │ │ +0003b380: 2d2b 0a7c 6931 3620 3a20 5220 3d20 532f -+.|i16 : R = S/ │ │ │ │ +0003b390: 6964 6561 6c20 6666 2020 2020 2020 2020 ideal ff │ │ │ │ +0003b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b3d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b3d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b420: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b430: 3136 203d 2052 2020 2020 2020 2020 2020 16 = R │ │ │ │ +0003b420: 207c 0a7c 6f31 3620 3d20 5220 2020 2020 |.|o16 = R │ │ │ │ +0003b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b470: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b470: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b4c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b4d0: 3136 203a 2051 756f 7469 656e 7452 696e 16 : QuotientRin │ │ │ │ -0003b4e0: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ +0003b4c0: 207c 0a7c 6f31 3620 3a20 5175 6f74 6965 |.|o16 : Quotie │ │ │ │ +0003b4d0: 6e74 5269 6e67 2020 2020 2020 2020 2020 ntRing │ │ │ │ +0003b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b510: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b510: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b570: 3137 203a 204d 203d 2068 6967 6853 797a 17 : M = highSyz │ │ │ │ -0003b580: 7967 7920 2852 5e31 2f69 6465 616c 2261 ygy (R^1/ideal"a │ │ │ │ -0003b590: 3262 3222 2920 2020 2020 2020 2020 2020 2b2") │ │ │ │ +0003b560: 2d2b 0a7c 6931 3720 3a20 4d20 3d20 6869 -+.|i17 : M = hi │ │ │ │ +0003b570: 6768 5379 7a79 6779 2028 525e 312f 6964 ghSyzygy (R^1/id │ │ │ │ +0003b580: 6561 6c22 6132 6232 2229 2020 2020 2020 eal"a2b2") │ │ │ │ +0003b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b5b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b5b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b600: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b610: 3137 203d 2063 6f6b 6572 6e65 6c20 7b36 17 = cokernel {6 │ │ │ │ -0003b620: 7d20 7c20 6232 2030 202d 6132 2030 207c } | b2 0 -a2 0 | │ │ │ │ +0003b600: 207c 0a7c 6f31 3720 3d20 636f 6b65 726e |.|o17 = cokern │ │ │ │ +0003b610: 656c 207b 367d 207c 2062 3220 3020 2d61 el {6} | b2 0 -a │ │ │ │ +0003b620: 3220 3020 7c20 2020 2020 2020 2020 2020 2 0 | │ │ │ │ 0003b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b650: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003b660: 2020 2020 2020 2020 2020 2020 2020 7b37 {7 │ │ │ │ -0003b670: 7d20 7c20 6120 2062 2030 2020 2030 207c } | a b 0 0 | │ │ │ │ +0003b650: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b660: 2020 207b 377d 207c 2061 2020 6220 3020 {7} | a b 0 │ │ │ │ +0003b670: 2020 3020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ 0003b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b6a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003b6b0: 2020 2020 2020 2020 2020 2020 2020 7b37 {7 │ │ │ │ -0003b6c0: 7d20 7c20 3020 2030 2062 2020 2061 207c } | 0 0 b a | │ │ │ │ +0003b6a0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003b6b0: 2020 207b 377d 207c 2030 2020 3020 6220 {7} | 0 0 b │ │ │ │ +0003b6c0: 2020 6120 7c20 2020 2020 2020 2020 2020 a | │ │ │ │ 0003b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b6f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b6f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b740: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b750: 3137 203a 2052 2d6d 6f64 756c 652c 2071 17 : R-module, q │ │ │ │ -0003b760: 756f 7469 6520 2020 2020 2020 3320 2020 uotie 3 │ │ │ │ +0003b740: 207c 0a7c 6f31 3720 3a20 522d 6d6f 6475 |.|o17 : R-modu │ │ │ │ +0003b750: 6c65 2c20 7175 6f74 6965 2020 2020 2020 le, quotie │ │ │ │ +0003b760: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0003b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b790: 2020 2020 2020 2020 2020 2020 7c0a 7c6e |.|n │ │ │ │ -0003b7a0: 7420 6f66 2052 2020 2020 2020 2020 2020 t of R │ │ │ │ +0003b790: 207c 0a7c 6e74 206f 6620 5220 2020 2020 |.|nt of R │ │ │ │ +0003b7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b7e0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b7e0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b840: 3138 203a 2063 6f6d 706c 6578 6974 7920 18 : complexity │ │ │ │ -0003b850: 4d20 2020 2020 2020 2020 2020 2020 2020 M │ │ │ │ +0003b830: 2d2b 0a7c 6931 3820 3a20 636f 6d70 6c65 -+.|i18 : comple │ │ │ │ +0003b840: 7869 7479 204d 2020 2020 2020 2020 2020 xity M │ │ │ │ +0003b850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b880: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b880: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b8d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003b8e0: 3138 203d 2032 2020 2020 2020 2020 2020 18 = 2 │ │ │ │ +0003b8d0: 207c 0a7c 6f31 3820 3d20 3220 2020 2020 |.|o18 = 2 │ │ │ │ +0003b8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b920: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0003b920: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0003b930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003b960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003b970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003b980: 3139 203a 206d 6620 3d20 6d61 7472 6978 19 : mf = matrix │ │ │ │ -0003b990: 4661 6374 6f72 697a 6174 696f 6e20 2866 Factorization (f │ │ │ │ -0003b9a0: 662c 204d 2920 2020 2020 2020 2020 2020 f, M) │ │ │ │ +0003b970: 2d2b 0a7c 6931 3920 3a20 6d66 203d 206d -+.|i19 : mf = m │ │ │ │ +0003b980: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +0003b990: 6f6e 2028 6666 2c20 4d29 2020 2020 2020 on (ff, M) │ │ │ │ +0003b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003b9c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003b9c0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003b9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ba00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003ba10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003ba20: 3139 203d 207b 7b37 7d20 7c20 2d61 202d 19 = {{7} | -a - │ │ │ │ -0003ba30: 3336 6220 3020 6120 7c2c 207b 387d 207c 36b 0 a |, {8} | │ │ │ │ -0003ba40: 2033 3561 3220 2034 3862 2020 3020 2020 35a2 48b 0 │ │ │ │ -0003ba50: 2020 2d33 3362 2030 2020 2020 207c 2c20 -33b 0 |, │ │ │ │ -0003ba60: 7b36 7d20 7c20 3020 2020 3336 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+0003cfa0: 6d61 7472 6978 4661 6374 6f72 697a 6174 matrixFactorizat │ │ │ │ +0003cfb0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 636f ion. * *note co │ │ │ │ +0003cfc0: 6d70 6c65 7869 7479 3a20 636f 6d70 6c65 mplexity: comple │ │ │ │ +0003cfd0: 7869 7479 2c20 2d2d 2063 6f6d 706c 6578 xity, -- complex │ │ │ │ +0003cfe0: 6974 7920 6f66 2061 206d 6f64 756c 6520 ity of a module │ │ │ │ +0003cff0: 6f76 6572 2061 2063 6f6d 706c 6574 650a over a complete. │ │ │ │ +0003d000: 2020 2020 696e 7465 7273 6563 7469 6f6e intersection │ │ │ │ +0003d010: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ +0003d020: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d030: 6f6e 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d on:.============ │ │ │ │ 0003d040: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003d050: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003d060: 0a0a 2020 2a20 226d 616b 6546 696e 6974 .. * "makeFinit │ │ │ │ -0003d070: 6552 6573 6f6c 7574 696f 6e28 4d61 7472 eResolution(Matr │ │ │ │ -0003d080: 6978 2c4c 6973 7429 220a 0a46 6f72 2074 ix,List)"..For t │ │ │ │ -0003d090: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -0003d0a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0003d0b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -0003d0c0: 7465 206d 616b 6546 696e 6974 6552 6573 te makeFiniteRes │ │ │ │ -0003d0d0: 6f6c 7574 696f 6e3a 206d 616b 6546 696e olution: makeFin │ │ │ │ -0003d0e0: 6974 6552 6573 6f6c 7574 696f 6e2c 2069 iteResolution, i │ │ │ │ -0003d0f0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ -0003d100: 0a66 756e 6374 696f 6e3a 2028 4d61 6361 .function: (Maca │ │ │ │ -0003d110: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -0003d120: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +0003d050: 3d3d 3d3d 3d0a 0a20 202a 2022 6d61 6b65 =====.. * "make │ │ │ │ +0003d060: 4669 6e69 7465 5265 736f 6c75 7469 6f6e FiniteResolution │ │ │ │ +0003d070: 284d 6174 7269 782c 4c69 7374 2922 0a0a (Matrix,List)".. │ │ │ │ +0003d080: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +0003d090: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +0003d0a0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +0003d0b0: 7420 2a6e 6f74 6520 6d61 6b65 4669 6e69 t *note makeFini │ │ │ │ +0003d0c0: 7465 5265 736f 6c75 7469 6f6e 3a20 6d61 teResolution: ma │ │ │ │ +0003d0d0: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d0e0: 6f6e 2c20 6973 2061 202a 6e6f 7465 206d on, is a *note m │ │ │ │ +0003d0f0: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ +0003d100: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +0003d110: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0003d120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d170: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -0003d180: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -0003d190: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -0003d1a0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -0003d1b0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -0003d1c0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -0003d1d0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -0003d1e0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -0003d1f0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -0003d200: 6e73 2e6d 323a 3238 3939 3a30 2e0a 1f0a ns.m2:2899:0.... │ │ │ │ -0003d210: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -0003d220: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0003d230: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -0003d240: 3a20 6d61 6b65 4669 6e69 7465 5265 736f : makeFiniteReso │ │ │ │ -0003d250: 6c75 7469 6f6e 436f 6469 6d32 2c20 4e65 lutionCodim2, Ne │ │ │ │ -0003d260: 7874 3a20 6d61 6b65 486f 6d6f 746f 7069 xt: makeHomotopi │ │ │ │ -0003d270: 6573 2c20 5072 6576 3a20 6d61 6b65 4669 es, Prev: makeFi │ │ │ │ -0003d280: 6e69 7465 5265 736f 6c75 7469 6f6e 2c20 niteResolution, │ │ │ │ -0003d290: 5570 3a20 546f 700a 0a6d 616b 6546 696e Up: Top..makeFin │ │ │ │ -0003d2a0: 6974 6552 6573 6f6c 7574 696f 6e43 6f64 iteResolutionCod │ │ │ │ -0003d2b0: 696d 3220 2d2d 204d 6170 7320 6173 736f im2 -- Maps asso │ │ │ │ -0003d2c0: 6369 6174 6564 2074 6f20 7468 6520 6669 ciated to the fi │ │ │ │ -0003d2d0: 6e69 7465 2072 6573 6f6c 7574 696f 6e20 nite resolution │ │ │ │ -0003d2e0: 6f66 2061 2068 6967 6820 7379 7a79 6779 of a high syzygy │ │ │ │ -0003d2f0: 206d 6f64 756c 6520 696e 2063 6f64 696d module in codim │ │ │ │ -0003d300: 2032 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2.************* │ │ │ │ +0003d160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +0003d170: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +0003d180: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +0003d190: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +0003d1a0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +0003d1b0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +0003d1c0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +0003d1d0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +0003d1e0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +0003d1f0: 6c75 7469 6f6e 732e 6d32 3a32 3839 393a lutions.m2:2899: │ │ │ │ +0003d200: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +0003d210: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0003d220: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +0003d230: 204e 6f64 653a 206d 616b 6546 696e 6974 Node: makeFinit │ │ │ │ +0003d240: 6552 6573 6f6c 7574 696f 6e43 6f64 696d eResolutionCodim │ │ │ │ +0003d250: 322c 204e 6578 743a 206d 616b 6548 6f6d 2, Next: makeHom │ │ │ │ +0003d260: 6f74 6f70 6965 732c 2050 7265 763a 206d otopies, Prev: m │ │ │ │ +0003d270: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ +0003d280: 696f 6e2c 2055 703a 2054 6f70 0a0a 6d61 ion, Up: Top..ma │ │ │ │ +0003d290: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d2a0: 6f6e 436f 6469 6d32 202d 2d20 4d61 7073 onCodim2 -- Maps │ │ │ │ +0003d2b0: 2061 7373 6f63 6961 7465 6420 746f 2074 associated to t │ │ │ │ +0003d2c0: 6865 2066 696e 6974 6520 7265 736f 6c75 he finite resolu │ │ │ │ +0003d2d0: 7469 6f6e 206f 6620 6120 6869 6768 2073 tion of a high s │ │ │ │ +0003d2e0: 797a 7967 7920 6d6f 6475 6c65 2069 6e20 yzygy module in │ │ │ │ +0003d2f0: 636f 6469 6d20 320a 2a2a 2a2a 2a2a 2a2a codim 2.******** │ │ │ │ +0003d300: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d310: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d320: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d330: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d340: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0003d350: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0003d360: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -0003d370: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -0003d380: 2020 6d61 7073 203d 206d 616b 6546 696e maps = makeFin │ │ │ │ -0003d390: 6974 6552 6573 6f6c 7574 696f 6e43 6f64 iteResolutionCod │ │ │ │ -0003d3a0: 696d 3228 6666 2c6d 6629 0a20 202a 2049 im2(ff,mf). * I │ │ │ │ -0003d3b0: 6e70 7574 733a 0a20 2020 2020 202a 206d nputs:. * m │ │ │ │ -0003d3c0: 662c 2061 202a 6e6f 7465 206c 6973 743a f, a *note list: │ │ │ │ -0003d3d0: 2028 4d61 6361 756c 6179 3244 6f63 294c (Macaulay2Doc)L │ │ │ │ -0003d3e0: 6973 742c 2c20 6d61 7472 6978 2066 6163 ist,, matrix fac │ │ │ │ -0003d3f0: 746f 7269 7a61 7469 6f6e 0a20 2020 2020 torization. │ │ │ │ -0003d400: 202a 2066 662c 2061 202a 6e6f 7465 206d * ff, a *note m │ │ │ │ -0003d410: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ -0003d420: 3244 6f63 294d 6174 7269 782c 2c20 7265 2Doc)Matrix,, re │ │ │ │ -0003d430: 6775 6c61 7220 7365 7175 656e 6365 0a20 gular sequence. │ │ │ │ -0003d440: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ -0003d450: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ -0003d460: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ -0003d470: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ -0003d480: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ -0003d490: 2020 2020 202a 2043 6865 636b 203d 3e20 * Check => │ │ │ │ -0003d4a0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0003d4b0: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ -0003d4c0: 7075 7473 3a0a 2020 2020 2020 2a20 6d61 puts:. * ma │ │ │ │ -0003d4d0: 7073 2c20 6120 2a6e 6f74 6520 6861 7368 ps, a *note hash │ │ │ │ -0003d4e0: 2074 6162 6c65 3a20 284d 6163 6175 6c61 table: (Macaula │ │ │ │ -0003d4f0: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ -0003d500: 2c20 6d61 6e79 206d 6170 730a 0a44 6573 , many maps..Des │ │ │ │ -0003d510: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -0003d520: 3d3d 3d3d 0a0a 4769 7665 6e20 6120 636f ====..Given a co │ │ │ │ -0003d530: 6469 6d20 3220 6d61 7472 6978 2066 6163 dim 2 matrix fac │ │ │ │ -0003d540: 746f 7269 7a61 7469 6f6e 2c20 6d61 6b65 torization, make │ │ │ │ -0003d550: 7320 616c 6c20 7468 6520 636f 6d70 6f6e s all the compon │ │ │ │ -0003d560: 656e 7473 206f 6620 7468 650a 6469 6666 ents of the.diff │ │ │ │ -0003d570: 6572 656e 7469 616c 2061 6e64 206f 6620 erential and of │ │ │ │ -0003d580: 7468 6520 686f 6d6f 746f 7069 6573 2074 the homotopies t │ │ │ │ -0003d590: 6861 7420 6172 6520 7265 6c65 7661 6e74 hat are relevant │ │ │ │ -0003d5a0: 2074 6f20 7468 6520 6669 6e69 7465 2072 to the finite r │ │ │ │ -0003d5b0: 6573 6f6c 7574 696f 6e2c 0a61 7320 696e esolution,.as in │ │ │ │ -0003d5c0: 2034 2e32 2e33 206f 6620 4569 7365 6e62 4.2.3 of Eisenb │ │ │ │ -0003d5d0: 7564 2d50 6565 7661 2022 4d69 6e69 6d61 ud-Peeva "Minima │ │ │ │ -0003d5e0: 6c20 4672 6565 2052 6573 6f6c 7574 696f l Free Resolutio │ │ │ │ -0003d5f0: 6e73 2061 6e64 2048 6967 6865 7220 4d61 ns and Higher Ma │ │ │ │ -0003d600: 7472 6978 0a46 6163 746f 7269 7a61 7469 trix.Factorizati │ │ │ │ -0003d610: 6f6e 7322 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ons"..+--------- │ │ │ │ +0003d360: 2a0a 0a20 202a 2055 7361 6765 3a20 0a20 *.. * Usage: . │ │ │ │ +0003d370: 2020 2020 2020 206d 6170 7320 3d20 6d61 maps = ma │ │ │ │ +0003d380: 6b65 4669 6e69 7465 5265 736f 6c75 7469 keFiniteResoluti │ │ │ │ +0003d390: 6f6e 436f 6469 6d32 2866 662c 6d66 290a onCodim2(ff,mf). │ │ │ │ +0003d3a0: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +0003d3b0: 2020 2a20 6d66 2c20 6120 2a6e 6f74 6520 * mf, a *note │ │ │ │ +0003d3c0: 6c69 7374 3a20 284d 6163 6175 6c61 7932 list: (Macaulay2 │ │ │ │ +0003d3d0: 446f 6329 4c69 7374 2c2c 206d 6174 7269 Doc)List,, matri │ │ │ │ +0003d3e0: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ +0003d3f0: 2020 2020 2020 2a20 6666 2c20 6120 2a6e * ff, a *n │ │ │ │ +0003d400: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ +0003d410: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ +0003d420: 2c2c 2072 6567 756c 6172 2073 6571 7565 ,, regular seque │ │ │ │ +0003d430: 6e63 650a 2020 2a20 2a6e 6f74 6520 4f70 nce. * *note Op │ │ │ │ +0003d440: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ +0003d450: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ +0003d460: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ +0003d470: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ +0003d480: 732c 3a0a 2020 2020 2020 2a20 4368 6563 s,:. * Chec │ │ │ │ +0003d490: 6b20 3d3e 202e 2e2e 2c20 6465 6661 756c k => ..., defaul │ │ │ │ +0003d4a0: 7420 7661 6c75 6520 6661 6c73 650a 2020 t value false. │ │ │ │ +0003d4b0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0003d4c0: 202a 206d 6170 732c 2061 202a 6e6f 7465 * maps, a *note │ │ │ │ +0003d4d0: 2068 6173 6820 7461 626c 653a 2028 4d61 hash table: (Ma │ │ │ │ +0003d4e0: 6361 756c 6179 3244 6f63 2948 6173 6854 caulay2Doc)HashT │ │ │ │ +0003d4f0: 6162 6c65 2c2c 206d 616e 7920 6d61 7073 able,, many maps │ │ │ │ +0003d500: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0003d510: 3d3d 3d3d 3d3d 3d3d 3d0a 0a47 6976 656e =========..Given │ │ │ │ +0003d520: 2061 2063 6f64 696d 2032 206d 6174 7269 a codim 2 matri │ │ │ │ +0003d530: 7820 6661 6374 6f72 697a 6174 696f 6e2c x factorization, │ │ │ │ +0003d540: 206d 616b 6573 2061 6c6c 2074 6865 2063 makes all the c │ │ │ │ +0003d550: 6f6d 706f 6e65 6e74 7320 6f66 2074 6865 omponents of the │ │ │ │ +0003d560: 0a64 6966 6665 7265 6e74 6961 6c20 616e .differential an │ │ │ │ +0003d570: 6420 6f66 2074 6865 2068 6f6d 6f74 6f70 d of the homotop │ │ │ │ +0003d580: 6965 7320 7468 6174 2061 7265 2072 656c ies that are rel │ │ │ │ +0003d590: 6576 616e 7420 746f 2074 6865 2066 696e evant to the fin │ │ │ │ +0003d5a0: 6974 6520 7265 736f 6c75 7469 6f6e 2c0a ite resolution,. │ │ │ │ +0003d5b0: 6173 2069 6e20 342e 322e 3320 6f66 2045 as in 4.2.3 of E │ │ │ │ +0003d5c0: 6973 656e 6275 642d 5065 6576 6120 224d isenbud-Peeva "M │ │ │ │ +0003d5d0: 696e 696d 616c 2046 7265 6520 5265 736f inimal Free Reso │ │ │ │ +0003d5e0: 6c75 7469 6f6e 7320 616e 6420 4869 6768 lutions and High │ │ │ │ +0003d5f0: 6572 204d 6174 7269 780a 4661 6374 6f72 er Matrix.Factor │ │ │ │ +0003d600: 697a 6174 696f 6e73 220a 0a2b 2d2d 2d2d izations"..+---- │ │ │ │ +0003d610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d650: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -0003d660: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020 k=ZZ/101 │ │ │ │ +0003d640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0003d650: 3120 3a20 6b6b 3d5a 5a2f 3130 3120 2020 1 : kk=ZZ/101 │ │ │ │ +0003d660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d690: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003d680: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003d690: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003d6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d6d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003d6e0: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +0003d6d0: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +0003d6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d720: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d710: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003d720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d760: 2020 207c 0a7c 6f31 203a 2051 756f 7469 |.|o1 : Quoti │ │ │ │ -0003d770: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0003d750: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +0003d760: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0003d770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d7a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0003d790: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0003d7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d7e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0003d7f0: 2053 203d 206b 6b5b 612c 625d 2020 2020 S = kk[a,b] │ │ │ │ +0003d7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0003d7e0: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2c62 |i2 : S = kk[a,b │ │ │ │ +0003d7f0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ 0003d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003d820: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003d870: 0a7c 6f32 203d 2053 2020 2020 2020 2020 .|o2 = S │ │ │ │ +0003d860: 2020 2020 7c0a 7c6f 3220 3d20 5320 2020 |.|o2 = S │ │ │ │ +0003d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8b0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d8a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003d8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d8f0: 2020 2020 207c 0a7c 6f32 203a 2050 6f6c |.|o2 : Pol │ │ │ │ -0003d900: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0003d8e0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0003d8f0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0003d900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d930: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0003d920: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003d930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003d960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003d970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ -0003d980: 203a 2066 6620 3d20 6d61 7472 6978 2261 : ff = matrix"a │ │ │ │ -0003d990: 342c 6234 2220 2020 2020 2020 2020 2020 4,b4" │ │ │ │ +0003d970: 2b0a 7c69 3320 3a20 6666 203d 206d 6174 +.|i3 : ff = mat │ │ │ │ +0003d980: 7269 7822 6134 2c62 3422 2020 2020 2020 rix"a4,b4" │ │ │ │ +0003d990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d9b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003d9c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003d9b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003d9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003d9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003d9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da00: 207c 0a7c 6f33 203d 207c 2061 3420 6234 |.|o3 = | a4 b4 │ │ │ │ -0003da10: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003d9f0: 2020 2020 2020 7c0a 7c6f 3320 3d20 7c20 |.|o3 = | │ │ │ │ +0003da00: 6134 2062 3420 7c20 2020 2020 2020 2020 a4 b4 | │ │ │ │ +0003da10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003da20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0003da30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003da40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003da50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003da60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003da80: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -0003da90: 2020 2020 2020 2031 2020 2020 2020 3220 1 2 │ │ │ │ +0003da70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0003da80: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +0003da90: 2020 2032 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0003daa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dac0: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ -0003dad0: 3a20 4d61 7472 6978 2053 2020 3c2d 2d20 : Matrix S <-- │ │ │ │ -0003dae0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +0003dab0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003dac0: 0a7c 6f33 203a 204d 6174 7269 7820 5320 .|o3 : Matrix S │ │ │ │ +0003dad0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ +0003dae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003daf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db00: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0003db00: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0003db10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003db20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003db30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003db40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003db50: 2b0a 7c69 3420 3a20 5220 3d20 532f 6964 +.|i4 : R = S/id │ │ │ │ -0003db60: 6561 6c20 6666 2020 2020 2020 2020 2020 eal ff │ │ │ │ +0003db40: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 203d -----+.|i4 : R = │ │ │ │ +0003db50: 2053 2f69 6465 616c 2066 6620 2020 2020 S/ideal ff │ │ │ │ +0003db60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003db90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003db80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0003db90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dbd0: 2020 2020 2020 7c0a 7c6f 3420 3d20 5220 |.|o4 = R │ │ │ │ +0003dbc0: 2020 2020 2020 2020 2020 207c 0a7c 6f34 |.|o4 │ │ │ │ +0003dbd0: 203d 2052 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0003dbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003dc00: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003dc10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0003dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003dc60: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +0003dc50: 207c 0a7c 6f34 203a 2051 756f 7469 656e |.|o4 : Quotien │ │ │ │ +0003dc60: 7452 696e 6720 2020 2020 2020 2020 2020 tRing │ │ │ │ 0003dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dc90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003dca0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003dc90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003dca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003dcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003dcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dce0: 2d2d 2b0a 7c69 3520 3a20 4e20 3d20 525e --+.|i5 : N = R^ │ │ │ │ -0003dcf0: 312f 6964 6561 6c22 6132 2c20 6162 2c20 1/ideal"a2, ab, │ │ │ │ -0003dd00: 6233 2220 2020 2020 2020 2020 2020 2020 b3" │ │ │ │ -0003dd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd20: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003dcd0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 204e -------+.|i5 : N │ │ │ │ +0003dce0: 203d 2052 5e31 2f69 6465 616c 2261 322c = R^1/ideal"a2, │ │ │ │ +0003dcf0: 2061 622c 2062 3322 2020 2020 2020 2020 ab, b3" │ │ │ │ +0003dd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dd10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003dd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dd60: 2020 2020 2020 2020 7c0a 7c6f 3520 3d20 |.|o5 = │ │ │ │ -0003dd70: 636f 6b65 726e 656c 207c 2061 3220 6162 cokernel | a2 ab │ │ │ │ -0003dd80: 2062 3320 7c20 2020 2020 2020 2020 2020 b3 | │ │ │ │ +0003dd50: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003dd60: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ +0003dd70: 6132 2061 6220 6233 207c 2020 2020 2020 a2 ab b3 | │ │ │ │ +0003dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dda0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0003dda0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ddc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003ddd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dde0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003ddf0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003de00: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +0003dde0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003ddf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003de00: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0003de10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de30: 207c 0a7c 6f35 203a 2052 2d6d 6f64 756c |.|o5 : R-modul │ │ │ │ -0003de40: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ +0003de20: 2020 2020 2020 7c0a 7c6f 3520 3a20 522d |.|o5 : R- │ │ │ │ +0003de30: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0003de40: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ 0003de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003de70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003de60: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003de70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003de80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003de90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003deb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204e -------+.|i6 : N │ │ │ │ -0003dec0: 203d 2063 6f6b 6572 2076 6172 7320 5220 = coker vars R │ │ │ │ +0003dea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0003deb0: 3620 3a20 4e20 3d20 636f 6b65 7220 7661 6 : N = coker va │ │ │ │ +0003dec0: 7273 2052 2020 2020 2020 2020 2020 2020 rs R │ │ │ │ 0003ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003def0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003dee0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003def0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003df00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003df40: 6f36 203d 2063 6f6b 6572 6e65 6c20 7c20 o6 = cokernel | │ │ │ │ -0003df50: 6120 6220 7c20 2020 2020 2020 2020 2020 a b | │ │ │ │ +0003df30: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ +0003df40: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ +0003df50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003df80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003df70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003dfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003dfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003dfe0: 2020 3120 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -0003dff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e000: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ -0003e010: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ -0003e020: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ -0003e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003dfb0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0003dfc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dfd0: 2020 2020 2020 2031 2020 2020 2020 2020 1 │ │ │ │ +0003dfe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003dff0: 2020 2020 2020 2020 2020 207c 0a7c 6f36 |.|o6 │ │ │ │ +0003e000: 203a 2052 2d6d 6f64 756c 652c 2071 756f : R-module, quo │ │ │ │ +0003e010: 7469 656e 7420 6f66 2052 2020 2020 2020 tient of R │ │ │ │ +0003e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003e040: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0003e050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0003e090: 3720 3a20 4d20 3d20 6869 6768 5379 7a79 7 : M = highSyzy │ │ │ │ -0003e0a0: 6779 204e 2020 2020 2020 2020 2020 2020 gy N │ │ │ │ +0003e080: 2d2b 0a7c 6937 203a 204d 203d 2068 6967 -+.|i7 : M = hig │ │ │ │ +0003e090: 6853 797a 7967 7920 4e20 2020 2020 2020 hSyzygy N │ │ │ │ +0003e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e0c0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e0d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0003e0c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0003e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e110: 2020 7c0a 7c6f 3720 3d20 636f 6b65 726e |.|o7 = cokern │ │ │ │ -0003e120: 656c 207b 327d 207c 2030 202d 6233 2061 el {2} | 0 -b3 a │ │ │ │ -0003e130: 3320 3020 7c20 2020 2020 2020 2020 2020 3 0 | │ │ │ │ -0003e140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e150: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0003e160: 2020 2020 2020 7b34 7d20 7c20 6220 6120 {4} | b a │ │ │ │ -0003e170: 2020 3020 2030 207c 2020 2020 2020 2020 0 0 | │ │ │ │ -0003e180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e190: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0003e1a0: 2020 2020 2020 2020 207b 347d 207c 2030 {4} | 0 │ │ │ │ -0003e1b0: 2030 2020 2062 2020 6120 7c20 2020 2020 0 b a | │ │ │ │ +0003e100: 2020 2020 2020 207c 0a7c 6f37 203d 2063 |.|o7 = c │ │ │ │ +0003e110: 6f6b 6572 6e65 6c20 7b32 7d20 7c20 3020 okernel {2} | 0 │ │ │ │ +0003e120: 2d62 3320 6133 2030 207c 2020 2020 2020 -b3 a3 0 | │ │ │ │ +0003e130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e140: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e150: 2020 2020 2020 2020 2020 207b 347d 207c {4} | │ │ │ │ +0003e160: 2062 2061 2020 2030 2020 3020 7c20 2020 b a 0 0 | │ │ │ │ +0003e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e180: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003e190: 2020 2020 2020 2020 2020 2020 2020 7b34 {4 │ │ │ │ +0003e1a0: 7d20 7c20 3020 3020 2020 6220 2061 207c } | 0 0 b a | │ │ │ │ +0003e1b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e1c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e1d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0003e1d0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0003e1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e210: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0003e220: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003e230: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ +0003e210: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e230: 2020 3320 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 0003e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e260: 207c 0a7c 6f37 203a 2052 2d6d 6f64 756c |.|o7 : R-modul │ │ │ │ -0003e270: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ +0003e250: 2020 2020 2020 7c0a 7c6f 3720 3a20 522d |.|o7 : R- │ │ │ │ +0003e260: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0003e270: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ 0003e280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e2a0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003e290: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0003e2a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e2b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e2c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e2e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 204d -------+.|i8 : M │ │ │ │ -0003e2f0: 5320 3d20 7075 7368 466f 7277 6172 6428 S = pushForward( │ │ │ │ -0003e300: 6d61 7028 522c 5329 2c4d 2920 2020 2020 map(R,S),M) │ │ │ │ -0003e310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e320: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0003e2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0003e2e0: 3820 3a20 4d53 203d 2070 7573 6846 6f72 8 : MS = pushFor │ │ │ │ +0003e2f0: 7761 7264 286d 6170 2852 2c53 292c 4d29 ward(map(R,S),M) │ │ │ │ +0003e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e320: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0003e330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e360: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0003e370: 6f38 203d 2063 6f6b 6572 6e65 6c20 7b32 o8 = cokernel {2 │ │ │ │ -0003e380: 7d20 7c20 3020 6233 2061 3320 3020 3020 } | 0 b3 a3 0 0 │ │ │ │ -0003e390: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e3b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e3c0: 207b 347d 207c 2062 202d 6120 3020 2030 {4} | b -a 0 0 │ │ │ │ -0003e3d0: 2030 2020 7c20 2020 2020 2020 2020 2020 0 | │ │ │ │ -0003e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e3f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e400: 2020 2020 7b34 7d20 7c20 3020 3020 2062 {4} | 0 0 b │ │ │ │ -0003e410: 2020 6120 6234 207c 2020 2020 2020 2020 a b4 | │ │ │ │ -0003e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e430: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0003e360: 2020 7c0a 7c6f 3820 3d20 636f 6b65 726e |.|o8 = cokern │ │ │ │ +0003e370: 656c 207b 327d 207c 2030 2062 3320 6133 el {2} | 0 b3 a3 │ │ │ │ +0003e380: 2030 2030 2020 7c20 2020 2020 2020 2020 0 0 | │ │ │ │ +0003e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e3a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003e3b0: 2020 2020 2020 7b34 7d20 7c20 6220 2d61 {4} | b -a │ │ │ │ +0003e3c0: 2030 2020 3020 3020 207c 2020 2020 2020 0 0 0 | │ │ │ │ +0003e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e3e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0003e3f0: 2020 2020 2020 2020 207b 347d 207c 2030 {4} | 0 │ │ │ │ +0003e400: 2030 2020 6220 2061 2062 3420 7c20 2020 0 b a b4 | │ │ │ │ +0003e410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e420: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0003e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e470: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0003e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e490: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ +0003e460: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003e470: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003e480: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ +0003e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e4b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0003e4c0: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175 8 : S-module, qu │ │ │ │ -0003e4d0: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +0003e4b0: 207c 0a7c 6f38 203a 2053 2d6d 6f64 756c |.|o8 : S-modul │ │ │ │ +0003e4c0: 652c 2071 756f 7469 656e 7420 6f66 2053 e, quotient of S │ │ │ │ +0003e4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e4f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e500: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0003e4f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0003e500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e540: 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2020 3020 3020 2020 2d61 5} | 0 0 0 -a │ │ │ │ +0003e670: 2062 3320 7c20 207b 347d 207c 2031 2030 b3 | {4} | 1 0 │ │ │ │ +0003e680: 2020 3020 7c20 7c0a 7c20 2020 2020 2020 0 | |.| │ │ │ │ +0003e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e6a0: 2020 7b35 7d20 7c20 3020 2030 2061 3320 {5} | 0 0 a3 │ │ │ │ +0003e6b0: 2062 2020 3020 207c 2020 2020 2020 2020 b 0 | │ │ │ │ +0003e6c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0003e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e710: 2020 2020 2020 207c 0a7c 6f39 203a 204c |.|o9 : L │ │ │ │ -0003e720: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ +0003e700: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0003e710: 3920 3a20 4c69 7374 2020 2020 2020 2020 9 : List │ │ │ │ +0003e720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e750: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0003e740: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0003e750: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0003e760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0003e780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0003e790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -0003e7a0: 6931 3020 3a20 4720 3d20 6d61 6b65 4669 i10 : G = makeFi │ │ │ │ -0003e7b0: 6e69 7465 5265 736f 6c75 7469 6f6e 436f niteResolutionCo │ │ │ │ -0003e7c0: 6469 6d32 2866 662c 6d66 2920 2020 2020 dim2(ff,mf) │ │ │ │ -0003e7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e7e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e790: 2d2d 2b0a 7c69 3130 203a 2047 203d 206d --+.|i10 : G = m │ │ │ │ +0003e7a0: 616b 6546 696e 6974 6552 6573 6f6c 7574 akeFiniteResolut │ │ │ │ +0003e7b0: 696f 6e43 6f64 696d 3228 6666 2c6d 6629 ionCodim2(ff,mf) │ │ │ │ +0003e7c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e7d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0003e7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0003e800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e820: 2020 207c 0a7c 6f31 3020 3d20 4861 7368 |.|o10 = Hash │ │ │ │ -0003e830: 5461 626c 657b 2261 6c70 6861 2220 3d3e Table{"alpha" => │ │ │ │ -0003e840: 207b 357d 207c 2030 2020 2030 207c 2020 {5} | 0 0 | │ │ │ │ -0003e850: 2020 2020 2020 2020 2020 2020 7d20 2020 } │ │ │ │ -0003e860: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0003e870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e880: 2020 2020 7b35 7d20 7c20 2d62 3320 3020 {5} | -b3 0 │ │ │ │ -0003e890: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0003e8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0003e8b0: 2020 2020 2020 2020 2020 2020 2262 2220 "b" │ │ │ │ -0003e8c0: 3d3e 207b 347d 207c 2062 2061 207c 2020 => {4} | b a | │ │ │ │ +0003e810: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ +0003e820: 2048 6173 6854 6162 6c65 7b22 616c 7068 HashTable{"alph │ │ │ │ +0003e830: 6122 203d 3e20 7b35 7d20 7c20 3020 2020 a" => {5} | 0 │ │ │ │ +0003e840: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ +0003e850: 207d 2020 2020 2020 2020 207c 0a7c 2020 } |.| │ │ │ │ +0003e860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e870: 2020 2020 2020 2020 207b 357d 207c 202d {5} | - │ │ │ │ +0003e880: 6233 2030 207c 2020 2020 2020 2020 2020 b3 0 | │ │ │ │ +0003e890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0003e8a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003e8b0: 2022 6222 203d 3e20 7b34 7d20 7c20 6220 "b" => {4} | b │ │ │ │ +0003e8c0: 6120 7c20 2020 2020 2020 2020 2020 2020 a | │ │ │ │ 0003e8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e8e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0003e8f0: 2020 2020 2020 2020 2020 2020 2020 2022 " │ │ │ │ -0003e900: 6831 2722 203d 3e20 7b35 7d20 7c20 3020 h1'" => {5} | 0 │ │ │ │ -0003e910: 2020 3020 2030 2020 7c20 2020 2020 2020 0 0 | │ │ │ │ -0003e920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0003e930: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0003e940: 2020 2020 2020 2020 2020 207b 357d 207c {5} | │ │ │ │ -0003e950: 202d 6233 2030 2020 3020 207c 2020 2020 -b3 0 0 | │ │ │ │ -0003e960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e970: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -0003e980: 2020 2020 2020 2020 2020 2020 2020 7b35 {5 │ │ │ │ -0003e990: 7d20 7c20 3020 2020 2d61 2062 3320 7c20 } | 0 -a b3 | │ │ │ │ -0003e9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0003e9b0: 2020 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3e20 7b34 7d20 7c20 rtial" => {4} | │ │ │ │ -0003ebb0: 6120 2d62 207c 2020 2020 2020 2020 2020 a -b | │ │ │ │ -0003ebc0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0003e8e0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0003e8f0: 2020 2020 2268 3127 2220 3d3e 207b 357d "h1'" => {5} │ │ │ │ +0003e900: 207c 2030 2020 2030 2020 3020 207c 2020 | 0 0 0 | │ │ │ │ +0003e910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e920: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0003e930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e940: 7b35 7d20 7c20 2d62 3320 3020 2030 2020 {5} | -b3 0 0 │ │ │ │ +0003e950: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0003e960: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0003e970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0003e980: 2020 207b 357d 207c 2030 2020 202d 6120 {5} | 0 -a │ │ │ │ +0003e990: 6233 207c 2020 2020 2020 2020 2020 2020 b3 | │ │ │ │ +0003e9a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 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6f6d 706c 0....File: Compl │ │ │ │ +0003f590: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0003f5a0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +0003f5b0: 204e 6f64 653a 206d 616b 6548 6f6d 6f74 Node: makeHomot │ │ │ │ +0003f5c0: 6f70 6965 732c 204e 6578 743a 206d 616b opies, Next: mak │ │ │ │ +0003f5d0: 6548 6f6d 6f74 6f70 6965 7331 2c20 5072 eHomotopies1, Pr │ │ │ │ +0003f5e0: 6576 3a20 6d61 6b65 4669 6e69 7465 5265 ev: makeFiniteRe │ │ │ │ +0003f5f0: 736f 6c75 7469 6f6e 436f 6469 6d32 2c20 solutionCodim2, │ │ │ │ +0003f600: 5570 3a20 546f 700a 0a6d 616b 6548 6f6d Up: Top..makeHom │ │ │ │ +0003f610: 6f74 6f70 6965 7320 2d2d 2072 6574 7572 otopies -- retur │ │ │ │ +0003f620: 6e73 2061 2073 7973 7465 6d20 6f66 2068 ns a system of h │ │ │ │ +0003f630: 6967 6865 7220 686f 6d6f 746f 7069 6573 igher homotopies │ │ │ │ +0003f640: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0003f650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 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-0003f920: 7920 6f66 2068 6967 6865 7220 686f 6d6f y of higher homo │ │ │ │ -0003f930: 746f 7069 6573 206f 6e20 4620 666f 7220 topies on F for │ │ │ │ -0003f940: 7468 6520 656c 656d 656e 7473 206f 6620 the elements of │ │ │ │ -0003f950: 662c 2069 6e20 7468 6520 7365 6e73 650a f, in the sense. │ │ │ │ -0003f960: 6465 7363 7269 6265 642c 2066 6f72 2065 described, for e │ │ │ │ -0003f970: 7861 6d70 6c65 2c20 696e 2045 6973 656e xample, in Eisen │ │ │ │ -0003f980: 6275 6420 2245 6e72 6963 6865 6420 4672 bud "Enriched Fr │ │ │ │ -0003f990: 6565 2052 6573 6f6c 7574 696f 6e73 2061 ee Resolutions a │ │ │ │ -0003f9a0: 6e64 2043 6861 6e67 6520 6f66 0a52 696e nd Change of.Rin │ │ │ │ -0003f9b0: 6773 222e 0a0a 5468 6520 6f75 7470 7574 gs"...The output │ │ │ │ -0003f9c0: 2069 7320 6120 6861 7368 2074 6162 6c65 is a hash table │ │ │ │ -0003f9d0: 2077 6974 6820 656e 7472 6965 7320 6f66 with entries of │ │ │ │ -0003f9e0: 2074 6865 2066 6f72 6d20 245c 7b4a 2c69 the form $\{J,i │ │ │ │ -0003f9f0: 5c7d 3d3e 7324 2c20 7768 6572 6520 4a20 \}=>s$, where J │ │ │ │ -0003fa00: 6973 2061 0a6c 6973 7420 6f66 206e 6f6e is a.list of non │ │ │ │ -0003fa10: 2d6e 6567 6174 6976 6520 696e 7465 6765 -negative intege │ │ │ │ -0003fa20: 7273 2c20 6f66 206c 656e 6774 6820 6e20 rs, of length n │ │ │ │ -0003fa30: 616e 6420 2448 5c23 5c7b 4a2c 695c 7d3a and $H\#\{J,i\}: │ │ │ │ -0003fa40: 2046 5f69 2d3e 465f 7b69 2b32 7c4a 7c2d F_i->F_{i+2|J|- │ │ │ │ -0003fa50: 317d 240a 6172 6520 6d61 7073 2073 6174 1}$.are maps sat │ │ │ │ -0003fa60: 6973 6679 696e 6720 7468 6520 636f 6e64 isfying the cond │ │ │ │ -0003fa70: 6974 696f 6e73 2024 2420 485c 235c 7b65 itions $$ H\#\{e │ │ │ │ -0003fa80: 302c 695c 7d20 3d20 643b 2024 2420 616e 0,i\} = d; $$ an │ │ │ │ -0003fa90: 6420 2424 0a48 235c 7b65 302c 692b 315c d $$.H#\{e0,i+1\ │ │ │ │ -0003faa0: 7d2a 4823 5c7b 652c 695c 7d2b 4823 5c7b }*H#\{e,i\}+H#\{ │ │ │ │ -0003fab0: 652c 692d 315c 7d48 235c 7b65 302c 695c e,i-1\}H#\{e0,i\ │ │ │ │ -0003fac0: 7d20 3d20 665f 692c 2024 2420 7768 6572 } = f_i, $$ wher │ │ │ │ -0003fad0: 6520 2465 3020 3d0a 5c7b 302c 5c64 6f74 e $e0 =.\{0,\dot │ │ │ │ -0003fae0: 732c 305c 7d24 2061 6e64 2024 6524 2069 s,0\}$ and $e$ i │ │ │ │ -0003faf0: 7320 7468 6520 696e 6465 7820 6f66 2064 s the index of d │ │ │ │ -0003fb00: 6567 7265 6520 3120 7769 7468 2061 2031 egree 1 with a 1 │ │ │ │ -0003fb10: 2069 6e20 7468 6520 2469 242d 7468 2070 in the $i$-th p │ │ │ │ -0003fb20: 6c61 6365 3b0a 616e 642c 2066 6f72 2065 lace;.and, for e │ │ │ │ -0003fb30: 6163 6820 696e 6465 7820 6c69 7374 2049 ach index list I │ │ │ │ -0003fb40: 2077 6974 6820 7c49 7c3c 3d64 2c20 2424 with |I|<=d, $$ │ │ │ │ -0003fb50: 2073 756d 5f7b 4a3c 497d 2048 235c 7b49 sum_{Js$, whe │ │ │ │ +0003f9f0: 7265 204a 2069 7320 610a 6c69 7374 206f re J is a.list o │ │ │ │ +0003fa00: 6620 6e6f 6e2d 6e65 6761 7469 7665 2069 f non-negative i │ │ │ │ +0003fa10: 6e74 6567 6572 732c 206f 6620 6c65 6e67 ntegers, of leng │ │ │ │ +0003fa20: 7468 206e 2061 6e64 2024 485c 235c 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-00040430: 2030 2020 7c20 7c0a 7c20 2020 2020 2020 0 | |.| │ │ │ │ -00040440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040450: 2020 2020 2020 2020 2020 7b31 7d20 7c20 {1} | │ │ │ │ -00040460: 6120 2030 2020 2d63 2030 2020 2d64 2030 a 0 -c 0 -d 0 │ │ │ │ -00040470: 2020 7c20 7c0a 7c20 2020 2020 2020 2020 | |.| │ │ │ │ -00040480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040490: 2020 2020 2020 2020 7b31 7d20 7c20 3020 {1} | 0 │ │ │ │ -000404a0: 2061 2020 6220 2030 2020 3020 202d 6420 a b 0 0 -d │ │ │ │ -000404b0: 7c20 7c0a 7c20 2020 2020 2020 2020 2020 | |.| │ │ │ │ -000404c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000404d0: 2020 2020 2020 7b31 7d20 7c20 3020 2030 {1} | 0 0 │ │ │ │ -000404e0: 2020 3020 2061 2020 6220 2063 2020 7c20 0 a b c | │ │ │ │ -000404f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00040500: 2020 7b7b 302c 2030 2c20 307d 2c20 337d {{0, 0, 0}, 3} │ │ │ │ -00040510: 203d 3e20 7b32 7d20 7c20 6320 2064 2020 => {2} | c d │ │ │ │ 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│ -00040eb0: 2020 7b7b 312c 2030 2c20 307d 2c20 307d {{1, 0, 0}, 0} │ │ │ │ -00040ec0: 203d 3e20 7b31 7d20 7c20 3120 7c20 2020 => {1} | 1 | │ │ │ │ -00040ed0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00040ee0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00040ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040f00: 2020 7b31 7d20 7c20 3020 7c20 2020 2020 {1} | 0 | │ │ │ │ -00040f10: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00040dc0: 207b 337d 207c 2030 2030 2020 3020 3020 {3} | 0 0 0 0 │ │ │ │ +00040dd0: 2030 2031 207c 2020 2020 207c 0a7c 2020 0 1 | |.| │ │ │ │ +00040de0: 2020 2020 2020 2020 2020 2020 207b 7b30 {{0 │ │ │ │ +00040df0: 2c20 312c 2031 7d2c 202d 317d 203d 3e20 , 1, 1}, -1} => │ │ │ │ +00040e00: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +00040e10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00040e20: 2020 2020 2020 2020 2020 207b 7b30 2c20 {{0, │ │ │ │ +00040e30: 322c 2030 7d2c 202d 317d 203d 3e20 3020 2, 0}, -1} => 0 │ │ │ │ +00040e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040e50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00040e60: 2020 2020 2020 2020 207b 7b31 2c20 302c {{1, 0, │ │ │ │ +00040e70: 2030 7d2c 202d 317d 203d 3e20 3020 2020 0}, -1} => 0 │ │ │ │ +00040e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040e90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00040ea0: 2020 2020 2020 207b 7b31 2c20 302c 2030 {{1, 0, 0 │ │ │ │ +00040eb0: 7d2c 2030 7d20 3d3e 207b 317d 207c 2031 }, 0} => {1} | 1 │ │ │ │ +00040ec0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00040ed0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00040ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040ef0: 2020 2020 2020 207b 317d 207c 2030 207c {1} | 0 | │ │ │ │ +00040f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00040f10: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00040f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040f30: 2020 2020 2020 2020 2020 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2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00041010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041020: 2020 2020 2020 2020 2020 2020 207b 327d {2} │ │ │ │ +00041030: 207c 2030 2030 2030 2030 207c 2020 2020 | 0 0 0 0 | │ │ │ │ +00041040: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00041050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00041060: 2020 2020 2020 2020 2020 207b 327d 207c {2} | │ │ │ │ +00041070: 2030 2030 2030 2031 207c 2020 2020 2020 0 0 0 1 | │ │ │ │ +00041080: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00041090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000410a0: 2020 2020 2020 2020 207b 327d 207c 2030 {2} | 0 │ │ │ │ +000410b0: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ +000410c0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000410d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000410e0: 2020 2020 2020 207b 327d 207c 2030 2030 {2} | 0 0 │ │ │ │ +000410f0: 2030 2030 207c 2020 2020 2020 2020 2020 0 0 | │ │ │ │ +00041100: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041110: 2020 207b 7b31 2c20 302c 2030 7d2c 2032 {{1, 0, 0}, 2 │ │ │ │ +00041120: 7d20 3d3e 207b 337d 207c 2030 2030 2031 } => {3} | 0 0 1 │ │ │ │ +00041130: 2030 2030 2030 207c 2020 2020 2020 207c 0 0 0 | | │ │ │ │ +00041140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00041150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041160: 2020 2020 2020 2020 2020 2020 2020 7b33 {3 │ │ │ │ -00041170: 7d20 7c20 3020 3020 3020 3020 3120 3020 } | 0 0 0 0 1 0 │ │ │ │ -00041180: 7c20 2020 2020 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +00041160: 2020 207b 337d 207c 2030 2030 2030 2030 {3} | 0 0 0 0 │ │ │ │ +00041170: 2031 2030 207c 2020 2020 2020 207c 0a7c 1 0 | |.| │ │ │ │ +00041180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000411a0: 2020 2020 2020 2020 2020 2020 7b33 7d20 {3} │ │ │ │ -000411b0: 7c20 3020 3020 3020 3020 3020 3120 7c20 | 0 0 0 0 0 1 | │ │ │ │ -000411c0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -000411d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000411e0: 2020 2020 2020 2020 2020 7b33 7d20 7c20 {3} | │ │ │ │ -000411f0: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ -00041200: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00041210: 2020 2020 2020 7b7b 312c 2030 2c20 317d {{1, 0, 1} │ │ │ │ -00041220: 2c20 2d31 7d20 3d3e 2030 2020 2020 2020 , -1} => 0 │ │ │ │ -00041230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041240: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041250: 2020 2020 7b7b 312c 2031 2c20 307d 2c20 {{1, 1, 0}, │ │ │ │ -00041260: 2d31 7d20 3d3e 2030 2020 2020 2020 2020 -1} => 0 │ │ │ │ -00041270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041280: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00041290: 2020 7b7b 322c 2030 2c20 307d 2c20 2d31 {{2, 0, 0}, -1 │ │ │ │ -000412a0: 7d20 3d3e 2030 2020 2020 2020 2020 2020 } => 0 │ │ │ │ -000412b0: 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+00041330: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00041340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041370: 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6e20 7468 --------+..In th │ │ │ │ -00041380: 6973 2063 6173 6520 7468 6520 6869 6768 is case the high │ │ │ │ -00041390: 6572 2068 6f6d 6f74 6f70 6965 7320 6172 er homotopies ar │ │ │ │ -000413a0: 6520 303a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d e 0:..+--------- │ │ │ │ +00041360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +00041370: 496e 2074 6869 7320 6361 7365 2074 6865 In this case the │ │ │ │ +00041380: 2068 6967 6865 7220 686f 6d6f 746f 7069 higher homotopi │ │ │ │ +00041390: 6573 2061 7265 2030 3a0a 0a2b 2d2d 2d2d es are 0:..+---- │ │ │ │ +000413a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000413b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000413c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000413e0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204c -------+.|i6 : L │ │ │ │ -000413f0: 203d 2073 6f72 7420 7365 6c65 6374 286b = sort select(k │ │ │ │ -00041400: 6579 7320 686f 6d6f 742c 206b 2d3e 2868 eys homot, k->(h │ │ │ │ -00041410: 6f6d 6f74 236b 213d 3020 616e 6420 7375 omot#k!=0 and su │ │ │ │ -00041420: 6d28 6b5f 3029 3e31 2929 7c0a 7c20 2020 m(k_0)>1))|.| │ │ │ │ +000413d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000413e0: 3620 3a20 4c20 3d20 736f 7274 2073 656c 6 : L = sort sel │ │ │ │ +000413f0: 6563 7428 6b65 7973 2068 6f6d 6f74 2c20 ect(keys homot, │ │ │ │ +00041400: 6b2d 3e28 686f 6d6f 7423 6b21 3d30 2061 k->(homot#k!=0 a │ │ │ │ +00041410: 6e64 2073 756d 286b 5f30 293e 3129 297c nd sum(k_0)>1))| │ │ │ │ +00041420: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00041430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041470: 6f36 203d 207b 7d20 2020 2020 2020 2020 o6 = {} │ │ │ │ +00041460: 2020 7c0a 7c6f 3620 3d20 7b7d 2020 2020 |.|o6 = {} │ │ │ │ +00041470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414b0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000414a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000414b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000414c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000414d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000414f0: 2020 207c 0a7c 6f36 203a 204c 6973 7420 |.|o6 : List │ │ │ │ +000414e0: 2020 2020 2020 2020 7c0a 7c6f 3620 3a20 |.|o6 : │ │ │ │ +000414f0: 4c69 7374 2020 2020 2020 2020 2020 2020 List │ │ │ │ 00041500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041530: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00041520: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00041530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041570: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4f6e 2074 ---------+..On t │ │ │ │ -00041580: 6865 206f 7468 6572 2068 616e 642c 2069 he other hand, i │ │ │ │ -00041590: 6620 7765 2074 616b 6520 6120 636f 6d70 f we take a comp │ │ │ │ -000415a0: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -000415b0: 6e20 616e 6420 736f 6d65 7468 696e 6720 n and something │ │ │ │ -000415c0: 636f 6e74 6169 6e65 640a 696e 2069 7420 contained.in it │ │ │ │ -000415d0: 696e 2061 206d 6f72 6520 636f 6d70 6c69 in a more compli │ │ │ │ -000415e0: 6361 7465 6420 7369 7475 6174 696f 6e2c cated situation, │ │ │ │ -000415f0: 2074 6865 2070 726f 6772 616d 2067 6976 the program giv │ │ │ │ -00041600: 6573 206e 6f6e 7a65 726f 2068 6967 6865 es nonzero highe │ │ │ │ -00041610: 720a 686f 6d6f 746f 7069 6573 3a0a 0a2b r.homotopies:..+ │ │ │ │ +00041560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +00041570: 0a4f 6e20 7468 6520 6f74 6865 7220 6861 .On the other ha │ │ │ │ +00041580: 6e64 2c20 6966 2077 6520 7461 6b65 2061 nd, if we take a │ │ │ │ +00041590: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ +000415a0: 6563 7469 6f6e 2061 6e64 2073 6f6d 6574 ection and somet │ │ │ │ +000415b0: 6869 6e67 2063 6f6e 7461 696e 6564 0a69 hing contained.i │ │ │ │ +000415c0: 6e20 6974 2069 6e20 6120 6d6f 7265 2063 n it in a more c │ │ │ │ +000415d0: 6f6d 706c 6963 6174 6564 2073 6974 7561 omplicated situa │ │ │ │ +000415e0: 7469 6f6e 2c20 7468 6520 7072 6f67 7261 tion, the progra │ │ │ │ +000415f0: 6d20 6769 7665 7320 6e6f 6e7a 6572 6f20 m gives nonzero │ │ │ │ +00041600: 6869 6768 6572 0a68 6f6d 6f74 6f70 6965 higher.homotopie │ │ │ │ +00041610: 733a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d s:..+----------- │ │ │ │ 00041620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041670: 6937 203a 206b 6b3d 205a 5a2f 3332 3030 i7 : kk= ZZ/3200 │ │ │ │ -00041680: 333b 2020 2020 2020 2020 2020 2020 2020 3; │ │ │ │ +00041660: 2d2d 2b0a 7c69 3720 3a20 6b6b 3d20 5a5a --+.|i7 : kk= ZZ │ │ │ │ +00041670: 2f33 3230 3033 3b20 2020 2020 2020 2020 /32003; │ │ │ │ +00041680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000416a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000416b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000416b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 000416c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000416d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000416e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000416f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041710: 6938 203a 2053 203d 206b 6b5b 612c 622c i8 : S = kk[a,b, │ │ │ │ -00041720: 632c 645d 3b20 2020 2020 2020 2020 2020 c,d]; │ │ │ │ +00041700: 2d2d 2b0a 7c69 3820 3a20 5320 3d20 6b6b --+.|i8 : S = kk │ │ │ │ +00041710: 5b61 2c62 2c63 2c64 5d3b 2020 2020 2020 [a,b,c,d]; │ │ │ │ +00041720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041750: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00041750: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000417a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -000417b0: 6939 203a 204d 203d 2053 5e31 2f28 6964 i9 : M = S^1/(id │ │ │ │ -000417c0: 6561 6c22 6132 2c62 322c 6332 2c64 3222 eal"a2,b2,c2,d2" │ │ │ │ -000417d0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ +000417a0: 2d2d 2b0a 7c69 3920 3a20 4d20 3d20 535e --+.|i9 : M = S^ │ │ │ │ +000417b0: 312f 2869 6465 616c 2261 322c 6232 2c63 1/(ideal"a2,b2,c │ │ │ │ +000417c0: 322c 6432 2229 3b20 2020 2020 2020 2020 2,d2"); │ │ │ │ +000417d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000417e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000417f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +000417f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041850: 6931 3020 3a20 4620 3d20 6672 6565 5265 i10 : F = freeRe │ │ │ │ -00041860: 736f 6c75 7469 6f6e 204d 2020 2020 2020 solution M │ │ │ │ +00041840: 2d2d 2b0a 7c69 3130 203a 2046 203d 2066 --+.|i10 : F = f │ │ │ │ +00041850: 7265 6552 6573 6f6c 7574 696f 6e20 4d20 reeResolution M │ │ │ │ +00041860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041890: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00041890: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000418a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000418b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000418c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000418d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000418e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000418f0: 2020 2020 2020 2031 2020 2020 2020 3420 1 4 │ │ │ │ -00041900: 2020 2020 2036 2020 2020 2020 3420 2020 6 4 │ │ │ │ -00041910: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000418e0: 2020 7c0a 7c20 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +000418f0: 2020 2034 2020 2020 2020 3620 2020 2020 4 6 │ │ │ │ +00041900: 2034 2020 2020 2020 3120 2020 2020 2020 4 1 │ │ │ │ +00041910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041930: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041940: 6f31 3020 3d20 5320 203c 2d2d 2053 2020 o10 = S <-- S │ │ │ │ -00041950: 3c2d 2d20 5320 203c 2d2d 2053 2020 3c2d <-- S <-- S <- │ │ │ │ -00041960: 2d20 5320 2020 2020 2020 2020 2020 2020 - S │ │ │ │ +00041930: 2020 7c0a 7c6f 3130 203d 2053 2020 3c2d |.|o10 = S <- │ │ │ │ +00041940: 2d20 5320 203c 2d2d 2053 2020 3c2d 2d20 - S <-- S <-- │ │ │ │ +00041950: 5320 203c 2d2d 2053 2020 2020 2020 2020 S <-- S │ │ │ │ +00041960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041980: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041a70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041a80: 6f31 3020 3a20 436f 6d70 6c65 7820 2020 o10 : Complex │ │ │ │ +00041a70: 2020 7c0a 7c6f 3130 203a 2043 6f6d 706c |.|o10 : Compl │ │ │ │ +00041a80: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 00041a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041ac0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00041ac0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041b00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041b10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041b20: 6931 3120 3a20 7365 7452 616e 646f 6d53 i11 : setRandomS │ │ │ │ -00041b30: 6565 6420 3020 2020 2020 2020 2020 2020 eed 0 │ │ │ │ +00041b10: 2d2d 2b0a 7c69 3131 203a 2073 6574 5261 --+.|i11 : setRa │ │ │ │ +00041b20: 6e64 6f6d 5365 6564 2030 2020 2020 2020 ndomSeed 0 │ │ │ │ +00041b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041b60: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041b70: 202d 2d20 7365 7474 696e 6720 7261 6e64 -- setting rand │ │ │ │ -00041b80: 6f6d 2073 6565 6420 746f 2030 2020 2020 om seed to 0 │ │ │ │ +00041b60: 2020 7c0a 7c20 2d2d 2073 6574 7469 6e67 |.| -- setting │ │ │ │ +00041b70: 2072 616e 646f 6d20 7365 6564 2074 6f20 random seed to │ │ │ │ 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2020 2020 207c 0a2b |.+ │ │ │ │ +00041c50: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041cb0: 6931 3220 3a20 6620 3d20 7261 6e64 6f6d i12 : f = random │ │ │ │ -00041cc0: 2853 5e31 2c53 5e7b 323a 2d35 7d29 3b20 (S^1,S^{2:-5}); │ │ │ │ +00041ca0: 2d2d 2b0a 7c69 3132 203a 2066 203d 2072 --+.|i12 : f = r │ │ │ │ +00041cb0: 616e 646f 6d28 535e 312c 535e 7b32 3a2d andom(S^1,S^{2:- │ │ │ │ +00041cc0: 357d 293b 2020 2020 2020 2020 2020 2020 5}); │ │ │ │ 00041cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041cf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00041cf0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00041d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041d50: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ -00041d60: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00041d40: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041d50: 2020 2031 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ +00041d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041d90: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041da0: 6f31 3220 3a20 4d61 7472 6978 2053 2020 o12 : Matrix S │ │ │ │ -00041db0: 3c2d 2d20 5320 2020 2020 2020 2020 2020 <-- S │ │ │ │ +00041d90: 2020 7c0a 7c6f 3132 203a 204d 6174 7269 |.|o12 : Matri │ │ │ │ +00041da0: 7820 5320 203c 2d2d 2053 2020 2020 2020 x S <-- S │ │ │ │ +00041db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041de0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00041de0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00041df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00041e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00041e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00041e40: 6931 3320 3a20 686f 6d6f 7420 3d20 6d61 i13 : homot = ma │ │ │ │ -00041e50: 6b65 486f 6d6f 746f 7069 6573 2866 2c46 keHomotopies(f,F │ │ │ │ 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3133 203d 2048 6173 6854 |.|o13 = HashT │ │ │ │ +00041ee0: 6162 6c65 7b7b 7b30 2c20 307d 2c20 307d able{{{0, 0}, 0} │ │ │ │ +00041ef0: 203d 3e20 3020 2020 2020 2020 2020 2020 => 0 │ │ │ │ 00041f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041f20: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041f40: 7b7b 302c 2030 7d2c 2031 7d20 3d3e 207c {{0, 0}, 1} => | │ │ │ │ -00041f50: 2061 3220 2020 2020 2020 2020 2020 2020 a2 │ │ │ │ +00041f20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00041f30: 2020 2020 207b 7b30 2c20 307d 2c20 317d {{0, 0}, 1} │ │ │ │ +00041f40: 203d 3e20 7c20 6132 2020 2020 2020 2020 => | a2 │ │ │ │ +00041f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00041f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00041f70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00041f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ 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0a7c |.| │ │ │ │ +00042010: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00042020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042030: 2020 2020 2020 2020 2020 2020 2020 207b { │ │ │ │ -00042040: 327d 207c 2020 2020 2020 2020 2020 2020 2} | │ │ │ │ +00042030: 2020 2020 7b32 7d20 7c20 2020 2020 2020 {2} | │ │ │ │ +00042040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00042050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042060: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00042060: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00042070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00042080: 2020 2020 2020 2020 2020 2020 2020 207b { │ │ │ │ -00042090: 327d 207c 2020 2020 2020 2020 2020 2020 2} | │ │ │ │ +00042080: 2020 2020 7b32 7d20 7c20 2020 2020 2020 {2} | │ │ │ │ +00042090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000420a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000420b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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2020 2020 2020 2020 2020 2020 │ │ │ │ 00053530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053540: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00053540: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00053550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053590: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00053590: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 000535a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000535b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000535c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000535d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000535e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000535f0: 3633 6364 332b 3633 3431 6434 207c 2020 63cd3+6341d4 | │ │ │ │ +000535e0: 2020 7c0a 7c36 3363 6433 2b36 3334 3164 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2020 2020 2020 │ │ │ │ 000538c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000538d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000538e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000538f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053900: 207c 0a7c 6f31 3420 3a20 4c69 7374 2020 |.|o14 : List │ │ │ │ +000538f0: 2020 2020 2020 7c0a 7c6f 3134 203a 204c |.|o14 : L │ │ │ │ +00053900: 6973 7420 2020 2020 2020 2020 2020 2020 ist │ │ │ │ 00053910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053950: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00053940: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00053950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053970: 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2020 2020 │ │ │ │ -00053a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053a40: 207c 0a7c 6f31 3520 3d20 3620 2020 2020 |.|o15 = 6 │ │ │ │ +00053a30: 2020 2020 2020 7c0a 7c6f 3135 203d 2036 |.|o15 = 6 │ │ │ │ +00053a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053a90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00053a80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00053a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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│ │ -00053ea0: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -00053eb0: 2d2b 2d2b 2020 2020 2020 2020 2020 2020 -+-+ │ │ │ │ +00053e90: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00053ea0: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020 ------+-+ │ │ │ │ +00053eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053ef0: 207c 0a7c 2020 2020 2020 7c7b 322c 2030 |.| |{2, 0 │ │ │ │ -00053f00: 7d7c 317c 2020 2020 2020 2020 2020 2020 }|1| │ │ │ │ +00053ee0: 2020 2020 2020 7c0a 7c20 2020 2020 207c |.| | │ │ │ │ +00053ef0: 7b32 2c20 307d 7c31 7c20 2020 2020 2020 {2, 0}|1| │ │ │ │ +00053f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f40: 207c 0a7c 2020 2020 2020 2b2d 2d2d 2d2d |.| +----- │ │ │ │ -00053f50: 2d2b 2d2b 2020 2020 2020 2020 2020 2020 -+-+ │ │ │ │ +00053f30: 2020 2020 2020 7c0a 7c20 2020 2020 202b |.| + │ │ │ │ +00053f40: 2d2d 2d2d 2d2d 2b2d 2b20 2020 2020 2020 ------+-+ │ │ │ │ +00053f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053f90: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00053f80: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00053f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053fe0: 2d2b 0a0a 466f 7220 6578 616d 706c 6520 -+..For example │ │ │ │ -00053ff0: 7765 2068 6176 653a 0a0a 2b2d 2d2d 2d2d we have:..+----- │ │ │ │ +00053fd0: 2d2d 2d2d 2d2d 2b0a 0a46 6f72 2065 7861 ------+..For exa │ │ │ │ +00053fe0: 6d70 6c65 2077 6520 6861 7665 3a0a 0a2b mple we have:..+ │ │ │ │ +00053ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054010: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054040: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3137 203a --------+.|i17 : │ │ │ │ -00054050: 2068 6f6d 6f74 2328 4c5f 3029 2020 2020 homot#(L_0) │ │ │ │ +00054030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00054040: 6931 3720 3a20 686f 6d6f 7423 284c 5f30 i17 : homot#(L_0 │ │ │ │ +00054050: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ 00054060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054090: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00054080: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00054090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000540a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000540b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000540c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000540d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000540e0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ -000540f0: 207b 367d 207c 202d 3133 3739 3561 342b {6} | -13795a4+ │ │ │ │ -00054100: 3230 3139 6133 622b 3133 3736 3961 3262 2019a3b+13769a2b │ │ │ │ -00054110: 322b 3735 3836 6162 332b 3836 3439 6234 2+7586ab3+8649b4 │ │ │ │ -00054120: 2b36 3435 3461 3363 2d31 3031 3837 6132 +6454a3c-10187a2 │ │ │ │ -00054130: 6263 2d31 3738 3361 7c0a 7c20 2020 2020 bc-1783a|.| │ │ │ │ -00054140: 207b 367d 207c 2031 3131 3532 6134 2d31 {6} | 11152a4-1 │ │ │ │ -00054150: 3333 3661 3362 2b31 3138 3436 6132 6232 336a3b+11846a2b2 │ │ │ │ -00054160: 2b31 3032 3634 6162 332b 3631 3862 342d +10264ab3+618b4- │ │ │ │ -00054170: 3131 3035 3161 3363 2b31 3231 3239 6132 11051a3c+12129a2 │ │ │ │ -00054180: 6263 2b35 3932 3761 7c0a 7c20 2020 2020 bc+5927a|.| │ │ │ │ -00054190: 207b 367d 207c 202d 3633 3338 6134 2b31 {6} | -6338a4+1 │ │ │ │ -000541a0: 3030 3235 6133 622b 3134 3938 3761 3363 0025a3b+14987a3c │ │ │ │ -000541b0: 2d39 3935 3961 3262 632d 3131 3639 3161 -9959a2bc-11691a │ │ │ │ -000541c0: 3263 322b 3132 3333 3661 6263 322d 3737 2c2+12336abc2-77 │ │ │ │ -000541d0: 3836 6133 642d 3131 7c0a 7c20 2020 2020 86a3d-11|.| │ │ │ │ -000541e0: 207b 367d 207c 2032 3237 3561 342d 3233 {6} | 2275a4-23 │ │ │ │ -000541f0: 3961 3362 2b31 3435 3934 6132 6232 2d38 9a3b+14594a2b2-8 │ │ │ │ -00054200: 3135 3361 6233 2d31 3139 3435 6234 2d38 153ab3-11945b4-8 │ │ │ │ -00054210: 3431 3661 3363 2b36 3235 3161 3262 632d 416a3c+6251a2bc- │ │ │ │ -00054220: 3330 3233 6162 3263 7c0a 7c20 2020 2020 3023ab2c|.| │ │ │ │ -00054230: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +000540d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000540e0: 6f31 3720 3d20 7b36 7d20 7c20 2d31 3337 o17 = {6} | -137 │ │ │ │ +000540f0: 3935 6134 2b32 3031 3961 3362 2b31 3337 95a4+2019a3b+137 │ │ │ │ +00054100: 3639 6132 6232 2b37 3538 3661 6233 2b38 69a2b2+7586ab3+8 │ │ │ │ +00054110: 3634 3962 342b 3634 3534 6133 632d 3130 649b4+6454a3c-10 │ │ │ │ +00054120: 3138 3761 3262 632d 3137 3833 617c 0a7c 187a2bc-1783a|.| │ │ │ │ +00054130: 2020 2020 2020 7b36 7d20 7c20 3131 3135 {6} | 1115 │ │ │ │ +00054140: 3261 342d 3133 3336 6133 622b 3131 3834 2a4-1336a3b+1184 │ │ │ │ +00054150: 3661 3262 322b 3130 3236 3461 6233 2b36 6a2b2+10264ab3+6 │ │ │ │ +00054160: 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2d2d 2d2d ---------------- │ │ │ │ 00054240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054270: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 --------|.| │ │ │ │ -00054280: 2062 3263 2b39 3231 3962 3363 2b35 3531 b2c+9219b3c+551 │ │ │ │ -00054290: 3361 3263 322b 3130 3535 3861 6263 322b 3a2c2+10558abc2+ │ │ │ │ -000542a0: 3235 3930 6232 6332 2b31 3136 3234 6133 2590b2c2+11624a3 │ │ │ │ -000542b0: 642d 3536 3033 6132 6264 2b31 3430 3538 d-5603a2bd+14058 │ │ │ │ -000542c0: 6162 3264 2d31 3236 7c0a 7c20 2020 2020 ab2d-126|.| │ │ │ │ -000542d0: 2062 3263 2b34 3839 6233 632d 3135 3338 b2c+489b3c-1538 │ │ │ │ -000542e0: 3361 3263 322b 3530 3761 6263 322d 3133 3a2c2+507abc2-13 │ │ │ │ -000542f0: 3830 3462 3263 322d 3834 3136 6163 332b 804b2c2-8416ac3+ │ │ │ │ -00054300: 3932 6334 2d31 3130 3537 6133 642d 3531 92c4-11057a3d-51 │ │ │ │ -00054310: 3133 6132 6264 2d32 7c0a 7c20 2020 2020 13a2bd-2|.| │ │ │ │ -00054320: 2035 3661 3262 642b 3439 3630 6132 6364 56a2bd+4960a2cd │ │ │ │ -00054330: 2d35 3538 3961 6263 642d 3831 3633 6163 -5589abcd-8163ac │ │ │ │ -00054340: 3264 2d31 3839 3562 6332 642b 3934 3634 2d-1895bc2d+9464 │ │ │ │ -00054350: 6132 6432 2d37 3235 3361 6264 322b 3132 a2d2-7253abd2+12 │ │ │ │ -00054360: 3634 3261 6364 322d 7c0a 7c20 2020 2020 642acd2-|.| │ │ │ │ -00054370: 202b 3539 3333 6233 632b 3932 6132 6332 +5933b3c+92a2c2 │ │ │ │ -00054380: 2b35 3334 3361 6263 322b 3337 3938 6232 +5343abc2+3798b2 │ │ │ │ -00054390: 6332 2d31 3539 3638 6133 642b 3437 3361 c2-15968a3d+473a │ │ │ │ -000543a0: 3262 642b 3133 3239 3361 6232 642d 3337 2bd+13293ab2d-37 │ │ │ │ -000543b0: 3631 6233 642d 3737 7c0a 7c20 2020 2020 61b3d-77|.| │ │ │ │ -000543c0: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +00054260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00054270: 2020 2020 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+9464a2d2-7253ab │ │ │ │ +00054350: 6432 2b31 3236 3432 6163 6432 2d7c 0a7c d2+12642acd2-|.| │ │ │ │ +00054360: 2020 2020 2020 2b35 3933 3362 3363 2b39 +5933b3c+9 │ │ │ │ +00054370: 3261 3263 322b 3533 3433 6162 6332 2b33 2a2c2+5343abc2+3 │ │ │ │ +00054380: 3739 3862 3263 322d 3135 3936 3861 3364 798b2c2-15968a3d │ │ │ │ +00054390: 2b34 3733 6132 6264 2b31 3332 3933 6162 +473a2bd+13293ab │ │ │ │ +000543a0: 3264 2d33 3736 3162 3364 2d37 377c 0a7c 2d-3761b3d-77|.| │ │ │ │ +000543b0: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ +000543c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000543d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000543e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000543f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054400: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 --------|.| │ │ │ │ -00054410: 2031 3562 3364 2b37 3836 3961 3263 642d 15b3d+7869a2cd- │ │ │ │ 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3261 6332 642d 31b2cd+6042ac2d- │ │ │ │ +00054430: 3235 3631 6263 3264 2d38 3730 3961 3264 2561bc2d-8709a2d │ │ │ │ +00054440: 322d 3133 3231 3961 6264 322b 347c 0a7c 2-13219abd2+4|.| │ │ │ │ +00054450: 2020 2020 2020 3736 3261 6232 642b 3134 762ab2d+14 │ │ │ │ +00054460: 3039 3562 3364 2d31 3538 3861 3263 642b 095b3d-1588a2cd+ │ │ │ │ +00054470: 3230 3030 6162 6364 2d32 3038 3062 3263 2000abcd-2080b2c │ │ │ │ +00054480: 642b 3931 3735 6163 3264 2d36 3439 6263 d+9175ac2d-649bc │ │ │ │ +00054490: 3264 2b38 3832 3963 3364 2b32 317c 0a7c 2d+8829c3d+21|.| │ │ │ │ +000544a0: 2020 2020 2020 3139 3538 6263 6432 2020 1958bcd2 │ │ │ │ +000544b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000544c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000544d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000544e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000544f0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00054500: 2031 3761 3263 642d 3733 3839 6162 6364 17a2cd-7389abcd │ │ │ │ -00054510: 2b34 3732 3362 3263 642d 3133 3236 3261 +4723b2cd-13262a │ │ │ │ -00054520: 6332 642b 3534 3331 6263 3264 2b31 3132 c2d+5431bc2d+112 │ │ │ │ -00054530: 3734 6132 6432 2d32 3137 6162 6432 2b31 74a2d2-217abd2+1 │ │ │ │ -00054540: 3236 3162 3264 322b 7c0a 7c20 2020 2020 261b2d2+|.| │ │ │ │ -00054550: 202d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d --------------- │ │ │ │ +000544e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +000544f0: 2020 2020 2020 3137 6132 6364 2d37 3338 17a2cd-738 │ │ │ │ +00054500: 3961 6263 642b 3437 3233 6232 6364 2d31 9abcd+4723b2cd-1 │ │ │ │ +00054510: 3332 3632 6163 3264 2b35 3433 3162 6332 3262ac2d+5431bc2 │ │ │ │ +00054520: 642b 3131 3237 3461 3264 322d 3231 3761 d+11274a2d2-217a │ │ │ │ +00054530: 6264 322b 3132 3631 6232 6432 2b7c 0a7c bd2+1261b2d2+|.| │ │ │ │ +00054540: 2020 2020 2020 2d2d 2d2d 2d2d 2d2d 2d2d ---------- │ │ │ │ +00054550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00054570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00054590: 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 --------|.| │ │ │ │ -000545a0: 2032 3039 6232 6432 2b31 3232 3235 6163 209b2d2+12225ac │ │ │ │ -000545b0: 6432 2d32 3630 3562 6364 322d 3932 6332 d2-2605bcd2-92c2 │ │ │ │ -000545c0: 6432 2b31 3539 3638 6164 332b 3134 3836 d2+15968ad3+1486 │ │ │ │ -000545d0: 3062 6433 2d38 3832 3963 6433 2d31 3132 0bd3-8829cd3-112 │ │ │ │ -000545e0: 3734 6434 207c 2020 7c0a 7c20 2020 2020 74d4 | |.| │ │ │ │ -000545f0: 2036 3461 3264 322b 3836 3335 6162 6432 64a2d2+8635abd2 │ │ │ │ -00054600: 2d37 3136 3162 3264 322b 3939 3761 6364 -7161b2d2+997acd │ │ │ │ -00054610: 322b 3330 3135 6263 6432 2b31 3132 3734 2+3015bcd2+11274 │ │ │ │ -00054620: 6332 6432 2020 2020 2020 2020 2020 2020 c2d2 │ │ │ │ -00054630: 2020 2020 207c 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +00054580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ +00054590: 2020 2020 2020 3230 3962 3264 322b 3132 209b2d2+12 │ │ │ │ +000545a0: 3232 3561 6364 322d 3236 3035 6263 6432 225acd2-2605bcd2 │ │ │ │ +000545b0: 2d39 3263 3264 322b 3135 3936 3861 6433 -92c2d2+15968ad3 │ │ │ │ +000545c0: 2b31 3438 3630 6264 332d 3838 3239 6364 +14860bd3-8829cd │ │ │ │ +000545d0: 332d 3131 3237 3464 3420 7c20 207c 0a7c 3-11274d4 | |.| │ │ │ │ +000545e0: 2020 2020 2020 3634 6132 6432 2b38 3633 64a2d2+863 │ │ │ │ +000545f0: 3561 6264 322d 3731 3631 6232 6432 2b39 5abd2-7161b2d2+9 │ │ │ │ +00054600: 3937 6163 6432 2b33 3031 3562 6364 322b 97acd2+3015bcd2+ │ │ │ │ +00054610: 3131 3237 3463 3264 3220 2020 2020 2020 11274c2d2 │ │ │ │ +00054620: 2020 2020 2020 2020 2020 7c20 207c 0a7c | |.| │ │ │ │ +00054630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054680: 2020 2020 207c 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ -00054690: 2038 3230 3161 6364 322d 3134 3038 3062 8201acd2-14080b │ │ │ │ -000546a0: 6364 3220 2020 2020 2020 2020 2020 2020 cd2 │ │ │ │ +00054670: 2020 2020 2020 2020 2020 7c20 207c 0a7c | |.| │ │ │ │ +00054680: 2020 2020 2020 3832 3031 6163 6432 2d31 8201acd2-1 │ │ │ │ +00054690: 3430 3830 6263 6432 2020 2020 2020 2020 4080bcd2 │ │ │ │ +000546a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000546b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000546c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000546d0: 2020 2020 207c 2020 7c0a 7c20 2020 2020 | |.| │ │ │ │ +000546c0: 2020 2020 2020 2020 2020 7c20 207c 0a7c | |.| │ │ │ │ +000546d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000546e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000546f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054700: 2020 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207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00054980: 2020 2020 2020 2020 207c 0a7c 6f31 3920 |.|o19 │ │ │ │ +00054990: 3a20 4c69 7374 2020 2020 2020 2020 2020 : List │ │ │ │ +000549a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000549b0: 2020 2020 2020 2020 7c0a 2b2d 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 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000549f0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ -00054a00: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ -00054a10: 6520 6d61 6b65 486f 6d6f 746f 7069 6573 e makeHomotopies │ │ │ │ -00054a20: 313a 206d 616b 6548 6f6d 6f74 6f70 6965 1: makeHomotopie │ │ │ │ -00054a30: 7331 2c20 2d2d 2072 6574 7572 6e73 2061 s1, -- returns a │ │ │ │ -00054a40: 2073 7973 7465 6d20 6f66 2066 6972 7374 system of first │ │ │ │ -00054a50: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a . homotopies. │ │ │ │ -00054a60: 0a57 6179 7320 746f 2075 7365 206d 616b .Ways to use mak │ │ │ │ -00054a70: 6548 6f6d 6f74 6f70 6965 733a 0a3d 3d3d eHomotopies:.=== │ │ │ │ -00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00054a90: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ -00054aa0: 616b 6548 6f6d 6f74 6f70 6965 7328 4d61 akeHomotopies(Ma │ │ │ │ -00054ab0: 7472 6978 2c43 6f6d 706c 6578 2922 0a20 trix,Complex)". │ │ │ │ -00054ac0: 202a 2022 6d61 6b65 486f 6d6f 746f 7069 * "makeHomotopi │ │ │ │ -00054ad0: 6573 284d 6174 7269 782c 436f 6d70 6c65 es(Matrix,Comple │ │ │ │ -00054ae0: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -00054af0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00054b00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00054b10: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00054b20: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ -00054b30: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ -00054b40: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00054b50: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00054b60: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00054b70: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +000549e0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ +000549f0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ +00054a00: 202a 6e6f 7465 206d 616b 6548 6f6d 6f74 *note makeHomot │ │ │ │ +00054a10: 6f70 6965 7331 3a20 6d61 6b65 486f 6d6f opies1: makeHomo │ │ │ │ +00054a20: 746f 7069 6573 312c 202d 2d20 7265 7475 topies1, -- retu │ │ │ │ +00054a30: 726e 7320 6120 7379 7374 656d 206f 6620 rns a system of │ │ │ │ +00054a40: 6669 7273 740a 2020 2020 686f 6d6f 746f first. homoto │ │ │ │ +00054a50: 7069 6573 0a0a 5761 7973 2074 6f20 7573 pies..Ways to us │ │ │ │ +00054a60: 6520 6d61 6b65 486f 6d6f 746f 7069 6573 e makeHomotopies │ │ │ │ +00054a70: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +00054a80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00054a90: 202a 2022 6d61 6b65 486f 6d6f 746f 7069 * "makeHomotopi │ │ │ │ +00054aa0: 6573 284d 6174 7269 782c 436f 6d70 6c65 es(Matrix,Comple │ │ │ │ +00054ab0: 7829 220a 2020 2a20 226d 616b 6548 6f6d x)". * "makeHom │ │ │ │ +00054ac0: 6f74 6f70 6965 7328 4d61 7472 6978 2c43 otopies(Matrix,C │ │ │ │ +00054ad0: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +00054ae0: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00054af0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00054b00: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00054b10: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +00054b20: 6965 733a 206d 616b 6548 6f6d 6f74 6f70 ies: 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2020 2a20 482c puts:. * H, │ │ │ │ -00054ed0: 2061 202a 6e6f 7465 2068 6173 6820 7461 a *note hash ta │ │ │ │ -00054ee0: 626c 653a 2028 4d61 6361 756c 6179 3244 ble: (Macaulay2D │ │ │ │ -00054ef0: 6f63 2948 6173 6854 6162 6c65 2c2c 2067 oc)HashTable,, g │ │ │ │ -00054f00: 6976 6573 2074 6865 2068 6f6d 6f74 6f70 ives the homotop │ │ │ │ -00054f10: 790a 2020 2020 2020 2020 6672 6f6d 2046 y. from F │ │ │ │ -00054f20: 5f69 2063 6f72 7265 7370 6f6e 6469 6e67 _i corresponding │ │ │ │ -00054f30: 2074 6f20 665f 6a20 6173 2074 6865 2076 to f_j as the v │ │ │ │ -00054f40: 616c 7565 2024 4823 5c7b 6a2c 695c 7d24 alue $H#\{j,i\}$ │ │ │ │ -00054f50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -00054f60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a53 616d 6520 =========..Same │ │ │ │ -00054f70: 6173 206d 616b 6548 6f6d 6f74 6f70 6965 as makeHomotopie │ │ │ │ -00054f80: 732c 2062 7574 206f 6e6c 7920 636f 6d70 s, but only comp │ │ │ │ -00054f90: 7574 6573 2074 6865 206f 7264 696e 6172 utes the ordinar │ │ │ │ -00054fa0: 7920 686f 6d6f 746f 7069 6573 2c20 6e6f y homotopies, no │ │ │ │ -00054fb0: 7420 7468 650a 6869 6768 6572 206f 6e65 t the.higher one │ │ │ │ -00054fc0: 732e 2055 7365 6420 696e 2065 7874 6572 s. Used in exter │ │ │ │ -00054fd0: 696f 7254 6f72 4d6f 6475 6c65 0a0a 5365 iorTorModule..Se │ │ │ │ -00054fe0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -00054ff0: 0a20 202a 202a 6e6f 7465 206d 616b 6548 . * *note makeH │ │ │ │ -00055000: 6f6d 6f74 6f70 6965 733a 206d 616b 6548 omotopies: makeH │ │ │ │ -00055010: 6f6d 6f74 6f70 6965 732c 202d 2d20 7265 omotopies, -- re │ │ │ │ -00055020: 7475 726e 7320 6120 7379 7374 656d 206f turns a system o │ │ │ │ -00055030: 6620 6869 6768 6572 0a20 2020 2068 6f6d f higher. hom │ │ │ │ -00055040: 6f74 6f70 6965 730a 2020 2a20 2a6e 6f74 otopies. * *not │ │ │ │ -00055050: 6520 6578 7465 7269 6f72 546f 724d 6f64 e exteriorTorMod │ │ │ │ -00055060: 756c 653a 2065 7874 6572 696f 7254 6f72 ule: exteriorTor │ │ │ │ -00055070: 4d6f 6475 6c65 2c20 2d2d 2054 6f72 2061 Module, -- Tor a │ │ │ │ -00055080: 7320 6120 6d6f 6475 6c65 206f 7665 7220 s a module over │ │ │ │ -00055090: 616e 0a20 2020 2065 7874 6572 696f 7220 an. exterior │ │ │ │ -000550a0: 616c 6765 6272 6120 6f72 2062 6967 7261 algebra or bigra │ │ │ │ -000550b0: 6465 6420 616c 6765 6272 610a 0a57 6179 ded algebra..Way │ │ │ │ -000550c0: 7320 746f 2075 7365 206d 616b 6548 6f6d s to use makeHom │ │ │ │ -000550d0: 6f74 6f70 6965 7331 3a0a 3d3d 3d3d 3d3d otopies1:.====== │ │ │ │ -000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000550f0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ -00055100: 6548 6f6d 6f74 6f70 6965 7331 284d 6174 eHomotopies1(Mat │ │ │ │ -00055110: 7269 782c 436f 6d70 6c65 7829 220a 2020 rix,Complex)". │ │ │ │ -00055120: 2a20 226d 616b 6548 6f6d 6f74 6f70 6965 * "makeHomotopie │ │ │ │ -00055130: 7331 284d 6174 7269 782c 436f 6d70 6c65 s1(Matrix,Comple │ │ │ │ -00055140: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -00055150: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -00055160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -00055170: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -00055180: 6d61 6b65 486f 6d6f 746f 7069 6573 313a makeHomotopies1: │ │ │ │ -00055190: 206d 616b 6548 6f6d 6f74 6f70 6965 7331 makeHomotopies1 │ │ │ │ -000551a0: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -000551b0: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -000551c0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -000551d0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +00054d40: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +00054d50: 6765 3a20 0a20 2020 2020 2020 2048 203d ge: . H = │ │ │ │ +00054d60: 206d 616b 6548 6f6d 6f74 6f70 6965 7331 makeHomotopies1 │ │ │ │ +00054d70: 2866 2c46 2c64 290a 2020 2a20 496e 7075 (f,F,d). * Inpu │ │ │ │ +00054d80: 7473 3a0a 2020 2020 2020 2a20 662c 2061 ts:. * f, a │ │ │ │ +00054d90: 202a 6e6f 7465 206d 6174 7269 783a 2028 *note matrix: ( │ │ │ │ +00054da0: 4d61 6361 756c 6179 3244 6f63 294d 6174 Macaulay2Doc)Mat │ │ │ │ +00054db0: 7269 782c 2c20 3178 6e20 6d61 7472 6978 rix,, 1xn matrix │ │ │ │ +00054dc0: 206f 6620 656c 656d 656e 7473 206f 6620 of elements of │ │ │ │ +00054dd0: 530a 2020 2020 2020 2a20 462c 2061 202a S. * F, a * │ │ │ │ +00054de0: 6e6f 7465 2063 6f6d 706c 6578 3a20 2843 note complex: (C │ │ │ │ +00054df0: 6f6d 706c 6578 6573 2943 6f6d 706c 6578 omplexes)Complex │ │ │ │ +00054e00: 2c2c 2061 646d 6974 7469 6e67 2068 6f6d ,, admitting hom │ │ │ │ +00054e10: 6f74 6f70 6965 7320 666f 7220 7468 650a otopies for the. │ │ │ │ +00054e20: 2020 2020 2020 2020 656e 7472 6965 7320 entries │ │ │ │ +00054e30: 6f66 2066 0a20 2020 2020 202a 2064 2c20 of f. * d, │ │ │ │ +00054e40: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +00054e50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00054e60: 5a5a 2c2c 2068 6f77 2066 6172 2062 6163 ZZ,, how far bac │ │ │ │ +00054e70: 6b20 746f 2063 6f6d 7075 7465 2074 6865 k to compute the │ │ │ │ +00054e80: 0a20 2020 2020 2020 2068 6f6d 6f74 6f70 . homotop │ │ │ │ +00054e90: 6965 7320 2864 6566 6175 6c74 7320 746f ies (defaults to │ │ │ │ +00054ea0: 206c 656e 6774 6820 6f66 2046 290a 2020 length of F). │ │ │ │ +00054eb0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00054ec0: 202a 2048 2c20 6120 2a6e 6f74 6520 6861 * H, a *note ha │ │ │ │ +00054ed0: 7368 2074 6162 6c65 3a20 284d 6163 6175 sh table: (Macau │ │ │ │ +00054ee0: 6c61 7932 446f 6329 4861 7368 5461 626c lay2Doc)HashTabl │ │ │ │ +00054ef0: 652c 2c20 6769 7665 7320 7468 6520 686f e,, gives the ho │ │ │ │ +00054f00: 6d6f 746f 7079 0a20 2020 2020 2020 2066 motopy. f │ │ │ │ +00054f10: 726f 6d20 465f 6920 636f 7272 6573 706f rom F_i correspo │ │ │ │ +00054f20: 6e64 696e 6720 746f 2066 5f6a 2061 7320 nding to f_j as │ │ │ │ +00054f30: 7468 6520 7661 6c75 6520 2448 235c 7b6a the value $H#\{j │ │ │ │ +00054f40: 2c69 5c7d 240a 0a44 6573 6372 6970 7469 ,i\}$..Descripti │ │ │ │ +00054f50: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +00054f60: 5361 6d65 2061 7320 6d61 6b65 486f 6d6f Same as makeHomo │ │ │ │ +00054f70: 746f 7069 6573 2c20 6275 7420 6f6e 6c79 topies, but only │ │ │ │ +00054f80: 2063 6f6d 7075 7465 7320 7468 6520 6f72 computes the or │ │ │ │ +00054f90: 6469 6e61 7279 2068 6f6d 6f74 6f70 6965 dinary homotopie │ │ │ │ +00054fa0: 732c 206e 6f74 2074 6865 0a68 6967 6865 s, not the.highe │ │ │ │ +00054fb0: 7220 6f6e 6573 2e20 5573 6564 2069 6e20 r ones. Used in │ │ │ │ +00054fc0: 6578 7465 7269 6f72 546f 724d 6f64 756c exteriorTorModul │ │ │ │ +00054fd0: 650a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d e..See also.==== │ │ │ │ +00054fe0: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00054ff0: 6d61 6b65 486f 6d6f 746f 7069 6573 3a20 makeHomotopies: │ │ │ │ +00055000: 6d61 6b65 486f 6d6f 746f 7069 6573 2c20 makeHomotopies, │ │ │ │ +00055010: 2d2d 2072 6574 7572 6e73 2061 2073 7973 -- returns a sys │ │ │ │ +00055020: 7465 6d20 6f66 2068 6967 6865 720a 2020 tem of higher. │ │ │ │ +00055030: 2020 686f 6d6f 746f 7069 6573 0a20 202a homotopies. * │ │ │ │ +00055040: 202a 6e6f 7465 2065 7874 6572 696f 7254 *note exteriorT │ │ │ │ +00055050: 6f72 4d6f 6475 6c65 3a20 6578 7465 7269 orModule: exteri │ │ │ │ +00055060: 6f72 546f 724d 6f64 756c 652c 202d 2d20 orTorModule, -- │ │ │ │ +00055070: 546f 7220 6173 2061 206d 6f64 756c 6520 Tor as a module │ │ │ │ +00055080: 6f76 6572 2061 6e0a 2020 2020 6578 7465 over an. exte │ │ │ │ +00055090: 7269 6f72 2061 6c67 6562 7261 206f 7220 rior algebra or │ │ │ │ +000550a0: 6269 6772 6164 6564 2061 6c67 6562 7261 bigraded algebra │ │ │ │ +000550b0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ +000550c0: 6b65 486f 6d6f 746f 7069 6573 313a 0a3d keHomotopies1:.= │ │ │ │ +000550d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000550e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +000550f0: 2022 6d61 6b65 486f 6d6f 746f 7069 6573 "makeHomotopies │ │ │ │ +00055100: 3128 4d61 7472 6978 2c43 6f6d 706c 6578 1(Matrix,Complex │ │ │ │ +00055110: 2922 0a20 202a 2022 6d61 6b65 486f 6d6f )". * "makeHomo │ │ │ │ +00055120: 746f 7069 6573 3128 4d61 7472 6978 2c43 topies1(Matrix,C │ │ │ │ +00055130: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +00055140: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +00055150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00055160: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +00055170: 6e6f 7465 206d 616b 6548 6f6d 6f74 6f70 note makeHomotop │ │ │ │ +00055180: 6965 7331 3a20 6d61 6b65 486f 6d6f 746f ies1: makeHomoto │ │ │ │ +00055190: 7069 6573 312c 2069 7320 6120 2a6e 6f74 pies1, is a *not │ │ │ │ +000551a0: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +000551b0: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +000551c0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +000551d0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 000551e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000551f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00055220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00055230: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00055240: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00055250: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00055260: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -00055270: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -00055280: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -00055290: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -000552a0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000552b0: 7469 6f6e 732e 6d32 3a33 3830 313a 302e tions.m2:3801:0. │ │ │ │ -000552c0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -000552d0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -000552e0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -000552f0: 6f64 653a 206d 616b 6548 6f6d 6f74 6f70 ode: makeHomotop │ │ │ │ -00055300: 6965 734f 6e48 6f6d 6f6c 6f67 792c 204e iesOnHomology, N │ │ │ │ -00055310: 6578 743a 206d 616b 654d 6f64 756c 652c ext: makeModule, │ │ │ │ -00055320: 2050 7265 763a 206d 616b 6548 6f6d 6f74 Prev: makeHomot │ │ │ │ -00055330: 6f70 6965 7331 2c20 5570 3a20 546f 700a opies1, Up: Top. │ │ │ │ -00055340: 0a6d 616b 6548 6f6d 6f74 6f70 6965 734f .makeHomotopiesO │ │ │ │ -00055350: 6e48 6f6d 6f6c 6f67 7920 2d2d 2048 6f6d nHomology -- Hom │ │ │ │ -00055360: 6f6c 6f67 7920 6f66 2061 2063 6f6d 706c ology of a compl │ │ │ │ -00055370: 6578 2061 7320 6578 7465 7269 6f72 206d ex as exterior m │ │ │ │ -00055380: 6f64 756c 650a 2a2a 2a2a 2a2a 2a2a 2a2a odule.********** │ │ │ │ +00055220: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00055230: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00055240: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00055250: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00055260: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +00055270: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00055280: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +00055290: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +000552a0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338 esolutions.m2:38 │ │ │ │ +000552b0: 3031 3a30 2e0a 1f0a 4669 6c65 3a20 436f 01:0....File: Co │ │ │ │ +000552c0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +000552d0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +000552e0: 666f 2c20 4e6f 6465 3a20 6d61 6b65 486f fo, Node: makeHo │ │ │ │ +000552f0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f motopiesOnHomolo │ │ │ │ +00055300: 6779 2c20 4e65 7874 3a20 6d61 6b65 4d6f gy, Next: makeMo │ │ │ │ +00055310: 6475 6c65 2c20 5072 6576 3a20 6d61 6b65 dule, Prev: make │ │ │ │ +00055320: 486f 6d6f 746f 7069 6573 312c 2055 703a Homotopies1, Up: │ │ │ │ +00055330: 2054 6f70 0a0a 6d61 6b65 486f 6d6f 746f Top..makeHomoto │ │ │ │ +00055340: 7069 6573 4f6e 486f 6d6f 6c6f 6779 202d piesOnHomology - │ │ │ │ +00055350: 2d20 486f 6d6f 6c6f 6779 206f 6620 6120 - Homology of a │ │ │ │ +00055360: 636f 6d70 6c65 7820 6173 2065 7874 6572 complex as exter │ │ │ │ +00055370: 696f 7220 6d6f 6475 6c65 0a2a 2a2a 2a2a ior module.***** │ │ │ │ +00055380: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00055390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000553a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000553c0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -000553d0: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -000553e0: 2848 2c68 2920 3d20 6d61 6b65 486f 6d6f (H,h) = makeHomo │ │ │ │ -000553f0: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779 topiesOnHomology │ │ │ │ -00055400: 2866 662c 2043 290a 2020 2a20 496e 7075 (ff, C). * Inpu │ │ │ │ -00055410: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ -00055420: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -00055430: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -00055440: 7472 6978 2c2c 206d 6174 7269 7820 6f66 trix,, matrix of │ │ │ │ -00055450: 2065 6c65 6d65 6e74 7320 686f 6d6f 746f elements homoto │ │ │ │ -00055460: 7069 630a 2020 2020 2020 2020 746f 2030 pic. to 0 │ │ │ │ -00055470: 206f 6e20 430a 2020 2020 2020 2a20 432c on C. * C, │ │ │ │ -00055480: 2061 202a 6e6f 7465 2063 6f6d 706c 6578 a *note complex │ │ │ │ -00055490: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d : (Complexes)Com │ │ │ │ -000554a0: 706c 6578 2c2c 200a 2020 2a20 4f75 7470 plex,, . * Outp │ │ │ │ -000554b0: 7574 733a 0a20 2020 2020 202a 2048 2c20 uts:. * H, │ │ │ │ -000554c0: 6120 2a6e 6f74 6520 6861 7368 2074 6162 a *note hash tab │ │ │ │ -000554d0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -000554e0: 6329 4861 7368 5461 626c 652c 2c20 486f c)HashTable,, Ho │ │ │ │ -000554f0: 6d6f 6c6f 6779 206f 6620 432c 2069 6e64 mology of C, ind │ │ │ │ -00055500: 6578 6564 0a20 2020 2020 2020 2062 7920 exed. by │ │ │ │ -00055510: 706c 6163 6573 2069 6e20 7468 6520 430a places in the C. │ │ │ │ -00055520: 2020 2020 2020 2a20 682c 2061 202a 6e6f * h, a *no │ │ │ │ -00055530: 7465 2068 6173 6820 7461 626c 653a 2028 te hash table: ( │ │ │ │ -00055540: 4d61 6361 756c 6179 3244 6f63 2948 6173 Macaulay2Doc)Has │ │ │ │ -00055550: 6854 6162 6c65 2c2c 2068 6f6d 6f74 6f70 hTable,, homotop │ │ │ │ -00055560: 6965 7320 666f 720a 2020 2020 2020 2020 ies for. │ │ │ │ -00055570: 656c 656d 656e 7473 206f 6620 6620 6f6e elements of f on │ │ │ │ -00055580: 2074 6865 2068 6f6d 6f6c 6f67 7920 6f66 the homology of │ │ │ │ -00055590: 2043 0a0a 4465 7363 7269 7074 696f 6e0a C..Description. │ │ │ │ -000555a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -000555b0: 2073 6372 6970 7420 6361 6c6c 7320 6d61 script calls ma │ │ │ │ -000555c0: 6b65 486f 6d6f 746f 7069 6573 3120 746f keHomotopies1 to │ │ │ │ -000555d0: 2070 726f 6475 6365 2068 6f6d 6f74 6f70 produce homotop │ │ │ │ -000555e0: 6965 7320 666f 7220 7468 6520 6666 5f69 ies for the ff_i │ │ │ │ -000555f0: 206f 6e20 432c 2061 6e64 0a74 6865 6e20 on C, and.then │ │ │ │ -00055600: 636f 6d70 7574 6573 2074 6865 6972 2061 computes their a │ │ │ │ -00055610: 6374 696f 6e20 6f6e 2074 6865 2048 6f6d ction on the Hom │ │ │ │ -00055620: 6f6c 6f67 7920 6f66 2043 2e0a 0a53 6565 ology of C...See │ │ │ │ -00055630: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00055640: 2020 2a20 2a6e 6f74 6520 6578 7465 7269 * *note exteri │ │ │ │ -00055650: 6f72 546f 724d 6f64 756c 653a 2065 7874 orTorModule: ext │ │ │ │ -00055660: 6572 696f 7254 6f72 4d6f 6475 6c65 2c20 eriorTorModule, │ │ │ │ -00055670: 2d2d 2054 6f72 2061 7320 6120 6d6f 6475 -- Tor as a modu │ │ │ │ -00055680: 6c65 206f 7665 7220 616e 0a20 2020 2065 le over an. e │ │ │ │ -00055690: 7874 6572 696f 7220 616c 6765 6272 6120 xterior algebra │ │ │ │ -000556a0: 6f72 2062 6967 7261 6465 6420 616c 6765 or bigraded alge │ │ │ │ -000556b0: 6272 610a 2020 2a20 2a6e 6f74 6520 6578 bra. * *note ex │ │ │ │ -000556c0: 7465 7269 6f72 4578 744d 6f64 756c 653a teriorExtModule: │ │ │ │ -000556d0: 2065 7874 6572 696f 7245 7874 4d6f 6475 exteriorExtModu │ │ │ │ -000556e0: 6c65 2c20 2d2d 2045 7874 284d 2c6b 2920 le, -- Ext(M,k) │ │ │ │ -000556f0: 6f72 2045 7874 284d 2c4e 2920 6173 2061 or Ext(M,N) as a │ │ │ │ -00055700: 0a20 2020 206d 6f64 756c 6520 6f76 6572 . module over │ │ │ │ -00055710: 2061 6e20 6578 7465 7269 6f72 2061 6c67 an exterior alg │ │ │ │ -00055720: 6562 7261 0a0a 5761 7973 2074 6f20 7573 ebra..Ways to us │ │ │ │ -00055730: 6520 6d61 6b65 486f 6d6f 746f 7069 6573 e makeHomotopies │ │ │ │ -00055740: 4f6e 486f 6d6f 6c6f 6779 3a0a 3d3d 3d3d OnHomology:.==== │ │ │ │ +000553b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +000553c0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +000553d0: 2020 2020 2028 482c 6829 203d 206d 616b (H,h) = mak │ │ │ │ +000553e0: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d eHomotopiesOnHom │ │ │ │ +000553f0: 6f6c 6f67 7928 6666 2c20 4329 0a20 202a ology(ff, C). * │ │ │ │ +00055400: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ +00055410: 2066 662c 2061 202a 6e6f 7465 206d 6174 ff, a *note mat │ │ │ │ +00055420: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +00055430: 6f63 294d 6174 7269 782c 2c20 6d61 7472 oc)Matrix,, matr │ │ │ │ +00055440: 6978 206f 6620 656c 656d 656e 7473 2068 ix of elements h │ │ │ │ +00055450: 6f6d 6f74 6f70 6963 0a20 2020 2020 2020 omotopic. │ │ │ │ +00055460: 2074 6f20 3020 6f6e 2043 0a20 2020 2020 to 0 on C. │ │ │ │ +00055470: 202a 2043 2c20 6120 2a6e 6f74 6520 636f * C, a *note co │ │ │ │ +00055480: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ +00055490: 7329 436f 6d70 6c65 782c 2c20 0a20 202a s)Complex,, . * │ │ │ │ +000554a0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +000554b0: 2a20 482c 2061 202a 6e6f 7465 2068 6173 * H, a *note has │ │ │ │ +000554c0: 6820 7461 626c 653a 2028 4d61 6361 756c h table: (Macaul │ │ │ │ +000554d0: 6179 3244 6f63 2948 6173 6854 6162 6c65 ay2Doc)HashTable │ │ │ │ +000554e0: 2c2c 2048 6f6d 6f6c 6f67 7920 6f66 2043 ,, Homology of C │ │ │ │ +000554f0: 2c20 696e 6465 7865 640a 2020 2020 2020 , indexed. │ │ │ │ +00055500: 2020 6279 2070 6c61 6365 7320 696e 2074 by places in t │ │ │ │ +00055510: 6865 2043 0a20 2020 2020 202a 2068 2c20 he C. * h, │ │ │ │ +00055520: 6120 2a6e 6f74 6520 6861 7368 2074 6162 a *note hash tab │ │ │ │ +00055530: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ +00055540: 6329 4861 7368 5461 626c 652c 2c20 686f c)HashTable,, ho │ │ │ │ +00055550: 6d6f 746f 7069 6573 2066 6f72 0a20 2020 motopies for. │ │ │ │ +00055560: 2020 2020 2065 6c65 6d65 6e74 7320 6f66 elements of │ │ │ │ +00055570: 2066 206f 6e20 7468 6520 686f 6d6f 6c6f f on the homolo │ │ │ │ +00055580: 6779 206f 6620 430a 0a44 6573 6372 6970 gy of C..Descrip │ │ │ │ +00055590: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ +000555a0: 0a0a 5468 6520 7363 7269 7074 2063 616c ..The script cal │ │ │ │ +000555b0: 6c73 206d 616b 6548 6f6d 6f74 6f70 6965 ls makeHomotopie │ │ │ │ +000555c0: 7331 2074 6f20 7072 6f64 7563 6520 686f s1 to produce ho │ │ │ │ +000555d0: 6d6f 746f 7069 6573 2066 6f72 2074 6865 motopies for the │ │ │ │ +000555e0: 2066 665f 6920 6f6e 2043 2c20 616e 640a ff_i on C, and. │ │ │ │ +000555f0: 7468 656e 2063 6f6d 7075 7465 7320 7468 then computes th │ │ │ │ +00055600: 6569 7220 6163 7469 6f6e 206f 6e20 7468 eir action on th │ │ │ │ +00055610: 6520 486f 6d6f 6c6f 6779 206f 6620 432e e Homology of C. │ │ │ │ +00055620: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00055630: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2065 ===.. * *note e │ │ │ │ +00055640: 7874 6572 696f 7254 6f72 4d6f 6475 6c65 xteriorTorModule │ │ │ │ +00055650: 3a20 6578 7465 7269 6f72 546f 724d 6f64 : exteriorTorMod │ │ │ │ +00055660: 756c 652c 202d 2d20 546f 7220 6173 2061 ule, -- Tor as a │ │ │ │ +00055670: 206d 6f64 756c 6520 6f76 6572 2061 6e0a module over an. │ │ │ │ +00055680: 2020 2020 6578 7465 7269 6f72 2061 6c67 exterior alg │ │ │ │ +00055690: 6562 7261 206f 7220 6269 6772 6164 6564 ebra or bigraded │ │ │ │ +000556a0: 2061 6c67 6562 7261 0a20 202a 202a 6e6f algebra. * *no │ │ │ │ +000556b0: 7465 2065 7874 6572 696f 7245 7874 4d6f te exteriorExtMo │ │ │ │ +000556c0: 6475 6c65 3a20 6578 7465 7269 6f72 4578 dule: exteriorEx │ │ │ │ +000556d0: 744d 6f64 756c 652c 202d 2d20 4578 7428 tModule, -- Ext( │ │ │ │ +000556e0: 4d2c 6b29 206f 7220 4578 7428 4d2c 4e29 M,k) or Ext(M,N) │ │ │ │ +000556f0: 2061 7320 610a 2020 2020 6d6f 6475 6c65 as a. module │ │ │ │ +00055700: 206f 7665 7220 616e 2065 7874 6572 696f over an exterio │ │ │ │ +00055710: 7220 616c 6765 6272 610a 0a57 6179 7320 r algebra..Ways │ │ │ │ +00055720: 746f 2075 7365 206d 616b 6548 6f6d 6f74 to use makeHomot │ │ │ │ +00055730: 6f70 6965 734f 6e48 6f6d 6f6c 6f67 793a opiesOnHomology: │ │ │ │ +00055740: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ 00055750: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00055760: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00055770: 3d0a 0a20 202a 2022 6d61 6b65 486f 6d6f =.. * "makeHomo │ │ │ │ -00055780: 746f 7069 6573 4f6e 486f 6d6f 6c6f 6779 topiesOnHomology │ │ │ │ -00055790: 284d 6174 7269 782c 436f 6d70 6c65 7829 (Matrix,Complex) │ │ │ │ -000557a0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ -000557b0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ -000557c0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ -000557d0: 6a65 6374 202a 6e6f 7465 206d 616b 6548 ject *note makeH │ │ │ │ -000557e0: 6f6d 6f74 6f70 6965 734f 6e48 6f6d 6f6c omotopiesOnHomol │ │ │ │ -000557f0: 6f67 793a 206d 616b 6548 6f6d 6f74 6f70 ogy: makeHomotop │ │ │ │ -00055800: 6965 734f 6e48 6f6d 6f6c 6f67 792c 2069 iesOnHomology, i │ │ │ │ -00055810: 7320 6120 2a6e 6f74 650a 6d65 7468 6f64 s a *note.method │ │ │ │ -00055820: 2066 756e 6374 696f 6e3a 2028 4d61 6361 function: (Maca │ │ │ │ -00055830: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00055840: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +00055760: 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d 616b ======.. * "mak │ │ │ │ +00055770: 6548 6f6d 6f74 6f70 6965 734f 6e48 6f6d eHomotopiesOnHom │ │ │ │ +00055780: 6f6c 6f67 7928 4d61 7472 6978 2c43 6f6d ology(Matrix,Com │ │ │ │ +00055790: 706c 6578 2922 0a0a 466f 7220 7468 6520 plex)"..For the │ │ │ │ +000557a0: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ +000557b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ +000557c0: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ +000557d0: 6d61 6b65 486f 6d6f 746f 7069 6573 4f6e makeHomotopiesOn │ │ │ │ +000557e0: 486f 6d6f 6c6f 6779 3a20 6d61 6b65 486f Homology: makeHo │ │ │ │ +000557f0: 6d6f 746f 7069 6573 4f6e 486f 6d6f 6c6f motopiesOnHomolo │ │ │ │ +00055800: 6779 2c20 6973 2061 202a 6e6f 7465 0a6d gy, is a *note.m │ │ │ │ +00055810: 6574 686f 6420 6675 6e63 7469 6f6e 3a20 ethod function: │ │ │ │ +00055820: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00055830: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +00055840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00055870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00055890: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ -000558a0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -000558b0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -000558c0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -000558d0: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -000558e0: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -000558f0: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00055900: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ -00055910: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00055920: 6e73 2e6d 323a 3236 3934 3a30 2e0a 1f0a ns.m2:2694:0.... │ │ │ │ -00055930: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ -00055940: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -00055950: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ -00055960: 3a20 6d61 6b65 4d6f 6475 6c65 2c20 4e65 : makeModule, Ne │ │ │ │ -00055970: 7874 3a20 6d61 6b65 542c 2050 7265 763a xt: makeT, Prev: │ │ │ │ -00055980: 206d 616b 6548 6f6d 6f74 6f70 6965 734f makeHomotopiesO │ │ │ │ -00055990: 6e48 6f6d 6f6c 6f67 792c 2055 703a 2054 nHomology, Up: T │ │ │ │ -000559a0: 6f70 0a0a 6d61 6b65 4d6f 6475 6c65 202d op..makeModule - │ │ │ │ -000559b0: 2d20 6d61 6b65 7320 6120 4d6f 6475 6c65 - makes a Module │ │ │ │ -000559c0: 206f 7574 206f 6620 6120 636f 6c6c 6563 out of a collec │ │ │ │ -000559d0: 7469 6f6e 206f 6620 6d6f 6475 6c65 7320 tion of modules │ │ │ │ -000559e0: 616e 6420 6d61 7073 0a2a 2a2a 2a2a 2a2a and maps.******* │ │ │ │ +00055880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ +00055890: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ +000558a0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ +000558b0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ +000558c0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ +000558d0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +000558e0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +000558f0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ +00055900: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00055910: 6c75 7469 6f6e 732e 6d32 3a32 3639 343a lutions.m2:2694: │ │ │ │ +00055920: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ +00055930: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +00055940: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ +00055950: 204e 6f64 653a 206d 616b 654d 6f64 756c Node: makeModul │ │ │ │ +00055960: 652c 204e 6578 743a 206d 616b 6554 2c20 e, Next: makeT, │ │ │ │ +00055970: 5072 6576 3a20 6d61 6b65 486f 6d6f 746f Prev: makeHomoto │ │ │ │ +00055980: 7069 6573 4f6e 486f 6d6f 6c6f 6779 2c20 piesOnHomology, │ │ │ │ +00055990: 5570 3a20 546f 700a 0a6d 616b 654d 6f64 Up: Top..makeMod │ │ │ │ +000559a0: 756c 6520 2d2d 206d 616b 6573 2061 204d ule -- makes a M │ │ │ │ +000559b0: 6f64 756c 6520 6f75 7420 6f66 2061 2063 odule out of a c │ │ │ │ +000559c0: 6f6c 6c65 6374 696f 6e20 6f66 206d 6f64 ollection of mod │ │ │ │ +000559d0: 756c 6573 2061 6e64 206d 6170 730a 2a2a ules and maps.** │ │ │ │ +000559e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000559f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00055a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00055a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00055a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ -00055a30: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ -00055a40: 2020 204d 203d 206d 616b 654d 6f64 756c M = makeModul │ │ │ │ -00055a50: 6528 482c 452c 7068 6929 0a20 202a 2049 e(H,E,phi). * I │ │ │ │ -00055a60: 6e70 7574 733a 0a20 2020 2020 202a 2048 nputs:. * H │ │ │ │ -00055a70: 2c20 6120 2a6e 6f74 6520 6861 7368 2074 , a *note hash t │ │ │ │ -00055a80: 6162 6c65 3a20 284d 6163 6175 6c61 7932 able: (Macaulay2 │ │ │ │ -00055a90: 446f 6329 4861 7368 5461 626c 652c 2c20 Doc)HashTable,, │ │ │ │ -00055aa0: 6772 6164 6564 2063 6f6d 706f 6e65 6e74 graded component │ │ │ │ -00055ab0: 7320 7468 6174 0a20 2020 2020 2020 2061 s that. a │ │ │ │ -00055ac0: 7265 206d 6f64 756c 6573 2c20 746f 206d re modules, to m │ │ │ │ -00055ad0: 616b 6520 696e 746f 2061 7320 7369 6e67 ake into as sing │ │ │ │ -00055ae0: 6c65 206d 6f64 756c 650a 2020 2020 2020 le module. │ │ │ │ -00055af0: 2a20 452c 2061 202a 6e6f 7465 206d 6174 * E, a *note mat │ │ │ │ -00055b00: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ -00055b10: 6f63 294d 6174 7269 782c 2c20 4d61 7472 oc)Matrix,, Matr │ │ │ │ -00055b20: 6978 206f 6620 7661 7269 6162 6c65 7320 ix of variables │ │ │ │ -00055b30: 7768 6f73 650a 2020 2020 2020 2020 6163 whose. ac │ │ │ │ -00055b40: 7469 6f6e 2077 696c 6c20 6465 6669 6e65 tion will define │ │ │ │ -00055b50: 640a 2020 2020 2020 2a20 7068 692c 2061 d. * phi, a │ │ │ │ -00055b60: 202a 6e6f 7465 2068 6173 6820 7461 626c *note hash tabl │ │ │ │ -00055b70: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -00055b80: 2948 6173 6854 6162 6c65 2c2c 206d 6170 )HashTable,, map │ │ │ │ -00055b90: 7320 6265 7477 6565 6e20 7468 650a 2020 s between the. │ │ │ │ -00055ba0: 2020 2020 2020 6772 6164 6564 2063 6f6d graded com │ │ │ │ -00055bb0: 706f 6e65 6e74 7320 7468 6174 2077 696c ponents that wil │ │ │ │ -00055bc0: 6c20 6265 2074 6865 2061 6374 696f 6e20 l be the action │ │ │ │ -00055bd0: 6f66 2074 6865 2076 6172 6961 626c 6573 of the variables │ │ │ │ -00055be0: 2069 6e20 450a 2020 2a20 4f75 7470 7574 in E. * Output │ │ │ │ -00055bf0: 733a 0a20 2020 2020 202a 204d 2c20 6120 s:. * M, a │ │ │ │ -00055c00: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ -00055c10: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ -00055c20: 6c65 2c2c 2067 7261 6465 6420 6d6f 6475 le,, graded modu │ │ │ │ -00055c30: 6c65 7320 7768 6f73 650a 2020 2020 2020 les whose. │ │ │ │ -00055c40: 2020 636f 6d70 6f6e 656e 7473 2061 7265 components are │ │ │ │ -00055c50: 2067 6976 656e 2062 7920 480a 0a44 6573 given by H..Des │ │ │ │ -00055c60: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ -00055c70: 3d3d 3d3d 0a0a 5468 6520 4861 7368 7461 ====..The Hashta │ │ │ │ -00055c80: 626c 6520 4820 7368 6f75 6c64 2068 6176 ble H should hav │ │ │ │ -00055c90: 6520 636f 6e73 6563 7574 6976 6520 696e e consecutive in │ │ │ │ -00055ca0: 7465 6765 7220 6b65 7973 2069 5f30 2e2e teger keys i_0.. │ │ │ │ -00055cb0: 695f 302c 2073 6179 2c20 7769 7468 2076 i_0, say, with v │ │ │ │ -00055cc0: 616c 7565 730a 4823 6920 7468 6174 2061 alues.H#i that a │ │ │ │ -00055cd0: 7265 206d 6f64 756c 6573 206f 7665 7220 re modules over │ │ │ │ -00055ce0: 6120 7269 6e67 2053 4520 7768 6f73 6520 a ring SE whose │ │ │ │ -00055cf0: 7661 7269 6162 6c65 7320 696e 636c 7564 variables includ │ │ │ │ -00055d00: 6520 7468 6520 656c 656d 656e 7473 206f e the elements o │ │ │ │ -00055d10: 6620 452e 0a45 3a20 5c6f 706c 7573 2053 f E..E: \oplus S │ │ │ │ -00055d20: 455e 7b64 5f69 7d20 5c74 6f20 5345 5e31 E^{d_i} \to SE^1 │ │ │ │ -00055d30: 2069 7320 6120 6d61 7472 6978 206f 6620 is a matrix of │ │ │ │ -00055d40: 6320 7661 7269 6162 6c65 7320 6672 6f6d c variables from │ │ │ │ -00055d50: 2053 4520 4820 6973 2061 2068 6173 6854 SE H is a hashT │ │ │ │ -00055d60: 6162 6c65 0a6f 6620 6d20 7061 6972 7320 able.of m pairs │ │ │ │ -00055d70: 7b69 2c20 745f 697d 2c20 7768 6572 6520 {i, t_i}, where │ │ │ │ -00055d80: 7468 6520 745f 6920 6172 6520 5245 2d6d the t_i are RE-m │ │ │ │ -00055d90: 6f64 756c 6573 2c20 616e 6420 7468 6520 odules, and the │ │ │ │ -00055da0: 6920 6172 6520 636f 6e73 6563 7574 6976 i are consecutiv │ │ │ │ -00055db0: 650a 696e 7465 6765 722e 2070 6869 2069 e.integer. phi i │ │ │ │ -00055dc0: 7320 6120 6861 7368 2d74 6162 6c65 206f s a hash-table o │ │ │ │ -00055dd0: 6620 686f 6d6f 6765 6e65 6f75 7320 6d61 f homogeneous ma │ │ │ │ -00055de0: 7073 2070 6869 237b 6a2c 697d 3a20 4823 ps phi#{j,i}: H# │ │ │ │ -00055df0: 692a 2a46 5f6a 5c74 6f20 4823 2869 2b31 i**F_j\to H#(i+1 │ │ │ │ -00055e00: 290a 7768 6572 6520 465f 6a20 3d20 736f ).where F_j = so │ │ │ │ -00055e10: 7572 6365 2028 455f 7b6a 7d20 3d20 6d61 urce (E_{j} = ma │ │ │ │ -00055e20: 7472 6978 207b 7b65 5f6a 7d7d 292e 2054 trix {{e_j}}). T │ │ │ │ -00055e30: 6875 7320 7468 6520 6d61 7073 2070 237b hus the maps p#{ │ │ │ │ -00055e40: 6a2c 697d 203d 2028 455f 6a20 7c7c 0a2d j,i} = (E_j ||.- │ │ │ │ -00055e50: 7068 6923 7b6a 2c69 7d29 3a20 745f 692a phi#{j,i}): t_i* │ │ │ │ -00055e60: 2a46 5f6a 205c 746f 2074 5f69 2b2b 745f *F_j \to t_i++t_ │ │ │ │ -00055e70: 7b28 692b 3129 7d2c 2061 7265 2068 6f6d {(i+1)}, are hom │ │ │ │ -00055e80: 6f67 656e 656f 7573 2e20 5468 6520 7363 ogeneous. The sc │ │ │ │ -00055e90: 7269 7074 2072 6574 7572 6e73 204d 0a3d ript returns M.= │ │ │ │ -00055ea0: 205c 6f70 6c75 735f 6920 545f 2061 7320 \oplus_i T_ as │ │ │ │ -00055eb0: 616e 2053 452d 6d6f 6475 6c65 2c20 636f an SE-module, co │ │ │ │ -00055ec0: 6d70 7574 6564 2061 7320 7468 6520 7175 mputed as the qu │ │ │ │ -00055ed0: 6f74 6965 6e74 206f 6620 5020 3a3d 205c otient of P := \ │ │ │ │ -00055ee0: 6f70 6c75 7320 545f 690a 6f62 7461 696e oplus T_i.obtain │ │ │ │ -00055ef0: 6564 2062 7920 6661 6374 6f72 696e 6720 ed by factoring │ │ │ │ -00055f00: 6f75 7420 7468 6520 7375 6d20 6f66 2074 out the sum of t │ │ │ │ -00055f10: 6865 2069 6d61 6765 7320 6f66 2074 6865 he images of the │ │ │ │ -00055f20: 206d 6170 7320 7023 7b6a 2c69 7d0a 0a54 maps p#{j,i}..T │ │ │ │ -00055f30: 6865 2048 6173 6874 6162 6c65 2070 6869 he Hashtable phi │ │ │ │ -00055f40: 2068 6173 206b 6579 7320 6f66 2074 6865 has keys of the │ │ │ │ -00055f50: 2066 6f72 6d20 7b6a 2c69 7d20 7768 6572 form {j,i} wher │ │ │ │ -00055f60: 6520 6a20 7275 6e73 2066 726f 6d20 3020 e j runs from 0 │ │ │ │ -00055f70: 746f 2063 2d31 2c20 6920 616e 640a 692b to c-1, i and.i+ │ │ │ │ -00055f80: 3120 6172 6520 6b65 7973 206f 6620 482c 1 are keys of H, │ │ │ │ -00055f90: 2061 6e64 2070 6869 237b 6a2c 697d 2069 and phi#{j,i} i │ │ │ │ -00055fa0: 7320 7468 6520 6d61 7020 6672 6f6d 2028 s the map from ( │ │ │ │ -00055fb0: 736f 7572 6365 2045 5f7b 697d 292a 2a48 source E_{i})**H │ │ │ │ -00055fc0: 2369 2074 6f20 4823 2869 2b31 290a 7468 #i to H#(i+1).th │ │ │ │ -00055fd0: 6174 2077 696c 6c20 6265 2069 6465 6e74 at will be ident │ │ │ │ -00055fe0: 6966 6965 6420 7769 7468 2074 6865 2061 ified with the a │ │ │ │ -00055ff0: 6374 696f 6e20 6f66 2045 5f7b 6a7d 2e0a ction of E_{j}.. │ │ │ │ -00056000: 0a54 6865 2073 6372 6970 7420 6973 2075 .The script is u │ │ │ │ -00056010: 7365 6420 696e 2062 6f74 6820 7468 6520 sed in both the │ │ │ │ -00056020: 7369 6e67 6c79 2067 7261 6465 6420 6361 singly graded ca │ │ │ │ -00056030: 7365 2c20 666f 7220 6578 616d 706c 6520 se, for example │ │ │ │ -00056040: 696e 0a65 7874 6572 696f 7254 6f72 4d6f in.exteriorTorMo │ │ │ │ -00056050: 6475 6c65 2866 662c 4d29 2061 6e64 2069 dule(ff,M) and i │ │ │ │ -00056060: 6e20 7468 6520 6269 6772 6164 6564 2063 n the bigraded c │ │ │ │ -00056070: 6173 652c 2066 6f72 2065 7861 6d70 6c65 ase, for example │ │ │ │ -00056080: 2069 6e0a 6578 7465 7269 6f72 546f 724d in.exteriorTorM │ │ │ │ -00056090: 6f64 756c 6528 6666 2c4d 2c4e 292e 0a0a odule(ff,M,N)... │ │ │ │ -000560a0: 496e 2074 6865 2066 6f6c 6c6f 7769 6e67 In the following │ │ │ │ -000560b0: 2077 6520 7573 6520 6d61 6b65 4d6f 6475 we use makeModu │ │ │ │ -000560c0: 6c65 2074 6f20 636f 6e73 7472 7563 7420 le to construct │ │ │ │ -000560d0: 6279 2068 616e 6420 6120 6672 6565 206d by hand a free m │ │ │ │ -000560e0: 6f64 756c 6520 6f66 2072 616e 6b20 310a odule of rank 1. │ │ │ │ -000560f0: 6f76 6572 2074 6865 2065 7874 6572 696f over the exterio │ │ │ │ -00056100: 7220 616c 6765 6272 6120 6f6e 2078 2c79 r algebra on x,y │ │ │ │ -00056110: 2c20 7374 6172 7469 6e67 2077 6974 6820 , starting with │ │ │ │ -00056120: 7468 6520 636f 6e73 7472 7563 7469 6f6e the construction │ │ │ │ -00056130: 206f 6620 6120 6d6f 6475 6c65 0a6f 7665 of a module.ove │ │ │ │ -00056140: 7220 6120 6269 686f 6d6f 6765 6e65 6f75 r a bihomogeneou │ │ │ │ -00056150: 7320 7269 6e67 2e0a 0a2b 2d2d 2d2d 2d2d s ring...+------ │ │ │ │ +00055a20: 2a2a 0a0a 2020 2a20 5573 6167 653a 200a **.. * Usage: . │ │ │ │ +00055a30: 2020 2020 2020 2020 4d20 3d20 6d61 6b65 M = make │ │ │ │ +00055a40: 4d6f 6475 6c65 2848 2c45 2c70 6869 290a Module(H,E,phi). │ │ │ │ +00055a50: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +00055a60: 2020 2a20 482c 2061 202a 6e6f 7465 2068 * H, a *note h │ │ │ │ +00055a70: 6173 6820 7461 626c 653a 2028 4d61 6361 ash table: (Maca │ │ │ │ +00055a80: 756c 6179 3244 6f63 2948 6173 6854 6162 ulay2Doc)HashTab │ │ │ │ +00055a90: 6c65 2c2c 2067 7261 6465 6420 636f 6d70 le,, graded comp │ │ │ │ +00055aa0: 6f6e 656e 7473 2074 6861 740a 2020 2020 onents that. │ │ │ │ +00055ab0: 2020 2020 6172 6520 6d6f 6475 6c65 732c are modules, │ │ │ │ +00055ac0: 2074 6f20 6d61 6b65 2069 6e74 6f20 6173 to make into as │ │ │ │ +00055ad0: 2073 696e 676c 6520 6d6f 6475 6c65 0a20 single module. │ │ │ │ +00055ae0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ +00055af0: 6520 6d61 7472 6978 3a20 284d 6163 6175 e matrix: (Macau │ │ │ │ +00055b00: 6c61 7932 446f 6329 4d61 7472 6978 2c2c lay2Doc)Matrix,, │ │ │ │ +00055b10: 204d 6174 7269 7820 6f66 2076 6172 6961 Matrix of varia │ │ │ │ +00055b20: 626c 6573 2077 686f 7365 0a20 2020 2020 bles whose. │ │ │ │ +00055b30: 2020 2061 6374 696f 6e20 7769 6c6c 2064 action will d │ │ │ │ +00055b40: 6566 696e 6564 0a20 2020 2020 202a 2070 efined. * p │ │ │ │ +00055b50: 6869 2c20 6120 2a6e 6f74 6520 6861 7368 hi, a *note hash │ │ │ │ +00055b60: 2074 6162 6c65 3a20 284d 6163 6175 6c61 table: (Macaula │ │ │ │ +00055b70: 7932 446f 6329 4861 7368 5461 626c 652c y2Doc)HashTable, │ │ │ │ +00055b80: 2c20 6d61 7073 2062 6574 7765 656e 2074 , maps between t │ │ │ │ +00055b90: 6865 0a20 2020 2020 2020 2067 7261 6465 he. grade │ │ │ │ +00055ba0: 6420 636f 6d70 6f6e 656e 7473 2074 6861 d components tha │ │ │ │ +00055bb0: 7420 7769 6c6c 2062 6520 7468 6520 6163 t will be the ac │ │ │ │ +00055bc0: 7469 6f6e 206f 6620 7468 6520 7661 7269 tion of the vari │ │ │ │ +00055bd0: 6162 6c65 7320 696e 2045 0a20 202a 204f ables in E. * O │ │ │ │ +00055be0: 7574 7075 7473 3a0a 2020 2020 2020 2a20 utputs:. * │ │ │ │ +00055bf0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ +00055c00: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ +00055c10: 294d 6f64 756c 652c 2c20 6772 6164 6564 )Module,, graded │ │ │ │ +00055c20: 206d 6f64 756c 6573 2077 686f 7365 0a20 modules whose. │ │ │ │ +00055c30: 2020 2020 2020 2063 6f6d 706f 6e65 6e74 component │ │ │ │ +00055c40: 7320 6172 6520 6769 7665 6e20 6279 2048 s are given by H │ │ │ │ +00055c50: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +00055c60: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2048 =========..The H │ │ │ │ +00055c70: 6173 6874 6162 6c65 2048 2073 686f 756c ashtable H shoul │ │ │ │ +00055c80: 6420 6861 7665 2063 6f6e 7365 6375 7469 d have consecuti │ │ │ │ +00055c90: 7665 2069 6e74 6567 6572 206b 6579 7320 ve integer keys │ │ │ │ +00055ca0: 695f 302e 2e69 5f30 2c20 7361 792c 2077 i_0..i_0, say, w │ │ │ │ +00055cb0: 6974 6820 7661 6c75 6573 0a48 2369 2074 ith values.H#i t │ │ │ │ +00055cc0: 6861 7420 6172 6520 6d6f 6475 6c65 7320 hat are modules │ │ │ │ +00055cd0: 6f76 6572 2061 2072 696e 6720 5345 2077 over a ring SE w │ │ │ │ +00055ce0: 686f 7365 2076 6172 6961 626c 6573 2069 hose variables i │ │ │ │ +00055cf0: 6e63 6c75 6465 2074 6865 2065 6c65 6d65 nclude the eleme │ │ │ │ +00055d00: 6e74 7320 6f66 2045 2e0a 453a 205c 6f70 nts of E..E: \op │ │ │ │ +00055d10: 6c75 7320 5345 5e7b 645f 697d 205c 746f lus SE^{d_i} \to │ │ │ │ +00055d20: 2053 455e 3120 6973 2061 206d 6174 7269 SE^1 is a matri │ │ │ │ +00055d30: 7820 6f66 2063 2076 6172 6961 626c 6573 x of c variables │ │ │ │ +00055d40: 2066 726f 6d20 5345 2048 2069 7320 6120 from SE H is a │ │ │ │ +00055d50: 6861 7368 5461 626c 650a 6f66 206d 2070 hashTable.of m p │ │ │ │ +00055d60: 6169 7273 207b 692c 2074 5f69 7d2c 2077 airs {i, t_i}, w │ │ │ │ +00055d70: 6865 7265 2074 6865 2074 5f69 2061 7265 here the t_i are │ │ │ │ +00055d80: 2052 452d 6d6f 6475 6c65 732c 2061 6e64 RE-modules, and │ │ │ │ +00055d90: 2074 6865 2069 2061 7265 2063 6f6e 7365 the i are conse │ │ │ │ +00055da0: 6375 7469 7665 0a69 6e74 6567 6572 2e20 cutive.integer. │ │ │ │ +00055db0: 7068 6920 6973 2061 2068 6173 682d 7461 phi is a hash-ta │ │ │ │ +00055dc0: 626c 6520 6f66 2068 6f6d 6f67 656e 656f ble of homogeneo │ │ │ │ +00055dd0: 7573 206d 6170 7320 7068 6923 7b6a 2c69 us maps phi#{j,i │ │ │ │ +00055de0: 7d3a 2048 2369 2a2a 465f 6a5c 746f 2048 }: H#i**F_j\to H │ │ │ │ +00055df0: 2328 692b 3129 0a77 6865 7265 2046 5f6a #(i+1).where F_j │ │ │ │ +00055e00: 203d 2073 6f75 7263 6520 2845 5f7b 6a7d = source (E_{j} │ │ │ │ +00055e10: 203d 206d 6174 7269 7820 7b7b 655f 6a7d = matrix {{e_j} │ │ │ │ +00055e20: 7d29 2e20 5468 7573 2074 6865 206d 6170 }). Thus the map │ │ │ │ +00055e30: 7320 7023 7b6a 2c69 7d20 3d20 2845 5f6a s p#{j,i} = (E_j │ │ │ │ +00055e40: 207c 7c0a 2d70 6869 237b 6a2c 697d 293a ||.-phi#{j,i}): │ │ │ │ +00055e50: 2074 5f69 2a2a 465f 6a20 5c74 6f20 745f t_i**F_j \to t_ │ │ │ │ +00055e60: 692b 2b74 5f7b 2869 2b31 297d 2c20 6172 i++t_{(i+1)}, ar │ │ │ │ +00055e70: 6520 686f 6d6f 6765 6e65 6f75 732e 2054 e homogeneous. T │ │ │ │ +00055e80: 6865 2073 6372 6970 7420 7265 7475 726e he script return │ │ │ │ +00055e90: 7320 4d0a 3d20 5c6f 706c 7573 5f69 2054 s M.= \oplus_i T │ │ │ │ +00055ea0: 5f20 6173 2061 6e20 5345 2d6d 6f64 756c _ as an SE-modul │ │ │ │ +00055eb0: 652c 2063 6f6d 7075 7465 6420 6173 2074 e, computed as t │ │ │ │ +00055ec0: 6865 2071 756f 7469 656e 7420 6f66 2050 he quotient of P │ │ │ │ +00055ed0: 203a 3d20 5c6f 706c 7573 2054 5f69 0a6f := \oplus T_i.o │ │ │ │ +00055ee0: 6274 6169 6e65 6420 6279 2066 6163 746f btained by facto │ │ │ │ +00055ef0: 7269 6e67 206f 7574 2074 6865 2073 756d ring out the sum │ │ │ │ +00055f00: 206f 6620 7468 6520 696d 6167 6573 206f of the images o │ │ │ │ +00055f10: 6620 7468 6520 6d61 7073 2070 237b 6a2c f the maps p#{j, │ │ │ │ +00055f20: 697d 0a0a 5468 6520 4861 7368 7461 626c i}..The Hashtabl │ │ │ │ +00055f30: 6520 7068 6920 6861 7320 6b65 7973 206f e phi has keys o │ │ │ │ +00055f40: 6620 7468 6520 666f 726d 207b 6a2c 697d f the form {j,i} │ │ │ │ +00055f50: 2077 6865 7265 206a 2072 756e 7320 6672 where j runs fr │ │ │ │ +00055f60: 6f6d 2030 2074 6f20 632d 312c 2069 2061 om 0 to c-1, i a │ │ │ │ +00055f70: 6e64 0a69 2b31 2061 7265 206b 6579 7320 nd.i+1 are keys │ │ │ │ +00055f80: 6f66 2048 2c20 616e 6420 7068 6923 7b6a of H, and phi#{j │ │ │ │ +00055f90: 2c69 7d20 6973 2074 6865 206d 6170 2066 ,i} is the map f │ │ │ │ +00055fa0: 726f 6d20 2873 6f75 7263 6520 455f 7b69 rom (source E_{i │ │ │ │ +00055fb0: 7d29 2a2a 4823 6920 746f 2048 2328 692b })**H#i to H#(i+ │ │ │ │ +00055fc0: 3129 0a74 6861 7420 7769 6c6c 2062 6520 1).that will be │ │ │ │ +00055fd0: 6964 656e 7469 6669 6564 2077 6974 6820 identified with │ │ │ │ +00055fe0: 7468 6520 6163 7469 6f6e 206f 6620 455f the action of E_ │ │ │ │ +00055ff0: 7b6a 7d2e 0a0a 5468 6520 7363 7269 7074 {j}...The script │ │ │ │ +00056000: 2069 7320 7573 6564 2069 6e20 626f 7468 is used in both │ │ │ │ +00056010: 2074 6865 2073 696e 676c 7920 6772 6164 the singly grad │ │ │ │ +00056020: 6564 2063 6173 652c 2066 6f72 2065 7861 ed case, for exa │ │ │ │ +00056030: 6d70 6c65 2069 6e0a 6578 7465 7269 6f72 mple in.exterior │ │ │ │ +00056040: 546f 724d 6f64 756c 6528 6666 2c4d 2920 TorModule(ff,M) │ │ │ │ +00056050: 616e 6420 696e 2074 6865 2062 6967 7261 and in the bigra │ │ │ │ +00056060: 6465 6420 6361 7365 2c20 666f 7220 6578 ded case, for ex │ │ │ │ +00056070: 616d 706c 6520 696e 0a65 7874 6572 696f ample in.exterio │ │ │ │ +00056080: 7254 6f72 4d6f 6475 6c65 2866 662c 4d2c rTorModule(ff,M, │ │ │ │ +00056090: 4e29 2e0a 0a49 6e20 7468 6520 666f 6c6c N)...In the foll │ │ │ │ +000560a0: 6f77 696e 6720 7765 2075 7365 206d 616b owing we use mak │ │ │ │ +000560b0: 654d 6f64 756c 6520 746f 2063 6f6e 7374 eModule to const │ │ │ │ +000560c0: 7275 6374 2062 7920 6861 6e64 2061 2066 ruct by hand a f │ │ │ │ +000560d0: 7265 6520 6d6f 6475 6c65 206f 6620 7261 ree module of ra │ │ │ │ +000560e0: 6e6b 2031 0a6f 7665 7220 7468 6520 6578 nk 1.over the ex │ │ │ │ +000560f0: 7465 7269 6f72 2061 6c67 6562 7261 206f terior algebra o │ │ │ │ +00056100: 6e20 782c 792c 2073 7461 7274 696e 6720 n x,y, starting │ │ │ │ +00056110: 7769 7468 2074 6865 2063 6f6e 7374 7275 with the constru │ │ │ │ +00056120: 6374 696f 6e20 6f66 2061 206d 6f64 756c ction of a modul │ │ │ │ +00056130: 650a 6f76 6572 2061 2062 6968 6f6d 6f67 e.over a bihomog │ │ │ │ +00056140: 656e 656f 7573 2072 696e 672e 0a0a 2b2d eneous ring...+- │ │ │ │ +00056150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000561a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2053 -------+.|i1 : S │ │ │ │ -000561b0: 4520 3d20 5a5a 2f31 3031 5b61 2c62 2c63 E = ZZ/101[a,b,c │ │ │ │ -000561c0: 2c78 2c79 2c44 6567 7265 6573 3d3e 746f ,x,y,Degrees=>to │ │ │ │ -000561d0: 4c69 7374 2833 3a7b 312c 307d 297c 746f List(3:{1,0})|to │ │ │ │ -000561e0: 4c69 7374 2832 3a7b 312c 317d 292c 2020 List(2:{1,1}), │ │ │ │ -000561f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000561a0: 3120 3a20 5345 203d 205a 5a2f 3130 315b 1 : SE = ZZ/101[ │ │ │ │ +000561b0: 612c 622c 632c 782c 792c 4465 6772 6565 a,b,c,x,y,Degree │ │ │ │ +000561c0: 733d 3e74 6f4c 6973 7428 333a 7b31 2c30 s=>toList(3:{1,0 │ │ │ │ +000561d0: 7d29 7c74 6f4c 6973 7428 323a 7b31 2c31 })|toList(2:{1,1 │ │ │ │ +000561e0: 7d29 2c20 2020 2020 2020 2020 7c0a 7c20 }), |.| │ │ │ │ +000561f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056240: 2020 2020 2020 207c 0a7c 6f31 203d 2053 |.|o1 = S │ │ │ │ -00056250: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +00056230: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00056240: 3120 3d20 5345 2020 2020 2020 2020 2020 1 = SE │ │ │ │ +00056250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056290: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056280: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000562a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000562b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000562c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000562d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000562e0: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ -000562f0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2032 olynomialRing, 2 │ │ │ │ -00056300: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ -00056310: 6520 7661 7269 6162 6c65 2873 2920 2020 e variable(s) │ │ │ │ -00056320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056330: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +000562d0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000562e0: 3120 3a20 506f 6c79 6e6f 6d69 616c 5269 1 : PolynomialRi │ │ │ │ +000562f0: 6e67 2c20 3220 736b 6577 2063 6f6d 6d75 ng, 2 skew commu │ │ │ │ +00056300: 7461 7469 7665 2076 6172 6961 626c 6528 tative variable( │ │ │ │ +00056310: 7329 2020 2020 2020 2020 2020 2020 2020 s) │ │ │ │ +00056320: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +00056330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056380: 2d2d 2d2d 2d2d 2d7c 0a7c 536b 6577 436f -------|.|SkewCo │ │ │ │ -00056390: 6d6d 7574 6174 6976 653d 3e7b 782c 797d mmutative=>{x,y} │ │ │ │ -000563a0: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00056370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c53 ------------|.|S │ │ │ │ +00056380: 6b65 7743 6f6d 6d75 7461 7469 7665 3d3e kewCommutative=> │ │ │ │ +00056390: 7b78 2c79 7d5d 2020 2020 2020 2020 2020 {x,y}] │ │ │ │ +000563a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000563b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000563c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000563d0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000563c0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000563d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000563e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000563f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056420: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -00056430: 4520 3d20 5345 2f69 6465 616c 2261 322c E = SE/ideal"a2, │ │ │ │ -00056440: 6232 2c63 3222 2020 2020 2020 2020 2020 b2,c2" │ │ │ │ +00056410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00056420: 3220 3a20 5245 203d 2053 452f 6964 6561 2 : RE = SE/idea │ │ │ │ +00056430: 6c22 6132 2c62 322c 6332 2220 2020 2020 l"a2,b2,c2" │ │ │ │ +00056440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056470: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056460: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000564a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000564b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000564c0: 2020 2020 2020 207c 0a7c 6f32 203d 2052 |.|o2 = R │ │ │ │ -000564d0: 4520 2020 2020 2020 2020 2020 2020 2020 E │ │ │ │ +000564b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000564c0: 3220 3d20 5245 2020 2020 2020 2020 2020 2 = RE │ │ │ │ +000564d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000564e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000564f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056560: 2020 2020 2020 207c 0a7c 6f32 203a 2051 |.|o2 : Q │ │ │ │ -00056570: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00056550: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00056560: 3220 3a20 5175 6f74 6965 6e74 5269 6e67 2 : QuotientRing │ │ │ │ +00056570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000565a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000565b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000565a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000565b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000565c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000565d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000565e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056600: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 2054 -------+.|i3 : T │ │ │ │ -00056610: 203d 2068 6173 6854 6162 6c65 207b 7b30 = hashTable {{0 │ │ │ │ -00056620: 2c52 455e 317d 2c7b 312c 5245 5e7b 323a ,RE^1},{1,RE^{2: │ │ │ │ -00056630: 7b20 2d31 2c2d 317d 7d7d 2c20 7b32 2c52 { -1,-1}}}, {2,R │ │ │ │ -00056640: 455e 7b7b 202d 322c 2d32 7d7d 7d7d 2020 E^{{ -2,-2}}}} │ │ │ │ -00056650: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000565f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00056600: 3320 3a20 5420 3d20 6861 7368 5461 626c 3 : T = hashTabl │ │ │ │ +00056610: 6520 7b7b 302c 5245 5e31 7d2c 7b31 2c52 e {{0,RE^1},{1,R │ │ │ │ +00056620: 455e 7b32 3a7b 202d 312c 2d31 7d7d 7d2c E^{2:{ -1,-1}}}, │ │ │ │ +00056630: 207b 322c 5245 5e7b 7b20 2d32 2c2d 327d {2,RE^{{ -2,-2} │ │ │ │ +00056640: 7d7d 7d20 2020 2020 2020 2020 7c0a 7c20 }}} |.| │ │ │ │ +00056650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000566b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566c0: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +00056690: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000566a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000566b0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +000566c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000566d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000566f0: 2020 2020 2020 207c 0a7c 6f33 203d 2048 |.|o3 = H │ │ │ │ -00056700: 6173 6854 6162 6c65 7b30 203d 3e20 5245 ashTable{0 => RE │ │ │ │ -00056710: 207d 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +000566e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000566f0: 3320 3d20 4861 7368 5461 626c 657b 3020 3 = HashTable{0 │ │ │ │ +00056700: 3d3e 2052 4520 7d20 2020 2020 2020 2020 => RE } │ │ │ │ +00056710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056740: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00056750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056760: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00056730: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00056750: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00056760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056790: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000567a0: 2020 2020 2020 2020 2031 203d 3e20 5245 1 => RE │ │ │ │ +00056780: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056790: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +000567a0: 3d3e 2052 4520 2020 2020 2020 2020 2020 => RE │ │ │ │ 000567b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000567c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000567d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000567e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000567f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056800: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +000567d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000567e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000567f0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00056800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056830: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00056840: 2020 2020 2020 2020 2032 203d 3e20 5245 2 => RE │ │ │ │ +00056820: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056830: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00056840: 3d3e 2052 4520 2020 2020 2020 2020 2020 => RE │ │ │ │ 00056850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056880: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056870: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000568a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000568b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000568c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000568d0: 2020 2020 2020 207c 0a7c 6f33 203a 2048 |.|o3 : H │ │ │ │ -000568e0: 6173 6854 6162 6c65 2020 2020 2020 2020 ashTable │ │ │ │ +000568c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000568d0: 3320 3a20 4861 7368 5461 626c 6520 2020 3 : HashTable │ │ │ │ +000568e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000568f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056920: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00056910: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00056920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00056950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00056970: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2045 -------+.|i4 : E │ │ │ │ -00056980: 203d 206d 6174 7269 787b 7b78 2c79 7d7d = matrix{{x,y}} │ │ │ │ +00056960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00056970: 3420 3a20 4520 3d20 6d61 7472 6978 7b7b 4 : E = matrix{{ │ │ │ │ +00056980: 782c 797d 7d20 2020 2020 2020 2020 2020 x,y}} │ │ │ │ 00056990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000569b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000569c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000569b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000569c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000569f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a10: 2020 2020 2020 207c 0a7c 6f34 203d 207c |.|o4 = | │ │ │ │ -00056a20: 2078 2079 207c 2020 2020 2020 2020 2020 x y | │ │ │ │ +00056a00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00056a10: 3420 3d20 7c20 7820 7920 7c20 2020 2020 4 = | x y | │ │ │ │ +00056a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00056a50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00056a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00056a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00056ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00056ac0: 2020 2020 2020 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2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00057270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00057280: 3720 3a20 6170 706c 7928 6b65 7973 2070 7 : apply(keys p │ │ │ │ +00057290: 6869 2c20 6b2d 3e69 7348 6f6d 6f67 656e hi, k->isHomogen │ │ │ │ +000572a0: 656f 7573 2070 6869 236b 2920 2020 2020 eous phi#k) │ │ │ │ +000572b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000572c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000572d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000572e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000572f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00057300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00057310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00057320: 2020 2020 2020 207c 0a7c 6f37 203d 207b |.|o7 = { │ │ │ │ -00057330: 7472 7565 2c20 7472 7565 2c20 7472 7565 true, true, true │ │ │ │ -00057340: 2c20 7472 7565 7d20 2020 2020 2020 2020 , true} │ │ │ │ +00057310: 2020 2020 2020 2020 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-00057fb0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00057fc0: 202a 6e6f 7465 2065 7874 6572 696f 7248 *note exteriorH │ │ │ │ -00057fd0: 6f6d 6f6c 6f67 794d 6f64 756c 653a 2065 omologyModule: e │ │ │ │ -00057fe0: 7874 6572 696f 7248 6f6d 6f6c 6f67 794d xteriorHomologyM │ │ │ │ -00057ff0: 6f64 756c 652c 202d 2d20 4d61 6b65 2074 odule, -- Make t │ │ │ │ -00058000: 6865 2068 6f6d 6f6c 6f67 790a 2020 2020 he homology. │ │ │ │ -00058010: 6f66 2061 2063 6f6d 706c 6578 2069 6e74 of a complex int │ │ │ │ -00058020: 6f20 6120 6d6f 6475 6c65 206f 7665 7220 o a module over │ │ │ │ -00058030: 616e 2065 7874 6572 696f 7220 616c 6765 an exterior alge │ │ │ │ -00058040: 6272 610a 2020 2a20 2a6e 6f74 6520 6578 bra. * *note ex │ │ │ │ -00058050: 7465 7269 6f72 546f 724d 6f64 756c 653a teriorTorModule: │ │ │ │ -00058060: 2065 7874 6572 696f 7254 6f72 4d6f 6475 exteriorTorModu │ │ │ │ -00058070: 6c65 2c20 2d2d 2054 6f72 2061 7320 6120 le, -- Tor as a │ │ │ │ -00058080: 6d6f 6475 6c65 206f 7665 7220 616e 0a20 module over an. │ │ │ │ -00058090: 2020 2065 7874 6572 696f 7220 616c 6765 exterior alge │ │ │ │ -000580a0: 6272 6120 6f72 2062 6967 7261 6465 6420 bra or bigraded │ │ │ │ -000580b0: 616c 6765 6272 610a 2020 2a20 2a6e 6f74 algebra. * *not │ │ │ │ -000580c0: 6520 6578 7465 7269 6f72 4578 744d 6f64 e exteriorExtMod │ │ │ │ -000580d0: 756c 653a 2065 7874 6572 696f 7245 7874 ule: exteriorExt │ │ │ │ -000580e0: 4d6f 6475 6c65 2c20 2d2d 2045 7874 284d Module, -- Ext(M │ │ │ │ -000580f0: 2c6b 2920 6f72 2045 7874 284d 2c4e 2920 ,k) or Ext(M,N) │ │ │ │ -00058100: 6173 2061 0a20 2020 206d 6f64 756c 6520 as a. module │ │ │ │ -00058110: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ -00058120: 2061 6c67 6562 7261 0a0a 5761 7973 2074 algebra..Ways t │ │ │ │ -00058130: 6f20 7573 6520 6d61 6b65 4d6f 6475 6c65 o use makeModule │ │ │ │ -00058140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -00058150: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ -00058160: 6d61 6b65 4d6f 6475 6c65 2848 6173 6854 makeModule(HashT │ │ │ │ -00058170: 6162 6c65 2c4d 6174 7269 782c 4861 7368 able,Matrix,Hash │ │ │ │ -00058180: 5461 626c 6529 220a 0a46 6f72 2074 6865 Table)"..For the │ │ │ │ -00058190: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -000581a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -000581b0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -000581c0: 206d 616b 654d 6f64 756c 653a 206d 616b makeModule: mak │ │ │ │ -000581d0: 654d 6f64 756c 652c 2069 7320 6120 2a6e eModule, is a *n │ │ │ │ -000581e0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -000581f0: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00058200: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00058210: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00057f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00057fa0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00057fb0: 0a0a 2020 2a20 2a6e 6f74 6520 6578 7465 .. * *note exte │ │ │ │ +00057fc0: 7269 6f72 486f 6d6f 6c6f 6779 4d6f 6475 riorHomologyModu │ │ │ │ +00057fd0: 6c65 3a20 6578 7465 7269 6f72 486f 6d6f le: exteriorHomo │ │ │ │ +00057fe0: 6c6f 6779 4d6f 6475 6c65 2c20 2d2d 204d logyModule, -- M │ │ │ │ +00057ff0: 616b 6520 7468 6520 686f 6d6f 6c6f 6779 ake the homology │ │ │ │ +00058000: 0a20 2020 206f 6620 6120 636f 6d70 6c65 . of a comple │ │ │ │ +00058010: 7820 696e 746f 2061 206d 6f64 756c 6520 x into a module │ │ │ │ +00058020: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ +00058030: 2061 6c67 6562 7261 0a20 202a 202a 6e6f algebra. * *no │ │ │ │ +00058040: 7465 2065 7874 6572 696f 7254 6f72 4d6f te exteriorTorMo │ │ │ │ +00058050: 6475 6c65 3a20 6578 7465 7269 6f72 546f dule: exteriorTo │ │ │ │ +00058060: 724d 6f64 756c 652c 202d 2d20 546f 7220 rModule, -- Tor │ │ 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323a nResolutions.m2: │ │ │ │ -000582f0: 3237 3539 3a30 2e0a 1f0a 4669 6c65 3a20 2759:0....File: │ │ │ │ -00058300: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00058310: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00058320: 696e 666f 2c20 4e6f 6465 3a20 6d61 6b65 info, Node: make │ │ │ │ -00058330: 542c 204e 6578 743a 206d 6174 7269 7846 T, Next: matrixF │ │ │ │ -00058340: 6163 746f 7269 7a61 7469 6f6e 2c20 5072 actorization, Pr │ │ │ │ -00058350: 6576 3a20 6d61 6b65 4d6f 6475 6c65 2c20 ev: makeModule, │ │ │ │ -00058360: 5570 3a20 546f 700a 0a6d 616b 6554 202d Up: Top..makeT - │ │ │ │ -00058370: 2d20 6d61 6b65 2074 6865 2043 4920 6f70 - make the CI op │ │ │ │ -00058380: 6572 6174 6f72 7320 6f6e 2061 2063 6f6d erators on a com │ │ │ │ -00058390: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a plex.*********** │ │ │ │ +00058250: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00058260: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00058270: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00058280: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00058290: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +000582a0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +000582b0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +000582c0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +000582d0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +000582e0: 732e 6d32 3a32 3735 393a 302e 0a1f 0a46 s.m2:2759:0....F │ │ │ │ +000582f0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00058300: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00058310: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00058320: 206d 616b 6554 2c20 4e65 7874 3a20 6d61 makeT, Next: ma │ │ │ │ +00058330: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +00058340: 6e2c 2050 7265 763a 206d 616b 654d 6f64 n, Prev: makeMod │ │ │ │ +00058350: 756c 652c 2055 703a 2054 6f70 0a0a 6d61 ule, Up: Top..ma │ │ │ │ +00058360: 6b65 5420 2d2d 206d 616b 6520 7468 6520 keT -- make the │ │ │ │ +00058370: 4349 206f 7065 7261 746f 7273 206f 6e20 CI operators on │ │ │ │ +00058380: 6120 636f 6d70 6c65 780a 2a2a 2a2a 2a2a a complex.****** │ │ │ │ +00058390: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000583a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000583b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000583c0: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -000583d0: 2020 2020 2020 5420 3d20 6d61 6b65 5428 T = makeT( │ │ │ │ -000583e0: 6666 2c46 2c69 290a 2020 2020 2020 2020 ff,F,i). │ │ │ │ -000583f0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c74 T = makeT(ff,F,t │ │ │ │ -00058400: 302c 6929 0a20 202a 2049 6e70 7574 733a 0,i). * Inputs: │ │ │ │ -00058410: 0a20 2020 2020 202a 2066 662c 2061 202a . * ff, a * │ │ │ │ -00058420: 6e6f 7465 206d 6174 7269 783a 2028 4d61 note matrix: (Ma │ │ │ │ -00058430: 6361 756c 6179 3244 6f63 294d 6174 7269 caulay2Doc)Matri │ │ │ │ -00058440: 782c 2c20 3178 6320 6d61 7472 6978 2077 x,, 1xc matrix w │ │ │ │ -00058450: 686f 7365 2065 6e74 7269 6573 2061 7265 hose entries are │ │ │ │ -00058460: 0a20 2020 2020 2020 2061 2063 6f6d 706c . a compl │ │ │ │ -00058470: 6574 6520 696e 7465 7273 6563 7469 6f6e ete intersection │ │ │ │ -00058480: 2069 6e20 530a 2020 2020 2020 2a20 462c in S. * F, │ │ │ │ -00058490: 2061 202a 6e6f 7465 2063 6f6d 706c 6578 a *note complex │ │ │ │ -000584a0: 3a20 2843 6f6d 706c 6578 6573 2943 6f6d : (Complexes)Com │ │ │ │ -000584b0: 706c 6578 2c2c 206f 7665 7220 532f 6964 plex,, over S/id │ │ │ │ -000584c0: 6561 6c20 6666 0a20 2020 2020 202a 2074 eal ff. * t │ │ │ │ -000584d0: 302c 2061 202a 6e6f 7465 206d 6174 7269 0, a *note matri │ │ │ │ -000584e0: 783a 2028 4d61 6361 756c 6179 3244 6f63 x: (Macaulay2Doc │ │ │ │ -000584f0: 294d 6174 7269 782c 2c20 4349 2d6f 7065 )Matrix,, CI-ope │ │ │ │ -00058500: 7261 746f 7220 6f6e 2046 2066 6f72 2066 rator on F for f │ │ │ │ -00058510: 665f 3020 746f 0a20 2020 2020 2020 2062 f_0 to. b │ │ │ │ -00058520: 6520 7072 6573 6572 7665 640a 2020 2020 e preserved. │ │ │ │ -00058530: 2020 2a20 692c 2061 6e20 2a6e 6f74 6520 * i, an *note │ │ │ │ -00058540: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -00058550: 6179 3244 6f63 295a 5a2c 2c20 6465 6669 ay2Doc)ZZ,, defi │ │ │ │ -00058560: 6e65 2043 4920 6f70 6572 6174 6f72 7320 ne CI operators │ │ │ │ -00058570: 6672 6f6d 2046 5f69 0a20 2020 2020 2020 from F_i. │ │ │ │ -00058580: 205c 746f 2046 5f7b 692d 327d 0a20 202a \to F_{i-2}. * │ │ │ │ -00058590: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -000585a0: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973 * L, a *note lis │ │ │ │ -000585b0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -000585c0: 294c 6973 742c 2c20 6f66 2043 4920 6f70 )List,, of CI op │ │ │ │ -000585d0: 6572 6174 6f72 7320 465f 6920 5c74 6f20 erators F_i \to │ │ │ │ -000585e0: 465f 7b69 2d32 7d0a 2020 2020 2020 2020 F_{i-2}. │ │ │ │ -000585f0: 636f 7272 6573 706f 6e64 696e 6720 746f corresponding to │ │ │ │ -00058600: 2065 6e74 7269 6573 206f 6620 6666 0a0a entries of ff.. │ │ │ │ -00058610: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00058620: 3d3d 3d3d 3d3d 3d0a 0a73 7562 7374 6974 =======..substit │ │ │ │ -00058630: 7574 6520 6d61 7472 6963 6573 206f 6620 ute matrices of │ │ │ │ -00058640: 7477 6f20 6469 6666 6572 656e 7469 616c two differential │ │ │ │ -00058650: 7320 6f66 2046 2069 6e74 6f20 5320 3d20 s of F into S = │ │ │ │ -00058660: 7269 6e67 2066 662c 2063 6f6d 706f 7365 ring ff, compose │ │ │ │ -00058670: 2074 6865 6d2c 0a61 6e64 2064 6976 6964 them,.and divid │ │ │ │ -00058680: 6520 6279 2065 6e74 7269 6573 206f 6620 e by entries of │ │ │ │ -00058690: 6666 2c20 696e 206f 7264 6572 2e20 4966 ff, in order. If │ │ │ │ -000586a0: 2074 6865 2073 6563 6f6e 6420 4d61 7472 the second Matr │ │ │ │ -000586b0: 6978 2061 7267 756d 656e 7420 7430 2069 ix argument t0 i │ │ │ │ -000586c0: 730a 7072 6573 656e 742c 2075 7365 2069 s.present, use i │ │ │ │ -000586d0: 7420 6173 2074 6865 2066 6972 7374 2043 t as the first C │ │ │ │ -000586e0: 4920 6f70 6572 6174 6f72 2e0a 0a54 6865 I operator...The │ │ │ │ -000586f0: 2064 6567 7265 6573 206f 6620 7468 6520 degrees of the │ │ │ │ -00058700: 7461 7267 6574 7320 6f66 2074 6865 2054 targets of the T │ │ │ │ -00058710: 5f6a 2061 7265 2063 6861 6e67 6564 2062 _j are changed b │ │ │ │ -00058720: 7920 7468 6520 6465 6772 6565 7320 6f66 y the degrees of │ │ │ │ -00058730: 2074 6865 2066 5f6a 2074 6f0a 6d61 6b65 the f_j to.make │ │ │ │ -00058740: 2074 6865 2054 5f6a 2068 6f6d 6f67 656e the T_j homogen │ │ │ │ -00058750: 656f 7573 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d eous...+-------- │ │ │ │ +000583b0: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +000583c0: 3a20 0a20 2020 2020 2020 2054 203d 206d : . T = m │ │ │ │ +000583d0: 616b 6554 2866 662c 462c 6929 0a20 2020 akeT(ff,F,i). │ │ │ │ +000583e0: 2020 2020 2054 203d 206d 616b 6554 2866 T = makeT(f │ │ │ │ +000583f0: 662c 462c 7430 2c69 290a 2020 2a20 496e f,F,t0,i). * In │ │ │ │ +00058400: 7075 7473 3a0a 2020 2020 2020 2a20 6666 puts:. * ff │ │ │ │ +00058410: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +00058420: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00058430: 4d61 7472 6978 2c2c 2031 7863 206d 6174 Matrix,, 1xc mat │ │ │ │ +00058440: 7269 7820 7768 6f73 6520 656e 7472 6965 rix whose entrie │ │ │ │ +00058450: 7320 6172 650a 2020 2020 2020 2020 6120 s are. a │ │ │ │ +00058460: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ +00058470: 6374 696f 6e20 696e 2053 0a20 2020 2020 ction in S. │ │ │ │ +00058480: 202a 2046 2c20 6120 2a6e 6f74 6520 636f * F, a *note co │ │ │ │ +00058490: 6d70 6c65 783a 2028 436f 6d70 6c65 7865 mplex: (Complexe │ │ │ │ +000584a0: 7329 436f 6d70 6c65 782c 2c20 6f76 6572 s)Complex,, over │ │ │ │ +000584b0: 2053 2f69 6465 616c 2066 660a 2020 2020 S/ideal ff. │ │ │ │ +000584c0: 2020 2a20 7430 2c20 6120 2a6e 6f74 6520 * t0, a *note │ │ │ │ +000584d0: 6d61 7472 6978 3a20 284d 6163 6175 6c61 matrix: (Macaula │ │ │ │ +000584e0: 7932 446f 6329 4d61 7472 6978 2c2c 2043 y2Doc)Matrix,, C │ │ │ │ +000584f0: 492d 6f70 6572 6174 6f72 206f 6e20 4620 I-operator on F │ │ │ │ +00058500: 666f 7220 6666 5f30 2074 6f0a 2020 2020 for ff_0 to. │ │ │ │ +00058510: 2020 2020 6265 2070 7265 7365 7276 6564 be preserved │ │ │ │ +00058520: 0a20 2020 2020 202a 2069 2c20 616e 202a . * i, an * │ │ │ │ +00058530: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +00058540: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +00058550: 2064 6566 696e 6520 4349 206f 7065 7261 define CI opera │ │ │ │ +00058560: 746f 7273 2066 726f 6d20 465f 690a 2020 tors from F_i. │ │ │ │ +00058570: 2020 2020 2020 5c74 6f20 465f 7b69 2d32 \to F_{i-2 │ │ │ │ +00058580: 7d0a 2020 2a20 4f75 7470 7574 733a 0a20 }. * Outputs:. │ │ │ │ +00058590: 2020 2020 202a 204c 2c20 6120 2a6e 6f74 * L, a *not │ │ │ │ +000585a0: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +000585b0: 7932 446f 6329 4c69 7374 2c2c 206f 6620 y2Doc)List,, of │ │ │ │ +000585c0: 4349 206f 7065 7261 746f 7273 2046 5f69 CI operators F_i │ │ │ │ +000585d0: 205c 746f 2046 5f7b 692d 327d 0a20 2020 \to F_{i-2}. │ │ │ │ +000585e0: 2020 2020 2063 6f72 7265 7370 6f6e 6469 correspondi │ │ │ │ +000585f0: 6e67 2074 6f20 656e 7472 6965 7320 6f66 ng to entries of │ │ │ │ +00058600: 2066 660a 0a44 6573 6372 6970 7469 6f6e ff..Description │ │ │ │ +00058610: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 7375 .===========..su │ │ │ │ +00058620: 6273 7469 7475 7465 206d 6174 7269 6365 bstitute matrice │ │ │ │ +00058630: 7320 6f66 2074 776f 2064 6966 6665 7265 s of two differe │ │ │ │ +00058640: 6e74 6961 6c73 206f 6620 4620 696e 746f ntials of F into │ │ │ │ +00058650: 2053 203d 2072 696e 6720 6666 2c20 636f S = ring ff, co │ │ │ │ +00058660: 6d70 6f73 6520 7468 656d 2c0a 616e 6420 mpose them,.and │ │ │ │ +00058670: 6469 7669 6465 2062 7920 656e 7472 6965 divide by entrie │ │ │ │ +00058680: 7320 6f66 2066 662c 2069 6e20 6f72 6465 s of ff, in orde │ │ │ │ +00058690: 722e 2049 6620 7468 6520 7365 636f 6e64 r. If the second │ │ │ │ +000586a0: 204d 6174 7269 7820 6172 6775 6d65 6e74 Matrix argument │ │ │ │ +000586b0: 2074 3020 6973 0a70 7265 7365 6e74 2c20 t0 is.present, │ │ │ │ +000586c0: 7573 6520 6974 2061 7320 7468 6520 6669 use it as the fi │ │ │ │ +000586d0: 7273 7420 4349 206f 7065 7261 746f 722e rst CI operator. │ │ │ │ +000586e0: 0a0a 5468 6520 6465 6772 6565 7320 6f66 ..The degrees of │ │ │ │ +000586f0: 2074 6865 2074 6172 6765 7473 206f 6620 the targets of │ │ │ │ +00058700: 7468 6520 545f 6a20 6172 6520 6368 616e the T_j are chan │ │ │ │ +00058710: 6765 6420 6279 2074 6865 2064 6567 7265 ged by the degre │ │ │ │ +00058720: 6573 206f 6620 7468 6520 665f 6a20 746f es of the f_j to │ │ │ │ +00058730: 0a6d 616b 6520 7468 6520 545f 6a20 686f .make the T_j ho │ │ │ │ +00058740: 6d6f 6765 6e65 6f75 732e 0a0a 2b2d 2d2d mogeneous...+--- │ │ │ │ +00058750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -00058790: 3120 3a20 5320 3d20 5a5a 2f31 3031 5b78 1 : S = ZZ/101[x │ │ │ │ -000587a0: 2c79 2c7a 5d3b 2020 2020 2020 2020 2020 ,y,z]; │ │ │ │ -000587b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000587c0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00058780: 2d2b 0a7c 6931 203a 2053 203d 205a 5a2f -+.|i1 : S = ZZ/ │ │ │ │ +00058790: 3130 315b 782c 792c 7a5d 3b20 2020 2020 101[x,y,z]; │ │ │ │ +000587a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000587b0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +000587c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000587d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000587f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 ----------+.|i2 │ │ │ │ -00058800: 3a20 6666 203d 206d 6174 7269 7822 7833 : ff = matrix"x3 │ │ │ │ -00058810: 2c79 332c 7a33 223b 2020 2020 2020 2020 ,y3,z3"; │ │ │ │ -00058820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058830: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000587e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000587f0: 0a7c 6932 203a 2066 6620 3d20 6d61 7472 .|i2 : ff = matr │ │ │ │ +00058800: 6978 2278 332c 7933 2c7a 3322 3b20 2020 ix"x3,y3,z3"; │ │ │ │ +00058810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058820: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00058830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058860: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00058870: 2020 2020 2020 2020 3120 2020 2020 2033 1 3 │ │ │ │ +00058850: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058860: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ +00058870: 2020 2020 3320 2020 2020 2020 2020 2020 3 │ │ │ │ 00058880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058890: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000588a0: 0a7c 6f32 203a 204d 6174 7269 7820 5320 .|o2 : Matrix S │ │ │ │ -000588b0: 203c 2d2d 2053 2020 2020 2020 2020 2020 <-- S │ │ │ │ -000588c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000588d0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00058890: 2020 2020 7c0a 7c6f 3220 3a20 4d61 7472 |.|o2 : Matr │ │ │ │ +000588a0: 6978 2053 2020 3c2d 2d20 5320 2020 2020 ix S <-- S │ │ │ │ +000588b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000588c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +000588d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000588e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000588f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00058910: 6933 203a 2052 203d 2053 2f69 6465 616c i3 : R = S/ideal │ │ │ │ -00058920: 2066 663b 2020 2020 2020 2020 2020 2020 ff; │ │ │ │ -00058930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00058900: 2d2d 2b0a 7c69 3320 3a20 5220 3d20 532f --+.|i3 : R = S/ │ │ │ │ +00058910: 6964 6561 6c20 6666 3b20 2020 2020 2020 ideal ff; │ │ │ │ +00058920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058930: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00058940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058960: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 -----------+.|i4 │ │ │ │ -00058980: 203a 204d 203d 2063 6f6b 6572 206d 6174 : M = coker mat │ │ │ │ -00058990: 7269 7822 782c 792c 7a3b 792c 7a2c 7822 rix"x,y,z;y,z,x" │ │ │ │ -000589a0: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ -000589b0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00058970: 2b0a 7c69 3420 3a20 4d20 3d20 636f 6b65 +.|i4 : M = coke │ │ │ │ +00058980: 7220 6d61 7472 6978 2278 2c79 2c7a 3b79 r matrix"x,y,z;y │ │ │ │ +00058990: 2c7a 2c78 223b 2020 2020 2020 2020 2020 ,z,x"; │ │ │ │ +000589a0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000589b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000589c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000589e0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -000589f0: 2062 6574 7469 2028 4620 3d20 6672 6565 betti (F = free │ │ │ │ -00058a00: 5265 736f 6c75 7469 6f6e 284d 2c20 4c65 Resolution(M, Le │ │ │ │ -00058a10: 6e67 7468 4c69 6d69 7420 3d3e 2033 2929 ngthLimit => 3)) │ │ │ │ -00058a20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000589d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000589e0: 7c69 3520 3a20 6265 7474 6920 2846 203d |i5 : betti (F = │ │ │ │ +000589f0: 2066 7265 6552 6573 6f6c 7574 696f 6e28 freeResolution( │ │ │ │ +00058a00: 4d2c 204c 656e 6774 684c 696d 6974 203d M, LengthLimit = │ │ │ │ +00058a10: 3e20 3329 297c 0a7c 2020 2020 2020 2020 > 3))|.| │ │ │ │ +00058a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058a50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00058a60: 2020 2020 2020 3020 3120 3220 3320 2020 0 1 2 3 │ │ │ │ +00058a40: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00058a50: 2020 2020 2020 2020 2020 2030 2031 2032 0 1 2 │ │ │ │ +00058a60: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00058a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058a80: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00058a90: 7c6f 3520 3d20 746f 7461 6c3a 2032 2033 |o5 = total: 2 3 │ │ │ │ -00058aa0: 2035 2036 2020 2020 2020 2020 2020 2020 5 6 │ │ │ │ -00058ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ac0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00058ad0: 2030 3a20 3220 3320 2e20 2e20 2020 2020 0: 2 3 . . │ │ │ │ +00058a80: 2020 207c 0a7c 6f35 203d 2074 6f74 616c |.|o5 = total │ │ │ │ +00058a90: 3a20 3220 3320 3520 3620 2020 2020 2020 : 2 3 5 6 │ │ │ │ +00058aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058ab0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00058ac0: 2020 2020 2020 303a 2032 2033 202e 202e 0: 2 3 . . │ │ │ │ +00058ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058af0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00058b00: 2020 2020 2020 2020 313a 202e 202e 2035 1: . . 5 │ │ │ │ -00058b10: 2036 2020 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -00058b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b30: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00058af0: 207c 0a7c 2020 2020 2020 2020 2031 3a20 |.| 1: │ │ │ │ +00058b00: 2e20 2e20 3520 3620 2020 2020 2020 2020 . . 5 6 │ │ │ │ +00058b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058b20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00058b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b60: 2020 2020 2020 2020 2020 7c0a 7c6f 3520 |.|o5 │ │ │ │ -00058b70: 3a20 4265 7474 6954 616c 6c79 2020 2020 : BettiTally │ │ │ │ +00058b50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00058b60: 0a7c 6f35 203a 2042 6574 7469 5461 6c6c .|o5 : BettiTall │ │ │ │ +00058b70: 7920 2020 2020 2020 2020 2020 2020 2020 y │ │ │ │ 00058b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ba0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00058b90: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00058ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -00058be0: 5420 3d20 6d61 6b65 5428 6666 2c46 2c33 T = makeT(ff,F,3 │ │ │ │ -00058bf0: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ -00058c00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00058c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00058bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00058bd0: 6936 203a 2054 203d 206d 616b 6554 2866 i6 : T = makeT(f │ │ │ │ +00058be0: 662c 462c 3329 3b20 2020 2020 2020 2020 f,F,3); │ │ │ │ +00058bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00058c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058c40: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 6e65 ------+.|i7 : ne │ │ │ │ -00058c50: 744c 6973 7420 5420 2020 2020 2020 2020 tList T │ │ │ │ +00058c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ +00058c40: 203a 206e 6574 4c69 7374 2054 2020 2020 : netList T │ │ │ │ +00058c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058c70: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058c70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00058c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00058c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058cb0: 2020 2020 7c0a 7c20 2020 2020 2b2d 2d2d |.| +--- │ │ │ │ -00058cc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058cd0: 2d2d 2d2d 2d2b 2020 2020 2020 2020 2020 -----+ │ │ │ │ -00058ce0: 2020 2020 2020 2020 2020 207c 0a7c 6f37 |.|o7 │ │ │ │ -00058cf0: 203d 207c 7b34 7d20 7c20 3020 3020 3020 = |{4} | 0 0 0 │ │ │ │ -00058d00: 3020 2031 2030 207c 2020 2020 7c20 2020 0 1 0 | | │ │ │ │ -00058d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058d20: 2020 7c0a 7c20 2020 2020 7c7b 347d 207c |.| |{4} | │ │ │ │ -00058d30: 2030 2030 2030 202d 3120 3020 3020 7c20 0 0 0 -1 0 0 | │ │ │ │ -00058d40: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00058d50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00058d60: 207c 7b34 7d20 7c20 3020 3020 3020 3020 |{4} | 0 0 0 0 │ │ │ │ -00058d70: 2030 2031 207c 2020 2020 7c20 2020 2020 0 1 | | │ │ │ │ -00058d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058d90: 7c0a 7c20 2020 2020 2b2d 2d2d 2d2d 2d2d |.| +------- │ │ │ │ -00058da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058db0: 2d2b 2020 2020 2020 2020 2020 2020 2020 -+ │ │ │ │ -00058dc0: 2020 2020 2020 207c 0a7c 2020 2020 207c |.| | │ │ │ │ -00058dd0: 7b34 7d20 7c20 3020 3120 3020 3020 3020 {4} | 0 1 0 0 0 │ │ │ │ -00058de0: 3020 7c20 2020 2020 7c20 2020 2020 2020 0 | | │ │ │ │ -00058df0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00058e00: 7c20 2020 2020 7c7b 347d 207c 2031 2030 | |{4} | 1 0 │ │ │ │ -00058e10: 2030 2030 2030 2030 207c 2020 2020 207c 0 0 0 0 | | │ │ │ │ -00058e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058e30: 2020 2020 207c 0a7c 2020 2020 207c 7b34 |.| |{4 │ │ │ │ -00058e40: 7d20 7c20 3020 3020 3120 3020 3020 3020 } | 0 0 1 0 0 0 │ │ │ │ -00058e50: 7c20 2020 2020 7c20 2020 2020 2020 2020 | | │ │ │ │ -00058e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00058e70: 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d +----------- │ │ │ │ -00058e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 -------------+ │ │ │ │ -00058e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058ea0: 2020 207c 0a7c 2020 2020 207c 7b34 7d20 |.| |{4} │ │ │ │ -00058eb0: 7c20 3020 202d 3120 3020 2030 202d 3120 | 0 -1 0 0 -1 │ │ │ │ -00058ec0: 3020 207c 7c20 2020 2020 2020 2020 2020 0 || │ │ │ │ -00058ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00058ee0: 2020 7c7b 347d 207c 202d 3120 3020 2030 |{4} | -1 0 0 │ │ │ │ -00058ef0: 2020 3120 3020 2030 2020 7c7c 2020 2020 1 0 0 || │ │ │ │ -00058f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058f10: 207c 0a7c 2020 2020 207c 7b34 7d20 7c20 |.| |{4} | │ │ │ │ -00058f20: 3020 2030 2020 2d31 2030 2030 2020 2d31 0 0 -1 0 0 -1 │ │ │ │ -00058f30: 207c 7c20 2020 2020 2020 2020 2020 2020 || │ │ │ │ -00058f40: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00058f50: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ -00058f60: 2d2d 2d2d 2d2d 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| │ │ │ │ +00058e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058e60: 207c 0a7c 2020 2020 202b 2d2d 2d2d 2d2d |.| +------ │ │ │ │ +00058e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00058e80: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ +00058e90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00058ea0: 7c7b 347d 207c 2030 2020 2d31 2030 2020 |{4} | 0 -1 0 │ │ │ │ +00058eb0: 3020 2d31 2030 2020 7c7c 2020 2020 2020 0 -1 0 || │ │ │ │ +00058ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00058ed0: 0a7c 2020 2020 207c 7b34 7d20 7c20 2d31 .| |{4} | -1 │ │ │ │ +00058ee0: 2030 2020 3020 2031 2030 2020 3020 207c 0 0 1 0 0 | │ │ │ │ +00058ef0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00058f00: 2020 2020 2020 7c0a 7c20 2020 2020 7c7b |.| |{ │ │ │ │ +00058f10: 347d 207c 2030 2020 3020 202d 3120 3020 4} | 0 0 -1 0 │ │ │ │ +00058f20: 3020 202d 3120 7c7c 2020 2020 2020 2020 0 -1 || │ │ │ │ +00058f30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058f40: 2020 2020 202b 2d2d 2d2d 2d2d 2d2d 2d2d +---------- │ │ │ │ +00058f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 --------------+ │ │ │ │ +00058f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00058f70: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00058f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00058f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00058fb0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6973 ------+.|i8 : is │ │ │ │ -00058fc0: 486f 6d6f 6765 6e65 6f75 7320 545f 3220 Homogeneous T_2 │ │ │ │ +00058fa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +00058fb0: 203a 2069 7348 6f6d 6f67 656e 656f 7573 : isHomogeneous │ │ │ │ +00058fc0: 2054 5f32 2020 2020 2020 2020 2020 2020 T_2 │ │ │ │ 00058fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00058fe0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00058fe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00058ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059020: 2020 2020 7c0a 7c6f 3820 3d20 7472 7565 |.|o8 = true │ │ │ │ +00059010: 2020 2020 2020 2020 207c 0a7c 6f38 203d |.|o8 = │ │ │ │ +00059020: 2074 7275 6520 2020 2020 2020 2020 2020 true │ │ │ │ 00059030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059050: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00059050: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00059060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059090: 2d2d 2b0a 0a43 6176 6561 740a 3d3d 3d3d --+..Caveat.==== │ │ │ │ -000590a0: 3d3d 0a0a 5363 7269 7074 2061 7373 756d ==..Script assum │ │ │ │ -000590b0: 6573 2074 6861 7420 7269 6e67 2046 203d es that ring F = │ │ │ │ -000590c0: 3d20 2872 696e 6720 6666 292f 2869 6465 = (ring ff)/(ide │ │ │ │ -000590d0: 616c 2066 6629 2e20 4974 206d 6967 6874 al ff). It might │ │ │ │ -000590e0: 2062 6520 6d6f 7265 2075 7365 6675 6c20 be more useful │ │ │ │ -000590f0: 746f 0a72 6574 7572 6e20 7468 6520 6f70 to.return the op │ │ │ │ -00059100: 6572 6174 6f72 7320 6173 206d 6174 7269 erators as matri │ │ │ │ -00059110: 6365 7320 6f76 6572 2053 2072 6174 6865 ces over S rathe │ │ │ │ -00059120: 7220 7468 616e 206f 7665 7220 522c 2073 r than over R, s │ │ │ │ -00059130: 696e 6365 2074 6869 7320 6973 2077 6861 ince this is wha │ │ │ │ -00059140: 740a 7765 2764 206e 6565 6420 666f 7220 t.we'd need for │ │ │ │ -00059150: 7468 696e 6773 206c 696b 6520 6d61 7472 things like matr │ │ │ │ -00059160: 6978 4661 6374 6f72 697a 6174 696f 6e20 ixFactorization │ │ │ │ -00059170: 2877 6865 7265 2074 6869 7320 7072 6f63 (where this proc │ │ │ │ -00059180: 6573 7320 6375 7272 656e 746c 790a 646f ess currently.do │ │ │ │ -00059190: 6e65 206f 6e20 7468 6520 666c 792c 206e ne on the fly, n │ │ │ │ -000591a0: 6f74 2063 616c 6c69 6e67 206d 616b 6554 ot calling makeT │ │ │ │ -000591b0: 290a 0a57 6179 7320 746f 2075 7365 206d )..Ways to use m │ │ │ │ -000591c0: 616b 6554 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d akeT:.========== │ │ │ │ -000591d0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 226d ========.. * "m │ │ │ │ -000591e0: 616b 6554 284d 6174 7269 782c 436f 6d70 akeT(Matrix,Comp │ │ │ │ -000591f0: 6c65 782c 5a5a 2922 0a0a 466f 7220 7468 lex,ZZ)"..For th │ │ │ │ -00059200: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00059210: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00059220: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00059230: 6520 6d61 6b65 543a 206d 616b 6554 2c20 e makeT: makeT, │ │ │ │ -00059240: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -00059250: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -00059260: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -00059270: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +00059080: 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 6174 -------+..Caveat │ │ │ │ +00059090: 0a3d 3d3d 3d3d 3d0a 0a53 6372 6970 7420 .======..Script │ │ │ │ +000590a0: 6173 7375 6d65 7320 7468 6174 2072 696e assumes that rin │ │ │ │ +000590b0: 6720 4620 3d3d 2028 7269 6e67 2066 6629 g F == (ring ff) │ │ │ │ +000590c0: 2f28 6964 6561 6c20 6666 292e 2049 7420 /(ideal ff). It │ │ │ │ +000590d0: 6d69 6768 7420 6265 206d 6f72 6520 7573 might be more us │ │ │ │ +000590e0: 6566 756c 2074 6f0a 7265 7475 726e 2074 eful to.return t │ │ │ │ +000590f0: 6865 206f 7065 7261 746f 7273 2061 7320 he operators as │ │ │ │ +00059100: 6d61 7472 6963 6573 206f 7665 7220 5320 matrices over S │ │ │ │ +00059110: 7261 7468 6572 2074 6861 6e20 6f76 6572 rather than over │ │ │ │ +00059120: 2052 2c20 7369 6e63 6520 7468 6973 2069 R, since this i │ │ │ │ +00059130: 7320 7768 6174 0a77 6527 6420 6e65 6564 s what.we'd need │ │ │ │ +00059140: 2066 6f72 2074 6869 6e67 7320 6c69 6b65 for things like │ │ │ │ +00059150: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +00059160: 7469 6f6e 2028 7768 6572 6520 7468 6973 tion (where this │ │ │ │ +00059170: 2070 726f 6365 7373 2063 7572 7265 6e74 process current │ │ │ │ +00059180: 6c79 0a64 6f6e 6520 6f6e 2074 6865 2066 ly.done on the f │ │ │ │ +00059190: 6c79 2c20 6e6f 7420 6361 6c6c 696e 6720 ly, not calling │ │ │ │ +000591a0: 6d61 6b65 5429 0a0a 5761 7973 2074 6f20 makeT)..Ways to │ │ │ │ +000591b0: 7573 6520 6d61 6b65 543a 0a3d 3d3d 3d3d use makeT:.===== │ │ │ │ +000591c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +000591d0: 202a 2022 6d61 6b65 5428 4d61 7472 6978 * "makeT(Matrix │ │ │ │ +000591e0: 2c43 6f6d 706c 6578 2c5a 5a29 220a 0a46 ,Complex,ZZ)"..F │ │ │ │ +000591f0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00059200: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00059210: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00059220: 202a 6e6f 7465 206d 616b 6554 3a20 6d61 *note makeT: ma │ │ │ │ +00059230: 6b65 542c 2069 7320 6120 2a6e 6f74 6520 keT, is a *note │ │ │ │ +00059240: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +00059250: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +00059260: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +00059270: 0a2d 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5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -00059350: 6f6e 732e 6d32 3a33 3534 383a 302e 0a1f ons.m2:3548:0... │ │ │ │ -00059360: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -00059370: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -00059380: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -00059390: 653a 206d 6174 7269 7846 6163 746f 7269 e: matrixFactori │ │ │ │ -000593a0: 7a61 7469 6f6e 2c20 4e65 7874 3a20 6d66 zation, Next: mf │ │ │ │ -000593b0: 426f 756e 642c 2050 7265 763a 206d 616b Bound, Prev: mak │ │ │ │ -000593c0: 6554 2c20 5570 3a20 546f 700a 0a6d 6174 eT, Up: Top..mat │ │ │ │ -000593d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -000593e0: 202d 2d20 4d61 7073 2069 6e20 6120 6869 -- Maps in a hi │ │ │ │ -000593f0: 6768 6572 2063 6f64 696d 656e 7369 6f6e gher codimension │ │ │ │ -00059400: 206d 6174 7269 7820 6661 6374 6f72 697a matrix factoriz │ │ │ │ -00059410: 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a ation.********** │ │ │ │ +000592c0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +000592d0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +000592e0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +000592f0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00059300: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00059310: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00059320: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +00059330: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +00059340: 6f6c 7574 696f 6e73 2e6d 323a 3335 3438 olutions.m2:3548 │ │ │ │ +00059350: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +00059360: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +00059370: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +00059380: 2c20 4e6f 6465 3a20 6d61 7472 6978 4661 , Node: matrixFa │ │ │ │ +00059390: 6374 6f72 697a 6174 696f 6e2c 204e 6578 ctorization, Nex │ │ │ │ +000593a0: 743a 206d 6642 6f75 6e64 2c20 5072 6576 t: mfBound, Prev │ │ │ │ +000593b0: 3a20 6d61 6b65 542c 2055 703a 2054 6f70 : makeT, Up: Top │ │ │ │ +000593c0: 0a0a 6d61 7472 6978 4661 6374 6f72 697a ..matrixFactoriz │ │ │ │ +000593d0: 6174 696f 6e20 2d2d 204d 6170 7320 696e ation -- Maps in │ │ │ │ +000593e0: 2061 2068 6967 6865 7220 636f 6469 6d65 a higher codime │ │ │ │ +000593f0: 6e73 696f 6e20 6d61 7472 6978 2066 6163 nsion matrix fac │ │ │ │ +00059400: 746f 7269 7a61 7469 6f6e 0a2a 2a2a 2a2a torization.***** │ │ │ │ +00059410: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059420: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059430: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00059440: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00059450: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00059460: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -00059470: 2020 2020 4d46 203d 206d 6174 7269 7846 MF = matrixF │ │ │ │ -00059480: 6163 746f 7269 7a61 7469 6f6e 2866 662c actorization(ff, │ │ │ │ -00059490: 4d29 0a20 202a 2049 6e70 7574 733a 0a20 M). * Inputs:. │ │ │ │ -000594a0: 2020 2020 202a 2066 662c 2061 202a 6e6f * ff, a *no │ │ │ │ -000594b0: 7465 206d 6174 7269 783a 2028 4d61 6361 te matrix: (Maca │ │ │ │ -000594c0: 756c 6179 3244 6f63 294d 6174 7269 782c ulay2Doc)Matrix, │ │ │ │ -000594d0: 2c20 6120 7375 6666 6963 6965 6e74 6c79 , a sufficiently │ │ │ │ -000594e0: 2067 656e 6572 616c 0a20 2020 2020 2020 general. │ │ │ │ -000594f0: 2072 6567 756c 6172 2073 6571 7565 6e63 regular sequenc │ │ │ │ -00059500: 6520 696e 2061 2072 696e 6720 530a 2020 e in a ring S. │ │ │ │ -00059510: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ -00059520: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ -00059530: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ -00059540: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d a maximal Cohen- │ │ │ │ -00059550: 4d61 6361 756c 6179 0a20 2020 2020 2020 Macaulay. │ │ │ │ -00059560: 206d 6f64 756c 6520 6f76 6572 2053 2f69 module over S/i │ │ │ │ -00059570: 6465 616c 2066 660a 2020 2a20 2a6e 6f74 deal ff. * *not │ │ │ │ -00059580: 6520 4f70 7469 6f6e 616c 2069 6e70 7574 e Optional input │ │ │ │ -00059590: 733a 2028 4d61 6361 756c 6179 3244 6f63 s: (Macaulay2Doc │ │ │ │ -000595a0: 2975 7369 6e67 2066 756e 6374 696f 6e73 )using functions │ │ │ │ -000595b0: 2077 6974 6820 6f70 7469 6f6e 616c 2069 with optional i │ │ │ │ -000595c0: 6e70 7574 732c 3a0a 2020 2020 2020 2a20 nputs,:. * │ │ │ │ -000595d0: 4175 676d 656e 7461 7469 6f6e 203d 3e20 Augmentation => │ │ │ │ -000595e0: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -000595f0: 7565 2074 7275 650a 2020 2020 2020 2a20 ue true. * │ │ │ │ -00059600: 4368 6563 6b20 3d3e 202e 2e2e 2c20 6465 Check => ..., de │ │ │ │ -00059610: 6661 756c 7420 7661 6c75 6520 6661 6c73 fault value fals │ │ │ │ -00059620: 650a 2020 2020 2020 2a20 4c61 7965 7265 e. * Layere │ │ │ │ -00059630: 6420 3d3e 202e 2e2e 2c20 6465 6661 756c d => ..., defaul │ │ │ │ -00059640: 7420 7661 6c75 6520 7472 7565 0a20 2020 t value true. │ │ │ │ -00059650: 2020 202a 2056 6572 626f 7365 203d 3e20 * Verbose => │ │ │ │ -00059660: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -00059670: 7565 2066 616c 7365 0a20 202a 204f 7574 ue false. * Out │ │ │ │ -00059680: 7075 7473 3a0a 2020 2020 2020 2a20 4d46 puts:. * MF │ │ │ │ -00059690: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -000596a0: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -000596b0: 7374 2c2c 205c 7b64 2c68 2c67 616d 6d61 st,, \{d,h,gamma │ │ │ │ -000596c0: 5c7d 2c20 7768 6572 6520 643a 415f 3120 \}, where d:A_1 │ │ │ │ -000596d0: 5c74 6f0a 2020 2020 2020 2020 415f 3020 \to. A_0 │ │ │ │ -000596e0: 616e 6420 683a 205c 6f70 6c75 7320 415f and h: \oplus A_ │ │ │ │ -000596f0: 3028 7029 205c 746f 2041 5f31 2069 7320 0(p) \to A_1 is │ │ │ │ -00059700: 7468 6520 6469 7265 6374 2073 756d 206f the direct sum o │ │ │ │ -00059710: 6620 7061 7274 6961 6c0a 2020 2020 2020 f partial. │ │ │ │ -00059720: 2020 686f 6d6f 746f 7069 6573 2c20 616e homotopies, an │ │ │ │ -00059730: 6420 6761 6d6d 613a 2041 5f30 202d 3e4d d gamma: A_0 ->M │ │ │ │ -00059740: 2069 7320 7468 6520 6175 676d 656e 7461 is the augmenta │ │ │ │ -00059750: 7469 6f6e 2028 7265 7475 726e 6564 206f tion (returned o │ │ │ │ -00059760: 6e6c 7920 6966 0a20 2020 2020 2020 2041 nly if. A │ │ │ │ -00059770: 7567 6d65 6e74 6174 696f 6e20 3d3e 7472 ugmentation =>tr │ │ │ │ -00059780: 7565 290a 0a44 6573 6372 6970 7469 6f6e ue)..Description │ │ │ │ -00059790: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 .===========..Th │ │ │ │ -000597a0: 6520 696e 7075 7420 6d6f 6475 6c65 204d e input module M │ │ │ │ -000597b0: 2073 686f 756c 6420 6265 2061 206d 6178 should be a max │ │ │ │ -000597c0: 696d 616c 2043 6f68 656e 2d4d 6163 6175 imal Cohen-Macau │ │ │ │ -000597d0: 6c61 7920 6d6f 6475 6c65 206f 7665 7220 lay module over │ │ │ │ -000597e0: 5220 3d20 532f 6964 6561 6c0a 6666 2e20 R = S/ideal.ff. │ │ │ │ -000597f0: 2049 6620 4d20 6973 2069 6e20 6661 6374 If M is in fact │ │ │ │ -00059800: 2061 2022 6869 6768 2073 797a 7967 7922 a "high syzygy" │ │ │ │ -00059810: 2c20 7468 656e 2074 6865 2066 756e 6374 , then the funct │ │ │ │ -00059820: 696f 6e0a 6d61 7472 6978 4661 6374 6f72 ion.matrixFactor │ │ │ │ -00059830: 697a 6174 696f 6e28 6666 2c4d 2c4c 6179 ization(ff,M,Lay │ │ │ │ -00059840: 6572 6564 3d3e 6661 6c73 6529 2075 7365 ered=>false) use │ │ │ │ -00059850: 7320 6120 6469 6666 6572 656e 742c 2066 s a different, f │ │ │ │ -00059860: 6173 7465 7220 616c 676f 7269 7468 6d0a aster algorithm. │ │ │ │ -00059870: 7768 6963 6820 6f6e 6c79 2077 6f72 6b73 which only works │ │ │ │ -00059880: 2069 6e20 7468 6520 6869 6768 2073 797a in the high syz │ │ │ │ -00059890: 7967 7920 6361 7365 2e0a 0a49 6e20 616c ygy case...In al │ │ │ │ -000598a0: 6c20 6578 616d 706c 6573 2077 6520 6b6e l examples we kn │ │ │ │ -000598b0: 6f77 2c20 4d20 6361 6e20 6265 2063 6f6e ow, M can be con │ │ │ │ -000598c0: 7369 6465 7265 6420 6120 2268 6967 6820 sidered a "high │ │ │ │ -000598d0: 7379 7a79 6779 2220 6173 206c 6f6e 6720 syzygy" as long │ │ │ │ -000598e0: 6173 0a45 7874 5e7b 6576 656e 7d5f 5228 as.Ext^{even}_R( │ │ │ │ -000598f0: 4d2c 6b29 2061 6e64 2045 7874 5e7b 6f64 M,k) and Ext^{od │ │ │ │ -00059900: 647d 5f52 284d 2c6b 2920 6861 7665 206e d}_R(M,k) have n │ │ │ │ -00059910: 6567 6174 6976 6520 7265 6775 6c61 7269 egative regulari │ │ │ │ -00059920: 7479 206f 7665 7220 7468 6520 7269 6e67 ty over the ring │ │ │ │ -00059930: 0a6f 6620 4349 206f 7065 7261 746f 7273 .of CI operators │ │ │ │ -00059940: 2028 7265 6772 6164 6564 2077 6974 6820 (regraded with │ │ │ │ -00059950: 7661 7269 6162 6c65 7320 6f66 2064 6567 variables of deg │ │ │ │ -00059960: 7265 6520 312e 2048 6f77 6576 6572 2c20 ree 1. However, │ │ │ │ -00059970: 7468 6520 6265 7374 2072 6573 756c 740a the best result. │ │ │ │ -00059980: 7765 2063 616e 2070 726f 7665 2069 7320 we can prove is │ │ │ │ -00059990: 7468 6174 2069 7420 7375 6666 6963 6573 that it suffices │ │ │ │ -000599a0: 2074 6f20 6861 7665 2072 6567 756c 6172 to have regular │ │ │ │ -000599b0: 6974 7920 3c20 2d28 322a 6469 6d20 522b ity < -(2*dim R+ │ │ │ │ -000599c0: 3129 2e0a 0a57 6865 6e20 7468 6520 6f70 1)...When the op │ │ │ │ -000599d0: 7469 6f6e 616c 2069 6e70 7574 2043 6865 tional input Che │ │ │ │ -000599e0: 636b 3d3d 7472 7565 2028 7468 6520 6465 ck==true (the de │ │ │ │ -000599f0: 6661 756c 7420 6973 2043 6865 636b 3d3d fault is Check== │ │ │ │ -00059a00: 6661 6c73 6529 2c20 7468 650a 7072 6f70 false), the.prop │ │ │ │ -00059a10: 6572 7469 6573 2069 6e20 7468 6520 6465 erties in the de │ │ │ │ -00059a20: 6669 6e69 7469 6f6e 206f 6620 4d61 7472 finition of Matr │ │ │ │ -00059a30: 6978 2046 6163 746f 7269 7a61 7469 6f6e ix Factorization │ │ │ │ -00059a40: 2061 7265 2076 6572 6966 6965 640a 0a54 are verified..T │ │ │ │ -00059a50: 6865 206f 7574 7075 7420 6973 2061 206c he output is a l │ │ │ │ -00059a60: 6973 7420 6f66 206d 6170 7320 5c7b 642c ist of maps \{d, │ │ │ │ -00059a70: 685c 7d20 6f72 205c 7b64 2c68 2c67 616d h\} or \{d,h,gam │ │ │ │ -00059a80: 6d61 5c7d 2c20 7768 6572 6520 6761 6d6d ma\}, where gamm │ │ │ │ -00059a90: 6120 6973 2061 6e0a 6175 676d 656e 7461 a is an.augmenta │ │ │ │ -00059aa0: 7469 6f6e 2c20 7468 6174 2069 732c 2061 tion, that is, a │ │ │ │ -00059ab0: 206d 6170 2066 726f 6d20 7461 7267 6574 map from target │ │ │ │ -00059ac0: 2064 2074 6f20 4d2e 0a0a 5468 6520 6d61 d to M...The ma │ │ │ │ -00059ad0: 7020 6420 6973 2061 2073 7065 6369 616c p d is a special │ │ │ │ -00059ae0: 206c 6966 7469 6e67 2074 6f20 5320 6f66 lifting to S of │ │ │ │ -00059af0: 2061 2070 7265 7365 6e74 6174 696f 6e20 a presentation │ │ │ │ -00059b00: 6f66 204d 206f 7665 7220 522e 2054 6f20 of M over R. To │ │ │ │ -00059b10: 6578 706c 6169 6e0a 7468 6520 636f 6e74 explain.the cont │ │ │ │ -00059b20: 656e 7473 2c20 7765 2069 6e74 726f 6475 ents, we introdu │ │ │ │ -00059b30: 6365 2073 6f6d 6520 6e6f 7461 7469 6f6e ce some notation │ │ │ │ -00059b40: 2028 6672 6f6d 2045 6973 656e 6275 6420 (from Eisenbud │ │ │ │ -00059b50: 616e 6420 5065 6576 612c 2022 4d69 6e69 and Peeva, "Mini │ │ │ │ -00059b60: 6d61 6c0a 6672 6565 2072 6573 6f6c 7574 mal.free resolut │ │ │ │ -00059b70: 696f 6e73 206f 7665 7220 636f 6d70 6c65 ions over comple │ │ │ │ -00059b80: 7465 2069 6e74 6572 7365 6374 696f 6e73 te intersections │ │ │ │ -00059b90: 2220 4c65 6374 7572 6520 4e6f 7465 7320 " Lecture Notes │ │ │ │ -00059ba0: 696e 204d 6174 6865 6d61 7469 6373 2c0a in Mathematics,. │ │ │ │ -00059bb0: 3231 3532 2e20 5370 7269 6e67 6572 2c20 2152. Springer, │ │ │ │ -00059bc0: 4368 616d 2c20 3230 3136 2e20 782b 3130 Cham, 2016. x+10 │ │ │ │ -00059bd0: 3720 7070 2e20 4953 424e 3a20 3937 382d 7 pp. ISBN: 978- │ │ │ │ -00059be0: 332d 3331 392d 3236 3433 362d 333b 0a39 3-319-26436-3;.9 │ │ │ │ -00059bf0: 3738 2d33 2d33 3139 2d32 3634 3337 2d30 78-3-319-26437-0 │ │ │ │ -00059c00: 292e 0a0a 5228 6929 203d 2053 2f28 6666 )...R(i) = S/(ff │ │ │ │ -00059c10: 5f30 2c2e 2e2c 6666 5f7b 692d 317d 292e _0,..,ff_{i-1}). │ │ │ │ -00059c20: 2048 6572 6520 303c 3d20 6920 3c3d 2063 Here 0<= i <= c │ │ │ │ -00059c30: 2c20 616e 6420 5220 3d20 5228 6329 2061 , and R = R(c) a │ │ │ │ -00059c40: 6e64 2053 203d 2052 2830 292e 0a0a 4228 nd S = R(0)...B( │ │ │ │ -00059c50: 6929 203d 2074 6865 206d 6174 7269 7820 i) = the matrix │ │ │ │ -00059c60: 286f 7665 7220 5329 2072 6570 7265 7365 (over S) represe │ │ │ │ -00059c70: 6e74 696e 6720 645f 693a 2042 5f31 2869 nting d_i: B_1(i │ │ │ │ -00059c80: 2920 5c74 6f20 425f 3028 6929 0a0a 6428 ) \to B_0(i)..d( │ │ │ │ -00059c90: 6929 3a20 415f 3128 6929 205c 746f 2041 i): A_1(i) \to A │ │ │ │ -00059ca0: 5f30 2869 2920 7468 6520 7265 7374 7269 _0(i) the restri │ │ │ │ -00059cb0: 6374 696f 6e20 6f66 2064 203d 2064 2863 ction of d = d(c │ │ │ │ -00059cc0: 292e 2077 6865 7265 2041 2869 2920 3d0a ). where A(i) =. │ │ │ │ -00059cd0: 5c6f 706c 7573 5f7b 693d 317d 5e70 2042 \oplus_{i=1}^p B │ │ │ │ -00059ce0: 2869 290a 0a0a 0a54 6865 206d 6170 2068 (i)....The map h │ │ │ │ -00059cf0: 2069 7320 6120 6469 7265 6374 2073 756d is a direct sum │ │ │ │ -00059d00: 206f 6620 6d61 7073 2074 6172 6765 7420 of maps target │ │ │ │ -00059d10: 6428 7029 205c 746f 2073 6f75 7263 6520 d(p) \to source │ │ │ │ -00059d20: 6428 7029 2074 6861 7420 6172 650a 686f d(p) that are.ho │ │ │ │ -00059d30: 6d6f 746f 7069 6573 2066 6f72 2066 665f motopies for ff_ │ │ │ │ -00059d40: 7020 6f6e 2074 6865 2072 6573 7472 6963 p on the restric │ │ │ │ -00059d50: 7469 6f6e 2064 2870 293a 206f 7665 7220 tion d(p): over │ │ │ │ -00059d60: 7468 6520 7269 6e67 2052 2328 702d 3129 the ring R#(p-1) │ │ │ │ -00059d70: 203d 0a53 2f28 6666 2331 2e2e 6666 2328 =.S/(ff#1..ff#( │ │ │ │ -00059d80: 702d 3129 2c20 736f 2064 2870 2920 2a20 p-1), so d(p) * │ │ │ │ -00059d90: 6823 7020 3d20 6666 2370 206d 6f64 2028 h#p = ff#p mod ( │ │ │ │ -00059da0: 6666 2331 2e2e 6666 2328 702d 3129 2e0a ff#1..ff#(p-1).. │ │ │ │ -00059db0: 0a49 6e20 6164 6469 7469 6f6e 2c20 6823 .In addition, h# │ │ │ │ -00059dc0: 7020 2a20 6428 7029 2069 6e64 7563 6573 p * d(p) induces │ │ │ │ -00059dd0: 2066 6623 7020 6f6e 2042 3123 7020 6d6f ff#p on B1#p mo │ │ │ │ -00059de0: 6420 2866 6623 312e 2e66 6623 2870 2d31 d (ff#1..ff#(p-1 │ │ │ │ -00059df0: 292e 0a0a 4865 7265 2069 7320 6120 7369 )...Here is a si │ │ │ │ -00059e00: 6d70 6c65 2065 7861 6d70 6c65 3a0a 0a2b mple example:..+ │ │ │ │ +00059450: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ +00059460: 0a20 2020 2020 2020 204d 4620 3d20 6d61 . MF = ma │ │ │ │ +00059470: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +00059480: 6e28 6666 2c4d 290a 2020 2a20 496e 7075 n(ff,M). * Inpu │ │ │ │ +00059490: 7473 3a0a 2020 2020 2020 2a20 6666 2c20 ts:. * ff, │ │ │ │ +000594a0: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ +000594b0: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ +000594c0: 7472 6978 2c2c 2061 2073 7566 6669 6369 trix,, a suffici │ │ │ │ +000594d0: 656e 746c 7920 6765 6e65 7261 6c0a 2020 ently general. │ │ │ │ +000594e0: 2020 2020 2020 7265 6775 6c61 7220 7365 regular se │ │ │ │ +000594f0: 7175 656e 6365 2069 6e20 6120 7269 6e67 quence in a ring │ │ │ │ +00059500: 2053 0a20 2020 2020 202a 204d 2c20 6120 S. * M, a │ │ │ │ +00059510: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ +00059520: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ +00059530: 6c65 2c2c 2061 206d 6178 696d 616c 2043 le,, a maximal C │ │ │ │ +00059540: 6f68 656e 2d4d 6163 6175 6c61 790a 2020 ohen-Macaulay. │ │ │ │ +00059550: 2020 2020 2020 6d6f 6475 6c65 206f 7665 module ove │ │ │ │ +00059560: 7220 532f 6964 6561 6c20 6666 0a20 202a r S/ideal ff. * │ │ │ │ +00059570: 202a 6e6f 7465 204f 7074 696f 6e61 6c20 *note Optional │ │ │ │ +00059580: 696e 7075 7473 3a20 284d 6163 6175 6c61 inputs: (Macaula │ │ │ │ +00059590: 7932 446f 6329 7573 696e 6720 6675 6e63 y2Doc)using func │ │ │ │ +000595a0: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ +000595b0: 6e61 6c20 696e 7075 7473 2c3a 0a20 2020 nal inputs,:. │ │ │ │ +000595c0: 2020 202a 2041 7567 6d65 6e74 6174 696f * Augmentatio │ │ │ │ +000595d0: 6e20 3d3e 202e 2e2e 2c20 6465 6661 756c n => ..., defaul │ │ │ │ +000595e0: 7420 7661 6c75 6520 7472 7565 0a20 2020 t value true. │ │ │ │ +000595f0: 2020 202a 2043 6865 636b 203d 3e20 2e2e * Check => .. │ │ │ │ +00059600: 2e2c 2064 6566 6175 6c74 2076 616c 7565 ., default value │ │ │ │ +00059610: 2066 616c 7365 0a20 2020 2020 202a 204c false. * L │ │ │ │ +00059620: 6179 6572 6564 203d 3e20 2e2e 2e2c 2064 ayered => ..., d │ │ │ │ +00059630: 6566 6175 6c74 2076 616c 7565 2074 7275 efault value tru │ │ │ │ +00059640: 650a 2020 2020 2020 2a20 5665 7262 6f73 e. * Verbos │ │ │ │ +00059650: 6520 3d3e 202e 2e2e 2c20 6465 6661 756c e => ..., defaul │ │ │ │ +00059660: 7420 7661 6c75 6520 6661 6c73 650a 2020 t value false. │ │ │ │ +00059670: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00059680: 202a 204d 462c 2061 202a 6e6f 7465 206c * MF, a *note l │ │ │ │ +00059690: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +000596a0: 6f63 294c 6973 742c 2c20 5c7b 642c 682c oc)List,, \{d,h, │ │ │ │ +000596b0: 6761 6d6d 615c 7d2c 2077 6865 7265 2064 gamma\}, where d │ │ │ │ +000596c0: 3a41 5f31 205c 746f 0a20 2020 2020 2020 :A_1 \to. │ │ │ │ +000596d0: 2041 5f30 2061 6e64 2068 3a20 5c6f 706c A_0 and h: \opl │ │ │ │ +000596e0: 7573 2041 5f30 2870 2920 5c74 6f20 415f us A_0(p) \to A_ │ │ │ │ +000596f0: 3120 6973 2074 6865 2064 6972 6563 7420 1 is the direct │ │ │ │ +00059700: 7375 6d20 6f66 2070 6172 7469 616c 0a20 sum of partial. │ │ │ │ +00059710: 2020 2020 2020 2068 6f6d 6f74 6f70 6965 homotopie │ │ │ │ +00059720: 732c 2061 6e64 2067 616d 6d61 3a20 415f s, and gamma: A_ │ │ │ │ +00059730: 3020 2d3e 4d20 6973 2074 6865 2061 7567 0 ->M is the aug │ │ │ │ +00059740: 6d65 6e74 6174 696f 6e20 2872 6574 7572 mentation (retur │ │ │ │ +00059750: 6e65 6420 6f6e 6c79 2069 660a 2020 2020 ned only if. │ │ │ │ +00059760: 2020 2020 4175 676d 656e 7461 7469 6f6e Augmentation │ │ │ │ +00059770: 203d 3e74 7275 6529 0a0a 4465 7363 7269 =>true)..Descri │ │ │ │ +00059780: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00059790: 3d0a 0a54 6865 2069 6e70 7574 206d 6f64 =..The input mod │ │ │ │ +000597a0: 756c 6520 4d20 7368 6f75 6c64 2062 6520 ule M should be │ │ │ │ +000597b0: 6120 6d61 7869 6d61 6c20 436f 6865 6e2d a maximal Cohen- │ │ │ │ +000597c0: 4d61 6361 756c 6179 206d 6f64 756c 6520 Macaulay module │ │ │ │ +000597d0: 6f76 6572 2052 203d 2053 2f69 6465 616c over R = S/ideal │ │ │ │ +000597e0: 0a66 662e 2020 4966 204d 2069 7320 696e .ff. If M is in │ │ │ │ +000597f0: 2066 6163 7420 6120 2268 6967 6820 7379 fact a "high sy │ │ │ │ +00059800: 7a79 6779 222c 2074 6865 6e20 7468 6520 zygy", then the │ │ │ │ +00059810: 6675 6e63 7469 6f6e 0a6d 6174 7269 7846 function.matrixF │ │ │ │ +00059820: 6163 746f 7269 7a61 7469 6f6e 2866 662c actorization(ff, │ │ │ │ +00059830: 4d2c 4c61 7965 7265 643d 3e66 616c 7365 M,Layered=>false │ │ │ │ +00059840: 2920 7573 6573 2061 2064 6966 6665 7265 ) uses a differe │ │ │ │ +00059850: 6e74 2c20 6661 7374 6572 2061 6c67 6f72 nt, faster algor │ │ │ │ +00059860: 6974 686d 0a77 6869 6368 206f 6e6c 7920 ithm.which only │ │ │ │ +00059870: 776f 726b 7320 696e 2074 6865 2068 6967 works in the hig │ │ │ │ +00059880: 6820 7379 7a79 6779 2063 6173 652e 0a0a h syzygy case... │ │ │ │ +00059890: 496e 2061 6c6c 2065 7861 6d70 6c65 7320 In all examples │ │ │ │ +000598a0: 7765 206b 6e6f 772c 204d 2063 616e 2062 we know, M can b │ │ │ │ +000598b0: 6520 636f 6e73 6964 6572 6564 2061 2022 e considered a " │ │ │ │ +000598c0: 6869 6768 2073 797a 7967 7922 2061 7320 high syzygy" as │ │ │ │ +000598d0: 6c6f 6e67 2061 730a 4578 745e 7b65 7665 long as.Ext^{eve │ │ │ │ +000598e0: 6e7d 5f52 284d 2c6b 2920 616e 6420 4578 n}_R(M,k) and Ex │ │ │ │ +000598f0: 745e 7b6f 6464 7d5f 5228 4d2c 6b29 2068 t^{odd}_R(M,k) h │ │ │ │ +00059900: 6176 6520 6e65 6761 7469 7665 2072 6567 ave negative reg │ │ │ │ +00059910: 756c 6172 6974 7920 6f76 6572 2074 6865 ularity over the │ │ │ │ +00059920: 2072 696e 670a 6f66 2043 4920 6f70 6572 ring.of CI oper │ │ │ │ +00059930: 6174 6f72 7320 2872 6567 7261 6465 6420 ators (regraded │ │ │ │ +00059940: 7769 7468 2076 6172 6961 626c 6573 206f with variables o │ │ │ │ +00059950: 6620 6465 6772 6565 2031 2e20 486f 7765 f degree 1. Howe │ │ │ │ +00059960: 7665 722c 2074 6865 2062 6573 7420 7265 ver, the best re │ │ │ │ +00059970: 7375 6c74 0a77 6520 6361 6e20 7072 6f76 sult.we can prov │ │ │ │ +00059980: 6520 6973 2074 6861 7420 6974 2073 7566 e is that it suf │ │ │ │ +00059990: 6669 6365 7320 746f 2068 6176 6520 7265 fices to have re │ │ │ │ +000599a0: 6775 6c61 7269 7479 203c 202d 2832 2a64 gularity < -(2*d │ │ │ │ +000599b0: 696d 2052 2b31 292e 0a0a 5768 656e 2074 im R+1)...When t │ │ │ │ +000599c0: 6865 206f 7074 696f 6e61 6c20 696e 7075 he optional inpu │ │ │ │ +000599d0: 7420 4368 6563 6b3d 3d74 7275 6520 2874 t Check==true (t │ │ │ │ +000599e0: 6865 2064 6566 6175 6c74 2069 7320 4368 he default is Ch │ │ │ │ +000599f0: 6563 6b3d 3d66 616c 7365 292c 2074 6865 eck==false), the │ │ │ │ +00059a00: 0a70 726f 7065 7274 6965 7320 696e 2074 .properties in t │ │ │ │ +00059a10: 6865 2064 6566 696e 6974 696f 6e20 6f66 he definition of │ │ │ │ +00059a20: 204d 6174 7269 7820 4661 6374 6f72 697a Matrix Factoriz │ │ │ │ +00059a30: 6174 696f 6e20 6172 6520 7665 7269 6669 ation are verifi │ │ │ │ +00059a40: 6564 0a0a 5468 6520 6f75 7470 7574 2069 ed..The output i │ │ │ │ +00059a50: 7320 6120 6c69 7374 206f 6620 6d61 7073 s a list of maps │ │ │ │ +00059a60: 205c 7b64 2c68 5c7d 206f 7220 5c7b 642c \{d,h\} or \{d, │ │ │ │ +00059a70: 682c 6761 6d6d 615c 7d2c 2077 6865 7265 h,gamma\}, where │ │ │ │ +00059a80: 2067 616d 6d61 2069 7320 616e 0a61 7567 gamma is an.aug │ │ │ │ +00059a90: 6d65 6e74 6174 696f 6e2c 2074 6861 7420 mentation, that │ │ │ │ +00059aa0: 6973 2c20 6120 6d61 7020 6672 6f6d 2074 is, a map from t │ │ │ │ +00059ab0: 6172 6765 7420 6420 746f 204d 2e0a 0a54 arget d to M...T │ │ │ │ +00059ac0: 6865 206d 6170 2064 2069 7320 6120 7370 he map d is a sp │ │ │ │ +00059ad0: 6563 6961 6c20 6c69 6674 696e 6720 746f ecial lifting to │ │ │ │ +00059ae0: 2053 206f 6620 6120 7072 6573 656e 7461 S of a presenta │ │ │ │ +00059af0: 7469 6f6e 206f 6620 4d20 6f76 6572 2052 tion of M over R │ │ │ │ +00059b00: 2e20 546f 2065 7870 6c61 696e 0a74 6865 . To explain.the │ │ │ │ +00059b10: 2063 6f6e 7465 6e74 732c 2077 6520 696e contents, we in │ │ │ │ +00059b20: 7472 6f64 7563 6520 736f 6d65 206e 6f74 troduce some not │ │ │ │ +00059b30: 6174 696f 6e20 2866 726f 6d20 4569 7365 ation (from Eise │ │ │ │ +00059b40: 6e62 7564 2061 6e64 2050 6565 7661 2c20 nbud and Peeva, │ │ │ │ +00059b50: 224d 696e 696d 616c 0a66 7265 6520 7265 "Minimal.free re │ │ │ │ +00059b60: 736f 6c75 7469 6f6e 7320 6f76 6572 2063 solutions over c │ │ │ │ +00059b70: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ +00059b80: 7469 6f6e 7322 204c 6563 7475 7265 204e tions" Lecture N │ │ │ │ +00059b90: 6f74 6573 2069 6e20 4d61 7468 656d 6174 otes in Mathemat │ │ │ │ +00059ba0: 6963 732c 0a32 3135 322e 2053 7072 696e ics,.2152. Sprin │ │ │ │ +00059bb0: 6765 722c 2043 6861 6d2c 2032 3031 362e ger, Cham, 2016. │ │ │ │ +00059bc0: 2078 2b31 3037 2070 702e 2049 5342 4e3a x+107 pp. ISBN: │ │ │ │ +00059bd0: 2039 3738 2d33 2d33 3139 2d32 3634 3336 978-3-319-26436 │ │ │ │ +00059be0: 2d33 3b0a 3937 382d 332d 3331 392d 3236 -3;.978-3-319-26 │ │ │ │ +00059bf0: 3433 372d 3029 2e0a 0a52 2869 2920 3d20 437-0)...R(i) = │ │ │ │ +00059c00: 532f 2866 665f 302c 2e2e 2c66 665f 7b69 S/(ff_0,..,ff_{i │ │ │ │ +00059c10: 2d31 7d29 2e20 4865 7265 2030 3c3d 2069 -1}). Here 0<= i │ │ │ │ +00059c20: 203c 3d20 632c 2061 6e64 2052 203d 2052 <= c, and R = R │ │ │ │ +00059c30: 2863 2920 616e 6420 5320 3d20 5228 3029 (c) and S = R(0) │ │ │ │ +00059c40: 2e0a 0a42 2869 2920 3d20 7468 6520 6d61 ...B(i) = the ma │ │ │ │ +00059c50: 7472 6978 2028 6f76 6572 2053 2920 7265 trix (over S) re │ │ │ │ +00059c60: 7072 6573 656e 7469 6e67 2064 5f69 3a20 presenting d_i: │ │ │ │ +00059c70: 425f 3128 6929 205c 746f 2042 5f30 2869 B_1(i) \to B_0(i │ │ │ │ +00059c80: 290a 0a64 2869 293a 2041 5f31 2869 2920 )..d(i): A_1(i) │ │ │ │ +00059c90: 5c74 6f20 415f 3028 6929 2074 6865 2072 \to A_0(i) the r │ │ │ │ +00059ca0: 6573 7472 6963 7469 6f6e 206f 6620 6420 estriction of d │ │ │ │ +00059cb0: 3d20 6428 6329 2e20 7768 6572 6520 4128 = d(c). where A( │ │ │ │ +00059cc0: 6929 203d 0a5c 6f70 6c75 735f 7b69 3d31 i) =.\oplus_{i=1 │ │ │ │ +00059cd0: 7d5e 7020 4228 6929 0a0a 0a0a 5468 6520 }^p B(i)....The │ │ │ │ +00059ce0: 6d61 7020 6820 6973 2061 2064 6972 6563 map h is a direc │ │ │ │ +00059cf0: 7420 7375 6d20 6f66 206d 6170 7320 7461 t sum of maps ta │ │ │ │ +00059d00: 7267 6574 2064 2870 2920 5c74 6f20 736f rget d(p) \to so │ │ │ │ +00059d10: 7572 6365 2064 2870 2920 7468 6174 2061 urce d(p) that a │ │ │ │ +00059d20: 7265 0a68 6f6d 6f74 6f70 6965 7320 666f re.homotopies fo │ │ │ │ +00059d30: 7220 6666 5f70 206f 6e20 7468 6520 7265 r ff_p on the re │ │ │ │ +00059d40: 7374 7269 6374 696f 6e20 6428 7029 3a20 striction d(p): │ │ │ │ +00059d50: 6f76 6572 2074 6865 2072 696e 6720 5223 over the ring R# │ │ │ │ +00059d60: 2870 2d31 2920 3d0a 532f 2866 6623 312e (p-1) =.S/(ff#1. │ │ │ │ +00059d70: 2e66 6623 2870 2d31 292c 2073 6f20 6428 .ff#(p-1), so d( │ │ │ │ +00059d80: 7029 202a 2068 2370 203d 2066 6623 7020 p) * h#p = ff#p │ │ │ │ +00059d90: 6d6f 6420 2866 6623 312e 2e66 6623 2870 mod (ff#1..ff#(p │ │ │ │ +00059da0: 2d31 292e 0a0a 496e 2061 6464 6974 696f -1)...In additio │ │ │ │ +00059db0: 6e2c 2068 2370 202a 2064 2870 2920 696e n, h#p * d(p) in │ │ │ │ +00059dc0: 6475 6365 7320 6666 2370 206f 6e20 4231 duces ff#p on B1 │ │ │ │ +00059dd0: 2370 206d 6f64 2028 6666 2331 2e2e 6666 #p mod (ff#1..ff │ │ │ │ +00059de0: 2328 702d 3129 2e0a 0a48 6572 6520 6973 #(p-1)...Here is │ │ │ │ +00059df0: 2061 2073 696d 706c 6520 6578 616d 706c a simple exampl │ │ │ │ +00059e00: 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d e:..+----------- │ │ │ │ 00059e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059e20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059e40: 2d2d 2b0a 7c69 3120 3a20 7365 7452 616e --+.|i1 : setRan │ │ │ │ -00059e50: 646f 6d53 6565 6420 3020 2020 2020 2020 domSeed 0 │ │ │ │ -00059e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059e70: 2020 2020 2020 207c 0a7c 202d 2d20 7365 |.| -- se │ │ │ │ -00059e80: 7474 696e 6720 7261 6e64 6f6d 2073 6565 tting random see │ │ │ │ -00059e90: 6420 746f 2030 2020 2020 2020 2020 2020 d to 0 │ │ │ │ -00059ea0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00059e30: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2073 -------+.|i1 : s │ │ │ │ +00059e40: 6574 5261 6e64 6f6d 5365 6564 2030 2020 etRandomSeed 0 │ │ │ │ +00059e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00059e60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00059e70: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ +00059e80: 6d20 7365 6564 2074 6f20 3020 2020 2020 m seed to 0 │ │ │ │ +00059e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00059ea0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00059eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059ee0: 207c 0a7c 6f31 203d 2030 2020 2020 2020 |.|o1 = 0 │ │ │ │ +00059ed0: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +00059ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f10: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00059f00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00059f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00059f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00059f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00059f50: 203a 206b 6b20 3d20 5a5a 2f31 3031 2020 : kk = ZZ/101 │ │ │ │ +00059f40: 2b0a 7c69 3220 3a20 6b6b 203d 205a 5a2f +.|i2 : kk = ZZ/ │ │ │ │ +00059f50: 3130 3120 2020 2020 2020 2020 2020 2020 101 │ │ │ │ 00059f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059f80: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00059f70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00059f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00059f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fb0: 2020 2020 207c 0a7c 6f32 203d 206b 6b20 |.|o2 = kk │ │ │ │ +00059fa0: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +00059fb0: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 00059fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00059fe0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00059fd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00059fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00059ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a010: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005a020: 0a7c 6f32 203a 2051 756f 7469 656e 7452 .|o2 : QuotientR │ │ │ │ -0005a030: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0005a040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a050: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005a010: 2020 2020 7c0a 7c6f 3220 3a20 5175 6f74 |.|o2 : Quot │ │ │ │ +0005a020: 6965 6e74 5269 6e67 2020 2020 2020 2020 ientRing │ │ │ │ +0005a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a040: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005a050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a080: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0005a090: 2053 203d 206b 6b5b 612c 622c 752c 765d S = kk[a,b,u,v] │ │ │ │ +0005a070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005a080: 7c69 3320 3a20 5320 3d20 6b6b 5b61 2c62 |i3 : S = kk[a,b │ │ │ │ +0005a090: 2c75 2c76 5d20 2020 2020 2020 2020 2020 ,u,v] │ │ │ │ 0005a0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a0b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005a0c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005a0b0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005a0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a0f0: 2020 207c 0a7c 6f33 203d 2053 2020 2020 |.|o3 = S │ │ │ │ +0005a0e0: 2020 2020 2020 2020 7c0a 7c6f 3320 3d20 |.|o3 = │ │ │ │ +0005a0f0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 0005a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a120: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0005a110: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005a120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a150: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0005a160: 6f33 203a 2050 6f6c 796e 6f6d 6961 6c52 o3 : PolynomialR │ │ │ │ -0005a170: 696e 6720 2020 2020 2020 2020 2020 2020 ing │ │ │ │ -0005a180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0005a150: 2020 7c0a 7c6f 3320 3a20 506f 6c79 6e6f |.|o3 : Polyno │ │ │ │ +0005a160: 6d69 616c 5269 6e67 2020 2020 2020 2020 mialRing │ │ │ │ +0005a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a180: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0005a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a1c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2066 -------+.|i4 : f │ │ │ │ -0005a1d0: 6620 3d20 6d61 7472 6978 2261 752c 6276 f = matrix"au,bv │ │ │ │ -0005a1e0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0005a1f0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0005a1c0: 3420 3a20 6666 203d 206d 6174 7269 7822 4 : ff = matrix" │ │ │ │ +0005a1d0: 6175 2c62 7622 2020 2020 2020 2020 2020 au,bv" │ │ │ │ +0005a1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a1f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a230: 207c 0a7c 6f34 203d 207c 2061 7520 6276 |.|o4 = | au bv │ │ │ │ -0005a240: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -0005a250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a260: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0005a220: 2020 2020 2020 7c0a 7c6f 3420 3d20 7c20 |.|o4 = | │ │ │ │ +0005a230: 6175 2062 7620 7c20 2020 2020 2020 2020 au bv | │ │ │ │ +0005a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a250: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0005a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a290: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0005a2a0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0005a2b0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0005a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a2d0: 7c0a 7c6f 3420 3a20 4d61 7472 6978 2053 |.|o4 : Matrix S │ │ │ │ -0005a2e0: 2020 3c2d 2d20 5320 2020 2020 2020 2020 <-- S │ │ │ │ -0005a2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a300: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0005a290: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005a2a0: 3120 2020 2020 2032 2020 2020 2020 2020 1 2 │ │ │ │ +0005a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a2c0: 2020 2020 207c 0a7c 6f34 203a 204d 6174 |.|o4 : Mat │ │ │ │ +0005a2d0: 7269 7820 5320 203c 2d2d 2053 2020 2020 rix S <-- S │ │ │ │ +0005a2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a2f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0005a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a330: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -0005a340: 3a20 5220 3d20 532f 6964 6561 6c20 6666 : R = S/ideal ff │ │ │ │ +0005a320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0005a330: 0a7c 6935 203a 2052 203d 2053 2f69 6465 .|i5 : R = S/ide │ │ │ │ +0005a340: 616c 2066 6620 2020 2020 2020 2020 2020 al ff │ │ │ │ 0005a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a360: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005a370: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0005a360: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a3a0: 2020 2020 7c0a 7c6f 3520 3d20 5220 2020 |.|o5 = R │ │ │ │ +0005a390: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ +0005a3a0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 0005a3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a3d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005a3c0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005a3d0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a400: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005a410: 7c6f 3520 3a20 5175 6f74 6965 6e74 5269 |o5 : QuotientRi │ │ │ │ -0005a420: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -0005a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a440: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005a400: 2020 207c 0a7c 6f35 203a 2051 756f 7469 |.|o5 : Quoti │ │ │ │ +0005a410: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0005a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a430: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005a440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a470: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ -0005a480: 4d30 203d 2052 5e31 2f69 6465 616c 2261 M0 = R^1/ideal"a │ │ │ │ -0005a490: 2c62 2220 2020 2020 2020 2020 2020 2020 ,b" │ │ │ │ -0005a4a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005a460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0005a470: 6936 203a 204d 3020 3d20 525e 312f 6964 i6 : M0 = R^1/id │ │ │ │ +0005a480: 6561 6c22 612c 6222 2020 2020 2020 2020 eal"a,b" │ │ │ │ +0005a490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a4a0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005a4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a4c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a4d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a4e0: 2020 7c0a 7c6f 3620 3d20 636f 6b65 726e |.|o6 = cokern │ │ │ │ -0005a4f0: 656c 207c 2061 2062 207c 2020 2020 2020 el | a b | │ │ │ │ -0005a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a510: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0005a4d0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ +0005a4e0: 6f6b 6572 6e65 6c20 7c20 6120 6220 7c20 okernel | a b | │ │ │ │ +0005a4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a500: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a540: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a540: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005a550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a560: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ -0005a570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a580: 207c 0a7c 6f36 203a 2052 2d6d 6f64 756c |.|o6 : R-modul │ │ │ │ -0005a590: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ -0005a5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a5b0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005a560: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0005a570: 2020 2020 2020 7c0a 7c6f 3620 3a20 522d |.|o6 : R- │ │ │ │ +0005a580: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ +0005a590: 206f 6620 5220 2020 2020 2020 2020 2020 of R │ │ │ │ +0005a5a0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0005a5b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ -0005a5f0: 203a 204d 203d 2068 6967 6853 797a 7967 : M = highSyzyg │ │ │ │ -0005a600: 7920 4d30 2020 2020 2020 2020 2020 2020 y M0 │ │ │ │ -0005a610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a620: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005a5e0: 2b0a 7c69 3720 3a20 4d20 3d20 6869 6768 +.|i7 : M = high │ │ │ │ +0005a5f0: 5379 7a79 6779 204d 3020 2020 2020 2020 Syzygy M0 │ │ │ │ +0005a600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a610: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0005a620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a650: 2020 2020 207c 0a7c 6f37 203d 2063 6f6b |.|o7 = cok │ │ │ │ -0005a660: 6572 6e65 6c20 7b32 7d20 7c20 6220 2d61 ernel {2} | b -a │ │ │ │ -0005a670: 2030 2030 207c 2020 2020 2020 2020 2020 0 0 | │ │ │ │ -0005a680: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0005a690: 2020 2020 2020 2020 2020 207b 327d 207c {2} | │ │ │ │ -0005a6a0: 2030 2030 2020 6120 6220 7c20 2020 2020 0 0 a b | │ │ │ │ -0005a6b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005a6c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -0005a6d0: 7b32 7d20 7c20 3020 7620 2030 2075 207c {2} | 0 v 0 u | │ │ │ │ -0005a6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a6f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005a640: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ +0005a650: 3d20 636f 6b65 726e 656c 207b 327d 207c = cokernel {2} | │ │ │ │ +0005a660: 2062 202d 6120 3020 3020 7c20 2020 2020 b -a 0 0 | │ │ │ │ +0005a670: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0005a680: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0005a690: 7b32 7d20 7c20 3020 3020 2061 2062 207c {2} | 0 0 a b | │ │ │ │ +0005a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a6b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005a6c0: 2020 2020 207b 327d 207c 2030 2076 2020 {2} | 0 v │ │ │ │ +0005a6d0: 3020 7520 7c20 2020 2020 2020 2020 2020 0 u | │ │ │ │ +0005a6e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005a6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a720: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -0005a730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a740: 2020 2020 2020 2020 3320 2020 2020 2020 3 │ │ │ │ -0005a750: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005a760: 7c6f 3720 3a20 522d 6d6f 6475 6c65 2c20 |o7 : R-module, │ │ │ │ -0005a770: 7175 6f74 6965 6e74 206f 6620 5220 2020 quotient of R │ │ │ │ -0005a780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a790: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005a710: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005a720: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005a730: 2020 2020 2020 2020 2020 2020 2033 2020 3 │ │ │ │ +0005a740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a750: 2020 207c 0a7c 6f37 203a 2052 2d6d 6f64 |.|o7 : R-mod │ │ │ │ +0005a760: 756c 652c 2071 756f 7469 656e 7420 6f66 ule, quotient of │ │ │ │ +0005a770: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0005a780: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005a790: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a7a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a7c0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 --------+.|i8 : │ │ │ │ -0005a7d0: 4d46 203d 206d 6174 7269 7846 6163 746f MF = matrixFacto │ │ │ │ -0005a7e0: 7269 7a61 7469 6f6e 2866 662c 4d29 3b20 rization(ff,M); │ │ │ │ -0005a7f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0005a7b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +0005a7c0: 6938 203a 204d 4620 3d20 6d61 7472 6978 i8 : MF = matrix │ │ │ │ +0005a7d0: 4661 6374 6f72 697a 6174 696f 6e28 6666 Factorization(ff │ │ │ │ +0005a7e0: 2c4d 293b 2020 2020 2020 2020 2020 2020 ,M); │ │ │ │ +0005a7f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0005a800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005a810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005a830: 2d2d 2b0a 7c69 3920 3a20 6e65 744c 6973 --+.|i9 : netLis │ │ │ │ -0005a840: 7420 4252 616e 6b73 204d 4620 2020 2020 t BRanks MF │ │ │ │ -0005a850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a860: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0005a820: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 206e -------+.|i9 : n │ │ │ │ +0005a830: 6574 4c69 7374 2042 5261 6e6b 7320 4d46 etList BRanks MF │ │ │ │ +0005a840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005a850: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005a860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a890: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0005a8a0: 2020 2020 2b2d 2b2d 2b20 2020 2020 2020 +-+-+ │ │ │ │ +0005a890: 207c 0a7c 2020 2020 202b 2d2b 2d2b 2020 |.| +-+-+ │ │ │ │ +0005a8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a8d0: 207c 0a7c 6f39 203d 207c 327c 327c 2020 |.|o9 = |2|2| │ │ │ │ +0005a8c0: 2020 2020 2020 7c0a 7c6f 3920 3d20 7c32 |.|o9 = |2 │ │ │ │ +0005a8d0: 7c32 7c20 2020 2020 2020 2020 2020 2020 |2| │ │ │ │ 0005a8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a900: 2020 2020 2020 7c0a 7c20 2020 2020 2b2d |.| +- │ │ │ │ -0005a910: 2b2d 2b20 2020 2020 2020 2020 2020 2020 +-+ │ │ │ │ +0005a8f0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0005a900: 2020 202b 2d2b 2d2b 2020 2020 2020 2020 +-+-+ │ │ │ │ +0005a910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005a920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005a930: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0005a940: 2020 207c 317c 327c 2020 2020 2020 2020 |1|2| │ │ │ │ +0005a930: 7c0a 7c20 2020 2020 7c31 7c32 7c20 2020 |.| |1|2| │ │ │ │ +0005a940: 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2062 207c 7c20 = |{2} | a b || │ │ │ │ +0005aa30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005aa40: 2020 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b20 +-----------+ │ │ │ │ +0005aa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005aa60: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005aa70: 7c6f 3130 203d 207c 7b32 7d20 7c20 6120 |o10 = |{2} | a │ │ │ │ +0005aa80: 6220 7c7c 2020 2020 2020 2020 2020 2020 b || │ │ │ │ 0005aa90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005aaa0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005aab0: 7c20 2020 2020 207c 7b32 7d20 7c20 3020 | |{2} | 0 │ │ │ │ -0005aac0: 7520 7c7c 2020 2020 2020 2020 2020 2020 u || │ │ │ │ -0005aad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005aae0: 2020 207c 0a7c 2020 2020 2020 2b2d 2d2d |.| +--- │ │ │ │ -0005aaf0: 2d2d 2d2d 2d2d 2d2d 2b20 2020 2020 2020 --------+ │ │ │ │ -0005ab00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ab10: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0005ab20: 207c 7b32 7d20 7c20 6220 6120 7c7c 2020 |{2} | b a || │ │ │ │ +0005aaa0: 2020 207c 0a7c 2020 2020 2020 7c7b 327d |.| |{2} │ │ │ │ +0005aab0: 207c 2030 2075 207c 7c20 2020 2020 2020 | 0 u || │ │ │ │ +0005aac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005aad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0005aae0: 202b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 2020 +-----------+ │ │ │ │ +0005aaf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ab00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005ab10: 2020 2020 2020 7c7b 327d 207c 2062 2061 |{2} | b a │ │ │ │ +0005ab20: 207c 7c20 2020 2020 2020 2020 2020 2020 || │ │ │ │ 0005ab30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ab40: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0005ab50: 2020 2020 2020 2b2d 2d2d 2d2d 2d2d 2d2d +--------- │ │ │ │ -0005ab60: 2d2d 2b20 2020 2020 2020 2020 2020 2020 --+ │ │ │ │ -0005ab70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ab80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +0005ab40: 2020 7c0a 7c20 2020 2020 202b 2d2d 2d2d |.| +---- │ │ │ │ +0005ab50: 2d2d 2d2d 2d2d 2d2b 2020 2020 2020 2020 -------+ │ │ │ │ +0005ab60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ab70: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0005ab80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ab90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005abb0: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 3a20 -------+.|i11 : │ │ │ │ -0005abc0: 6265 7474 6920 6672 6565 5265 736f 6c75 betti freeResolu │ │ │ │ -0005abd0: 7469 6f6e 284d 2c20 4c65 6e67 7468 4c69 tion(M, LengthLi │ │ │ │ -0005abe0: 6d69 7420 3d3e 2037 2920 2020 7c0a 7c20 mit => 7) |.| │ │ │ │ +0005aba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0005abb0: 3131 203a 2062 6574 7469 2066 7265 6552 11 : betti freeR │ │ │ │ +0005abc0: 6573 6f6c 7574 696f 6e28 4d2c 204c 656e esolution(M, Len │ │ │ │ +0005abd0: 6774 684c 696d 6974 203d 3e20 3729 2020 gthLimit => 7) │ │ │ │ +0005abe0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005abf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ac00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ac10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ac20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -0005ac30: 2030 2031 2032 2033 2034 2035 2036 2020 0 1 2 3 4 5 6 │ │ │ │ -0005ac40: 3720 2020 2020 2020 2020 2020 2020 2020 7 │ │ │ │ -0005ac50: 2020 2020 2020 7c0a 7c6f 3131 203d 2074 |.|o11 = t │ │ │ │ -0005ac60: 6f74 616c 3a20 3320 3420 3520 3620 3720 otal: 3 4 5 6 7 │ │ │ │ -0005ac70: 3820 3920 3130 2020 2020 2020 2020 2020 8 9 10 │ │ │ │ -0005ac80: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -0005ac90: 2020 2020 2020 2020 323a 2033 2034 2035 2: 3 4 5 │ │ │ │ -0005aca0: 2036 2037 2038 2039 2031 3020 2020 2020 6 7 8 9 10 │ │ │ │ -0005acb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005acc0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005ac10: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0005ac20: 2020 2020 2020 3020 3120 3220 3320 3420 0 1 2 3 4 │ │ │ │ +0005ac30: 3520 3620 2037 2020 2020 2020 2020 2020 5 6 7 │ │ │ │ +0005ac40: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0005ac50: 3120 3d20 746f 7461 6c3a 2033 2034 2035 1 = total: 3 4 5 │ │ │ │ +0005ac60: 2036 2037 2038 2039 2031 3020 2020 2020 6 7 8 9 10 │ │ │ │ +0005ac70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ac80: 7c0a 7c20 2020 2020 2020 2020 2032 3a20 |.| 2: │ │ │ │ +0005ac90: 3320 3420 3520 3620 3720 3820 3920 3130 3 4 5 6 7 8 9 10 │ │ │ │ +0005aca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005acb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0005acc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005acd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ace0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005acf0: 2020 2020 207c 0a7c 6f31 3120 3a20 4265 |.|o11 : Be │ │ │ │ -0005ad00: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ -0005ad10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ad20: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0005ace0: 2020 2020 2020 2020 2020 7c0a 7c6f 3131 |.|o11 │ │ │ │ +0005acf0: 203a 2042 6574 7469 5461 6c6c 7920 2020 : BettiTally │ │ │ │ +0005ad00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ad10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0005ad20: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0005ad30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ad40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ad50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0005ad60: 0a7c 6931 3220 3a20 696e 6669 6e69 7465 .|i12 : infinite │ │ │ │ -0005ad70: 4265 7474 694e 756d 6265 7273 2028 4d46 BettiNumbers (MF │ │ │ │ -0005ad80: 2c37 2920 2020 2020 2020 2020 2020 2020 ,7) │ │ │ │ -0005ad90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005ad50: 2d2d 2d2d 2b0a 7c69 3132 203a 2069 6e66 ----+.|i12 : inf │ │ │ │ +0005ad60: 696e 6974 6542 6574 7469 4e75 6d62 6572 initeBettiNumber │ │ │ │ +0005ad70: 7320 284d 462c 3729 2020 2020 2020 2020 s (MF,7) │ │ │ │ +0005ad80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005adc0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ -0005add0: 3d20 7b33 2c20 342c 2035 2c20 362c 2037 = {3, 4, 5, 6, 7 │ │ │ │ -0005ade0: 2c20 382c 2039 2c20 3130 7d20 2020 2020 , 8, 9, 10} │ │ │ │ -0005adf0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005ae00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005adb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005adc0: 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2020 2020 2020 |.| │ │ │ │ -0005af10: 2020 2020 2020 2030 2031 2032 2020 2020 0 1 2 │ │ │ │ +0005aef0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0005af00: 2020 2020 2020 2020 2020 2020 3020 3120 0 1 │ │ │ │ +0005af10: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0005af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005af30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0005af40: 3133 203d 2074 6f74 616c 3a20 3320 3520 13 = total: 3 5 │ │ │ │ -0005af50: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -0005af60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005af70: 207c 0a7c 2020 2020 2020 2020 2020 323a |.| 2: │ │ │ │ -0005af80: 2033 2034 202e 2020 2020 2020 2020 2020 3 4 . │ │ │ │ -0005af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005afa0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0005afb0: 2020 2033 3a20 2e20 3120 3220 2020 2020 3: . 1 2 │ │ │ │ +0005af30: 207c 0a7c 6f31 3320 3d20 746f 7461 6c3a |.|o13 = total: │ │ │ │ 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│ -0005b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b040: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0005b000: 2020 2020 207c 0a7c 6f31 3320 3a20 4265 |.|o13 : Be │ │ │ │ +0005b010: 7474 6954 616c 6c79 2020 2020 2020 2020 ttiTally │ │ │ │ +0005b020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005b030: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0005b040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b070: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 ----------+.|i14 │ │ │ │ -0005b080: 203a 2066 696e 6974 6542 6574 7469 4e75 : finiteBettiNu │ │ │ │ -0005b090: 6d62 6572 7320 4d46 2020 2020 2020 2020 mbers MF │ │ │ │ -0005b0a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005b0b0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0005b060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0005b070: 0a7c 6931 3420 3a20 6669 6e69 7465 4265 .|i14 : finiteBe │ │ │ │ +0005b080: 7474 694e 756d 6265 7273 204d 4620 2020 ttiNumbers MF │ │ │ │ +0005b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005b0a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b0e0: 2020 2020 7c0a 7c6f 3134 203d 207b 332c |.|o14 = {3, │ │ │ │ -0005b0f0: 2035 2c20 327d 2020 2020 2020 2020 2020 5, 2} │ │ │ │ -0005b100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b110: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005b0d0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +0005b0e0: 3d20 7b33 2c20 352c 2032 7d20 2020 2020 = {3, 5, 2} │ │ │ │ +0005b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005b100: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005b110: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005b120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005b130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b140: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005b150: 7c6f 3134 203a 204c 6973 7420 2020 2020 |o14 : List │ │ │ │ +0005b140: 2020 207c 0a7c 6f31 3420 3a20 4c69 7374 |.|o14 : List │ │ │ │ +0005b150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005b160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005b180: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005b170: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b1b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 --------+..See a │ │ │ │ -0005b1c0: 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 lso.========.. │ │ │ │ -0005b1d0: 2a20 2a6e 6f74 6520 6669 6e69 7465 4265 * *note finiteBe │ │ │ │ -0005b1e0: 7474 694e 756d 6265 7273 3a20 6669 6e69 ttiNumbers: fini │ │ │ │ -0005b1f0: 7465 4265 7474 694e 756d 6265 7273 2c20 teBettiNumbers, │ │ │ │ -0005b200: 2d2d 2062 6574 7469 206e 756d 6265 7273 -- betti numbers │ │ │ │ -0005b210: 206f 6620 6669 6e69 7465 0a20 2020 2072 of finite. r │ │ │ │ -0005b220: 6573 6f6c 7574 696f 6e20 636f 6d70 7574 esolution comput │ │ │ │ -0005b230: 6564 2066 726f 6d20 6120 6d61 7472 6978 ed from a matrix │ │ │ │ -0005b240: 2066 6163 746f 7269 7a61 7469 6f6e 0a20 factorization. │ │ │ │ -0005b250: 202a 202a 6e6f 7465 2069 6e66 696e 6974 * *note infinit │ │ │ │ -0005b260: 6542 6574 7469 4e75 6d62 6572 733a 2069 eBettiNumbers: i │ │ │ │ -0005b270: 6e66 696e 6974 6542 6574 7469 4e75 6d62 nfiniteBettiNumb │ │ │ │ -0005b280: 6572 732c 202d 2d20 6265 7474 6920 6e75 ers, -- betti nu │ │ │ │ -0005b290: 6d62 6572 7320 6f66 0a20 2020 2066 696e mbers of. fin │ │ │ │ -0005b2a0: 6974 6520 7265 736f 6c75 7469 6f6e 2063 ite resolution c │ │ │ │ -0005b2b0: 6f6d 7075 7465 6420 6672 6f6d 2061 206d omputed from a m │ │ │ │ -0005b2c0: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ -0005b2d0: 696f 6e0a 2020 2a20 2a6e 6f74 6520 6869 ion. * *note hi │ │ │ │ -0005b2e0: 6768 5379 7a79 6779 3a20 6869 6768 5379 ghSyzygy: highSy │ │ │ │ -0005b2f0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73 zygy, -- Returns │ │ │ │ -0005b300: 2061 2073 797a 7967 7920 6d6f 6475 6c65 a syzygy module │ │ │ │ -0005b310: 206f 6e65 2062 6579 6f6e 6420 7468 650a one beyond the. │ │ │ │ -0005b320: 2020 2020 7265 6775 6c61 7269 7479 206f regularity o │ │ │ │ -0005b330: 6620 4578 7428 4d2c 6b29 0a20 202a 202a f Ext(M,k). * * │ │ │ │ -0005b340: 6e6f 7465 2062 4d61 7073 3a20 624d 6170 note bMaps: bMap │ │ │ │ -0005b350: 732c 202d 2d20 6c69 7374 2074 6865 206d s, -- list the m │ │ │ │ -0005b360: 6170 7320 2064 5f70 3a42 5f31 2870 292d aps d_p:B_1(p)- │ │ │ │ -0005b370: 2d3e 425f 3028 7029 2069 6e20 610a 2020 ->B_0(p) in a. │ │ │ │ -0005b380: 2020 6d61 7472 6978 4661 6374 6f72 697a matrixFactoriz │ │ │ │ -0005b390: 6174 696f 6e0a 2020 2a20 2a6e 6f74 6520 ation. * *note │ │ │ │ -0005b3a0: 4252 616e 6b73 3a20 4252 616e 6b73 2c20 BRanks: BRanks, │ │ │ │ -0005b3b0: 2d2d 2072 616e 6b73 206f 6620 7468 6520 -- ranks of the │ │ │ │ -0005b3c0: 6d6f 6475 6c65 7320 425f 6928 6429 2069 modules B_i(d) i │ │ │ │ -0005b3d0: 6e20 610a 2020 2020 6d61 7472 6978 4661 n a. matrixFa │ │ │ │ -0005b3e0: 6374 6f72 697a 6174 696f 6e0a 0a57 6179 ctorization..Way │ │ │ │ -0005b3f0: 7320 746f 2075 7365 206d 6174 7269 7846 s to use matrixF │ │ │ │ -0005b400: 6163 746f 7269 7a61 7469 6f6e 3a0a 3d3d actorization:.== │ │ │ │ +0005b1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ +0005b1b0: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0005b1c0: 3d0a 0a20 202a 202a 6e6f 7465 2066 696e =.. * *note fin │ │ │ │ +0005b1d0: 6974 6542 6574 7469 4e75 6d62 6572 733a iteBettiNumbers: │ │ │ │ +0005b1e0: 2066 696e 6974 6542 6574 7469 4e75 6d62 finiteBettiNumb │ │ │ │ +0005b1f0: 6572 732c 202d 2d20 6265 7474 6920 6e75 ers, -- betti nu │ │ │ │ +0005b200: 6d62 6572 7320 6f66 2066 696e 6974 650a mbers of finite. │ │ │ │ +0005b210: 2020 2020 7265 736f 6c75 7469 6f6e 2063 resolution c │ │ │ │ +0005b220: 6f6d 7075 7465 6420 6672 6f6d 2061 206d omputed from a m │ │ │ │ +0005b230: 6174 7269 7820 6661 6374 6f72 697a 6174 atrix factorizat │ │ │ │ +0005b240: 696f 6e0a 2020 2a20 2a6e 6f74 6520 696e ion. * *note in │ │ │ │ +0005b250: 6669 6e69 7465 4265 7474 694e 756d 6265 finiteBettiNumbe │ │ │ │ +0005b260: 7273 3a20 696e 6669 6e69 7465 4265 7474 rs: infiniteBett │ │ │ │ +0005b270: 694e 756d 6265 7273 2c20 2d2d 2062 6574 iNumbers, -- bet │ │ │ │ +0005b280: 7469 206e 756d 6265 7273 206f 660a 2020 ti numbers of. │ │ │ │ +0005b290: 2020 6669 6e69 7465 2072 6573 6f6c 7574 finite resolut │ │ │ │ +0005b2a0: 696f 6e20 636f 6d70 7574 6564 2066 726f ion computed fro │ │ │ │ +0005b2b0: 6d20 6120 6d61 7472 6978 2066 6163 746f m a matrix facto │ │ │ │ +0005b2c0: 7269 7a61 7469 6f6e 0a20 202a 202a 6e6f rization. * *no │ │ │ │ +0005b2d0: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ +0005b2e0: 6967 6853 797a 7967 792c 202d 2d20 5265 ighSyzygy, -- Re │ │ │ │ +0005b2f0: 7475 726e 7320 6120 7379 7a79 6779 206d turns a syzygy m │ │ │ │ +0005b300: 6f64 756c 6520 6f6e 6520 6265 796f 6e64 odule one beyond │ │ │ │ +0005b310: 2074 6865 0a20 2020 2072 6567 756c 6172 the. regular │ │ │ │ +0005b320: 6974 7920 6f66 2045 7874 284d 2c6b 290a ity of Ext(M,k). │ │ │ │ +0005b330: 2020 2a20 2a6e 6f74 6520 624d 6170 733a * *note bMaps: │ │ │ │ +0005b340: 2062 4d61 7073 2c20 2d2d 206c 6973 7420 bMaps, -- list │ │ │ │ +0005b350: 7468 6520 6d61 7073 2020 645f 703a 425f the maps d_p:B_ │ │ │ │ +0005b360: 3128 7029 2d2d 3e42 5f30 2870 2920 696e 1(p)-->B_0(p) in │ │ │ │ +0005b370: 2061 0a20 2020 206d 6174 7269 7846 6163 a. matrixFac │ │ │ │ +0005b380: 746f 7269 7a61 7469 6f6e 0a20 202a 202a torization. * * │ │ │ │ +0005b390: 6e6f 7465 2042 5261 6e6b 733a 2042 5261 note BRanks: BRa │ │ │ │ +0005b3a0: 6e6b 732c 202d 2d20 7261 6e6b 7320 6f66 nks, -- ranks of │ │ │ │ +0005b3b0: 2074 6865 206d 6f64 756c 6573 2042 5f69 the modules B_i │ │ │ │ +0005b3c0: 2864 2920 696e 2061 0a20 2020 206d 6174 (d) in a. mat │ │ │ │ +0005b3d0: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +0005b3e0: 0a0a 5761 7973 2074 6f20 7573 6520 6d61 ..Ways to use ma │ │ │ │ +0005b3f0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +0005b400: 6e3a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d n:.============= │ │ │ │ 0005b410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0005b420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -0005b430: 2020 2a20 226d 6174 7269 7846 6163 746f * "matrixFacto │ │ │ │ -0005b440: 7269 7a61 7469 6f6e 284d 6174 7269 782c rization(Matrix, │ │ │ │ -0005b450: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ -0005b460: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0005b470: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0005b480: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0005b490: 6520 6d61 7472 6978 4661 6374 6f72 697a e matrixFactoriz │ │ │ │ -0005b4a0: 6174 696f 6e3a 206d 6174 7269 7846 6163 ation: matrixFac │ │ │ │ -0005b4b0: 746f 7269 7a61 7469 6f6e 2c20 6973 2061 torization, is a │ │ │ │ -0005b4c0: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -0005b4d0: 6e63 7469 6f6e 2077 6974 6820 6f70 7469 nction with opti │ │ │ │ -0005b4e0: 6f6e 733a 2028 4d61 6361 756c 6179 3244 ons: (Macaulay2D │ │ │ │ -0005b4f0: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -0005b500: 6e57 6974 684f 7074 696f 6e73 2c2e 0a0a nWithOptions,... │ │ │ │ +0005b420: 3d3d 3d0a 0a20 202a 2022 6d61 7472 6978 ===.. * "matrix │ │ │ │ +0005b430: 4661 6374 6f72 697a 6174 696f 6e28 4d61 Factorization(Ma │ │ │ │ +0005b440: 7472 6978 2c4d 6f64 756c 6529 220a 0a46 trix,Module)"..F │ │ │ │ +0005b450: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +0005b460: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +0005b470: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +0005b480: 202a 6e6f 7465 206d 6174 7269 7846 6163 *note matrixFac │ │ │ │ +0005b490: 746f 7269 7a61 7469 6f6e 3a20 6d61 7472 torization: matr │ │ │ │ +0005b4a0: 6978 4661 6374 6f72 697a 6174 696f 6e2c ixFactorization, │ │ │ │ +0005b4b0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +0005b4c0: 6f64 0a66 756e 6374 696f 6e20 7769 7468 od.function with │ │ │ │ +0005b4d0: 206f 7074 696f 6e73 3a20 284d 6163 6175 options: (Macau │ │ │ │ +0005b4e0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +0005b4f0: 6e63 7469 6f6e 5769 7468 4f70 7469 6f6e nctionWithOption │ │ │ │ +0005b500: 732c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d s,...----------- │ │ │ │ 0005b510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005b540: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005b550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0005b560: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0005b570: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0005b580: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0005b590: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0005b5a0: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0005b5b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0005b5c0: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ -0005b5d0: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -0005b5e0: 6c75 7469 6f6e 732e 6d32 3a34 3033 333a lutions.m2:4033: │ │ │ │ -0005b5f0: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ -0005b600: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0005b610: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ -0005b620: 204e 6f64 653a 206d 6642 6f75 6e64 2c20 Node: mfBound, │ │ │ │ -0005b630: 4e65 7874 3a20 6d6f 6475 6c65 4173 4578 Next: moduleAsEx │ │ │ │ -0005b640: 742c 2050 7265 763a 206d 6174 7269 7846 t, Prev: matrixF │ │ │ │ -0005b650: 6163 746f 7269 7a61 7469 6f6e 2c20 5570 actorization, Up │ │ │ │ -0005b660: 3a20 546f 700a 0a6d 6642 6f75 6e64 202d : Top..mfBound - │ │ │ │ -0005b670: 2d20 6465 7465 726d 696e 6573 2068 6f77 - determines how │ │ │ │ -0005b680: 2068 6967 6820 6120 7379 7a79 6779 2074 high a syzygy t │ │ │ │ -0005b690: 6f20 7461 6b65 2066 6f72 2022 6d61 7472 o take for "matr │ │ │ │ -0005b6a0: 6978 4661 6374 6f72 697a 6174 696f 6e22 ixFactorization" │ │ │ │ -0005b6b0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ +0005b550: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +0005b560: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +0005b570: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +0005b580: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +0005b590: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +0005b5a0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +0005b5b0: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ +0005b5c0: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +0005b5d0: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ +0005b5e0: 3430 3333 3a30 2e0a 1f0a 4669 6c65 3a20 4033:0....File: │ │ │ │ +0005b5f0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0005b600: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0005b610: 696e 666f 2c20 4e6f 6465 3a20 6d66 426f info, Node: mfBo │ │ │ │ +0005b620: 756e 642c 204e 6578 743a 206d 6f64 756c und, Next: modul │ │ │ │ +0005b630: 6541 7345 7874 2c20 5072 6576 3a20 6d61 eAsExt, Prev: ma │ │ │ │ +0005b640: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ +0005b650: 6e2c 2055 703a 2054 6f70 0a0a 6d66 426f n, Up: Top..mfBo │ │ │ │ +0005b660: 756e 6420 2d2d 2064 6574 6572 6d69 6e65 und -- determine │ │ │ │ +0005b670: 7320 686f 7720 6869 6768 2061 2073 797a s how high a syz │ │ │ │ +0005b680: 7967 7920 746f 2074 616b 6520 666f 7220 ygy to take for │ │ │ │ +0005b690: 226d 6174 7269 7846 6163 746f 7269 7a61 "matrixFactoriza │ │ │ │ +0005b6a0: 7469 6f6e 220a 2a2a 2a2a 2a2a 2a2a 2a2a tion".********** │ │ │ │ +0005b6b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005b6c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005b6d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005b6f0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ -0005b700: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ -0005b710: 7020 3d20 6d66 426f 756e 6420 4d0a 2020 p = mfBound M. │ │ │ │ -0005b720: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -0005b730: 2a20 4d2c 2061 202a 6e6f 7465 206d 6f64 * M, a *note mod │ │ │ │ -0005b740: 756c 653a 2028 4d61 6361 756c 6179 3244 ule: (Macaulay2D │ │ │ │ -0005b750: 6f63 294d 6f64 756c 652c 2c20 6f76 6572 oc)Module,, over │ │ │ │ -0005b760: 2061 2063 6f6d 706c 6574 6520 696e 7465 a complete inte │ │ │ │ -0005b770: 7273 6563 7469 6f6e 0a20 202a 204f 7574 rsection. * Out │ │ │ │ -0005b780: 7075 7473 3a0a 2020 2020 2020 2a20 702c puts:. * p, │ │ │ │ -0005b790: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ -0005b7a0: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ -0005b7b0: 295a 5a2c 2c20 0a0a 4465 7363 7269 7074 )ZZ,, ..Descript │ │ │ │ -0005b7c0: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -0005b7d0: 0a49 6620 7020 3d20 6d66 426f 756e 6420 .If p = mfBound │ │ │ │ -0005b7e0: 4d2c 2074 6865 6e20 7468 6520 702d 7468 M, then the p-th │ │ │ │ -0005b7f0: 2073 797a 7967 7920 6f66 204d 2c20 7768 syzygy of M, wh │ │ │ │ -0005b800: 6963 6820 6973 2063 6f6d 7075 7465 6420 ich is computed │ │ │ │ -0005b810: 6279 0a68 6967 6853 797a 7967 7928 4d29 by.highSyzygy(M) │ │ │ │ -0005b820: 2c20 7368 6f75 6c64 2028 7468 6973 2069 , should (this i │ │ │ │ -0005b830: 7320 6120 636f 6e6a 6563 7475 7265 2920 s a conjecture) │ │ │ │ -0005b840: 6265 2061 2022 6869 6768 2053 797a 7967 be a "high Syzyg │ │ │ │ -0005b850: 7922 2069 6e20 7468 6520 7365 6e73 650a y" in the sense. │ │ │ │ -0005b860: 7265 7175 6972 6564 2066 6f72 206d 6174 required for mat │ │ │ │ -0005b870: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0005b880: 2e20 496e 2065 7861 6d70 6c65 732c 2074 . In examples, t │ │ │ │ -0005b890: 6865 2065 7374 696d 6174 6520 7365 656d he estimate seem │ │ │ │ -0005b8a0: 7320 7368 6172 7020 2865 7863 6570 740a s sharp (except. │ │ │ │ -0005b8b0: 7768 656e 204d 2069 7320 616c 7265 6164 when M is alread │ │ │ │ -0005b8c0: 7920 6120 6869 6768 2073 797a 7967 7929 y a high syzygy) │ │ │ │ -0005b8d0: 2e0a 0a54 6865 2061 6374 7561 6c20 666f ...The actual fo │ │ │ │ -0005b8e0: 726d 756c 6120 7573 6564 2069 733a 0a0a rmula used is:.. │ │ │ │ -0005b8f0: 6d66 426f 756e 6420 4d20 3d20 6d61 7828 mfBound M = max( │ │ │ │ -0005b900: 322a 725f 7b65 7665 6e7d 2c20 312b 322a 2*r_{even}, 1+2* │ │ │ │ -0005b910: 725f 7b6f 6464 7d29 0a0a 7768 6572 6520 r_{odd})..where │ │ │ │ -0005b920: 725f 7b65 7665 6e7d 203d 2072 6567 756c r_{even} = regul │ │ │ │ -0005b930: 6172 6974 7920 6576 656e 4578 744d 6f64 arity evenExtMod │ │ │ │ -0005b940: 756c 6520 4d20 616e 6420 725f 7b6f 6464 ule M and r_{odd │ │ │ │ -0005b950: 7d20 3d20 7265 6775 6c61 7269 7479 0a6f } = regularity.o │ │ │ │ -0005b960: 6464 4578 744d 6f64 756c 6520 4d2e 2048 ddExtModule M. H │ │ │ │ -0005b970: 6572 6520 6576 656e 4578 744d 6f64 756c ere evenExtModul │ │ │ │ -0005b980: 6520 4d20 6973 2074 6865 2065 7665 6e20 e M is the even │ │ │ │ -0005b990: 6465 6772 6565 2070 6172 7420 6f66 2045 degree part of E │ │ │ │ -0005b9a0: 7874 284d 2c20 2872 6573 6964 7565 0a63 xt(M, (residue.c │ │ │ │ -0005b9b0: 6c61 7373 2066 6965 6c64 2929 2e0a 0a53 lass field))...S │ │ │ │ -0005b9c0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -0005b9d0: 0a0a 2020 2a20 2a6e 6f74 6520 6869 6768 .. * *note high │ │ │ │ -0005b9e0: 5379 7a79 6779 3a20 6869 6768 5379 7a79 Syzygy: highSyzy │ │ │ │ -0005b9f0: 6779 2c20 2d2d 2052 6574 7572 6e73 2061 gy, -- Returns a │ │ │ │ -0005ba00: 2073 797a 7967 7920 6d6f 6475 6c65 206f syzygy module o │ │ │ │ -0005ba10: 6e65 2062 6579 6f6e 6420 7468 650a 2020 ne beyond the. │ │ │ │ -0005ba20: 2020 7265 6775 6c61 7269 7479 206f 6620 regularity of │ │ │ │ -0005ba30: 4578 7428 4d2c 6b29 0a20 202a 202a 6e6f Ext(M,k). * *no │ │ │ │ -0005ba40: 7465 2065 7665 6e45 7874 4d6f 6475 6c65 te evenExtModule │ │ │ │ -0005ba50: 3a20 6576 656e 4578 744d 6f64 756c 652c : evenExtModule, │ │ │ │ -0005ba60: 202d 2d20 6576 656e 2070 6172 7420 6f66 -- even part of │ │ │ │ -0005ba70: 2045 7874 5e2a 284d 2c6b 2920 6f76 6572 Ext^*(M,k) over │ │ │ │ -0005ba80: 2061 0a20 2020 2063 6f6d 706c 6574 6520 a. complete │ │ │ │ -0005ba90: 696e 7465 7273 6563 7469 6f6e 2061 7320 intersection as │ │ │ │ -0005baa0: 6d6f 6475 6c65 206f 7665 7220 4349 206f module over CI o │ │ │ │ -0005bab0: 7065 7261 746f 7220 7269 6e67 0a20 202a perator ring. * │ │ │ │ -0005bac0: 202a 6e6f 7465 206f 6464 4578 744d 6f64 *note oddExtMod │ │ │ │ -0005bad0: 756c 653a 206f 6464 4578 744d 6f64 756c ule: oddExtModul │ │ │ │ -0005bae0: 652c 202d 2d20 6f64 6420 7061 7274 206f e, -- odd part o │ │ │ │ -0005baf0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -0005bb00: 7220 6120 636f 6d70 6c65 7465 0a20 2020 r a complete. │ │ │ │ -0005bb10: 2069 6e74 6572 7365 6374 696f 6e20 6173 intersection as │ │ │ │ -0005bb20: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -0005bb30: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -0005bb40: 2a20 2a6e 6f74 6520 6d61 7472 6978 4661 * *note matrixFa │ │ │ │ -0005bb50: 6374 6f72 697a 6174 696f 6e3a 206d 6174 ctorization: mat │ │ │ │ -0005bb60: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -0005bb70: 2c20 2d2d 204d 6170 7320 696e 2061 2068 , -- Maps in a h │ │ │ │ -0005bb80: 6967 6865 720a 2020 2020 636f 6469 6d65 igher. codime │ │ │ │ -0005bb90: 6e73 696f 6e20 6d61 7472 6978 2066 6163 nsion matrix fac │ │ │ │ -0005bba0: 746f 7269 7a61 7469 6f6e 0a0a 5761 7973 torization..Ways │ │ │ │ -0005bbb0: 2074 6f20 7573 6520 6d66 426f 756e 643a to use mfBound: │ │ │ │ -0005bbc0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005bbd0: 3d3d 3d3d 3d0a 0a20 202a 2022 6d66 426f =====.. * "mfBo │ │ │ │ -0005bbe0: 756e 6428 4d6f 6475 6c65 2922 0a0a 466f und(Module)"..Fo │ │ │ │ -0005bbf0: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0005bc00: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005bc10: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0005bc20: 2a6e 6f74 6520 6d66 426f 756e 643a 206d *note mfBound: m │ │ │ │ -0005bc30: 6642 6f75 6e64 2c20 6973 2061 202a 6e6f fBound, is a *no │ │ │ │ -0005bc40: 7465 206d 6574 686f 6420 6675 6e63 7469 te method functi │ │ │ │ -0005bc50: 6f6e 3a0a 284d 6163 6175 6c61 7932 446f on:.(Macaulay2Do │ │ │ │ -0005bc60: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ -0005bc70: 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ,...------------ │ │ │ │ +0005b6e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +0005b6f0: 0a20 202a 2055 7361 6765 3a20 0a20 2020 . * Usage: . │ │ │ │ +0005b700: 2020 2020 2070 203d 206d 6642 6f75 6e64 p = mfBound │ │ │ │ +0005b710: 204d 0a20 202a 2049 6e70 7574 733a 0a20 M. * Inputs:. │ │ │ │ +0005b720: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ +0005b730: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0005b740: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0005b750: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0005b760: 2069 6e74 6572 7365 6374 696f 6e0a 2020 intersection. │ │ │ │ +0005b770: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +0005b780: 202a 2070 2c20 616e 202a 6e6f 7465 2069 * p, an *note i │ │ │ │ +0005b790: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ +0005b7a0: 7932 446f 6329 5a5a 2c2c 200a 0a44 6573 y2Doc)ZZ,, ..Des │ │ │ │ +0005b7b0: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +0005b7c0: 3d3d 3d3d 0a0a 4966 2070 203d 206d 6642 ====..If p = mfB │ │ │ │ +0005b7d0: 6f75 6e64 204d 2c20 7468 656e 2074 6865 ound M, then the │ │ │ │ +0005b7e0: 2070 2d74 6820 7379 7a79 6779 206f 6620 p-th syzygy of │ │ │ │ +0005b7f0: 4d2c 2077 6869 6368 2069 7320 636f 6d70 M, which is comp │ │ │ │ +0005b800: 7574 6564 2062 790a 6869 6768 5379 7a79 uted by.highSyzy │ │ │ │ +0005b810: 6779 284d 292c 2073 686f 756c 6420 2874 gy(M), should (t │ │ │ │ +0005b820: 6869 7320 6973 2061 2063 6f6e 6a65 6374 his is a conject │ │ │ │ +0005b830: 7572 6529 2062 6520 6120 2268 6967 6820 ure) be a "high │ │ │ │ +0005b840: 5379 7a79 6779 2220 696e 2074 6865 2073 Syzygy" in the s │ │ │ │ +0005b850: 656e 7365 0a72 6571 7569 7265 6420 666f ense.required fo │ │ │ │ +0005b860: 7220 6d61 7472 6978 4661 6374 6f72 697a r matrixFactoriz │ │ │ │ +0005b870: 6174 696f 6e2e 2049 6e20 6578 616d 706c ation. In exampl │ │ │ │ +0005b880: 6573 2c20 7468 6520 6573 7469 6d61 7465 es, the estimate │ │ │ │ +0005b890: 2073 6565 6d73 2073 6861 7270 2028 6578 seems sharp (ex │ │ │ │ +0005b8a0: 6365 7074 0a77 6865 6e20 4d20 6973 2061 cept.when M is a │ │ │ │ +0005b8b0: 6c72 6561 6479 2061 2068 6967 6820 7379 lready a high sy │ │ │ │ +0005b8c0: 7a79 6779 292e 0a0a 5468 6520 6163 7475 zygy)...The actu │ │ │ │ +0005b8d0: 616c 2066 6f72 6d75 6c61 2075 7365 6420 al formula used │ │ │ │ +0005b8e0: 6973 3a0a 0a6d 6642 6f75 6e64 204d 203d is:..mfBound M = │ │ │ │ +0005b8f0: 206d 6178 2832 2a72 5f7b 6576 656e 7d2c max(2*r_{even}, │ │ │ │ +0005b900: 2031 2b32 2a72 5f7b 6f64 647d 290a 0a77 1+2*r_{odd})..w │ │ │ │ +0005b910: 6865 7265 2072 5f7b 6576 656e 7d20 3d20 here r_{even} = │ │ │ │ +0005b920: 7265 6775 6c61 7269 7479 2065 7665 6e45 regularity evenE │ │ │ │ +0005b930: 7874 4d6f 6475 6c65 204d 2061 6e64 2072 xtModule M and r │ │ │ │ +0005b940: 5f7b 6f64 647d 203d 2072 6567 756c 6172 _{odd} = regular │ │ │ │ +0005b950: 6974 790a 6f64 6445 7874 4d6f 6475 6c65 ity.oddExtModule │ │ │ │ +0005b960: 204d 2e20 4865 7265 2065 7665 6e45 7874 M. Here evenExt │ │ │ │ +0005b970: 4d6f 6475 6c65 204d 2069 7320 7468 6520 Module M is the │ │ │ │ +0005b980: 6576 656e 2064 6567 7265 6520 7061 7274 even degree part │ │ │ │ +0005b990: 206f 6620 4578 7428 4d2c 2028 7265 7369 of Ext(M, (resi │ │ │ │ +0005b9a0: 6475 650a 636c 6173 7320 6669 656c 6429 due.class field) │ │ │ │ +0005b9b0: 292e 0a0a 5365 6520 616c 736f 0a3d 3d3d )...See also.=== │ │ │ │ +0005b9c0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +0005b9d0: 2068 6967 6853 797a 7967 793a 2068 6967 highSyzygy: hig │ │ │ │ +0005b9e0: 6853 797a 7967 792c 202d 2d20 5265 7475 hSyzygy, -- Retu │ │ │ │ +0005b9f0: 726e 7320 6120 7379 7a79 6779 206d 6f64 rns a syzygy mod │ │ │ │ +0005ba00: 756c 6520 6f6e 6520 6265 796f 6e64 2074 ule one beyond t │ │ │ │ +0005ba10: 6865 0a20 2020 2072 6567 756c 6172 6974 he. regularit │ │ │ │ +0005ba20: 7920 6f66 2045 7874 284d 2c6b 290a 2020 y of Ext(M,k). │ │ │ │ +0005ba30: 2a20 2a6e 6f74 6520 6576 656e 4578 744d * *note evenExtM │ │ │ │ +0005ba40: 6f64 756c 653a 2065 7665 6e45 7874 4d6f odule: evenExtMo │ │ │ │ +0005ba50: 6475 6c65 2c20 2d2d 2065 7665 6e20 7061 dule, -- even pa │ │ │ │ +0005ba60: 7274 206f 6620 4578 745e 2a28 4d2c 6b29 rt of Ext^*(M,k) │ │ │ │ +0005ba70: 206f 7665 7220 610a 2020 2020 636f 6d70 over a. comp │ │ │ │ +0005ba80: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ +0005ba90: 6e20 6173 206d 6f64 756c 6520 6f76 6572 n as module over │ │ │ │ +0005baa0: 2043 4920 6f70 6572 6174 6f72 2072 696e CI operator rin │ │ │ │ +0005bab0: 670a 2020 2a20 2a6e 6f74 6520 6f64 6445 g. * *note oddE │ │ │ │ +0005bac0: 7874 4d6f 6475 6c65 3a20 6f64 6445 7874 xtModule: oddExt │ │ │ │ +0005bad0: 4d6f 6475 6c65 2c20 2d2d 206f 6464 2070 Module, -- odd p │ │ │ │ +0005bae0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +0005baf0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +0005bb00: 650a 2020 2020 696e 7465 7273 6563 7469 e. intersecti │ │ │ │ +0005bb10: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ +0005bb20: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +0005bb30: 6e67 0a20 202a 202a 6e6f 7465 206d 6174 ng. * *note mat │ │ │ │ +0005bb40: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ +0005bb50: 3a20 6d61 7472 6978 4661 6374 6f72 697a : matrixFactoriz │ │ │ │ +0005bb60: 6174 696f 6e2c 202d 2d20 4d61 7073 2069 ation, -- Maps i │ │ │ │ +0005bb70: 6e20 6120 6869 6768 6572 0a20 2020 2063 n a higher. c │ │ │ │ +0005bb80: 6f64 696d 656e 7369 6f6e 206d 6174 7269 odimension matri │ │ │ │ +0005bb90: 7820 6661 6374 6f72 697a 6174 696f 6e0a x factorization. │ │ │ │ +0005bba0: 0a57 6179 7320 746f 2075 7365 206d 6642 .Ways to use mfB │ │ │ │ +0005bbb0: 6f75 6e64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d ound:.========== │ │ │ │ +0005bbc0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0005bbd0: 226d 6642 6f75 6e64 284d 6f64 756c 6529 "mfBound(Module) │ │ │ │ +0005bbe0: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0005bbf0: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0005bc00: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0005bc10: 6a65 6374 202a 6e6f 7465 206d 6642 6f75 ject *note mfBou │ │ │ │ +0005bc20: 6e64 3a20 6d66 426f 756e 642c 2069 7320 nd: mfBound, is │ │ │ │ +0005bc30: 6120 2a6e 6f74 6520 6d65 7468 6f64 2066 a *note method f │ │ │ │ +0005bc40: 756e 6374 696f 6e3a 0a28 4d61 6361 756c unction:.(Macaul │ │ │ │ +0005bc50: 6179 3244 6f63 294d 6574 686f 6446 756e ay2Doc)MethodFun │ │ │ │ +0005bc60: 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d ction,...------- │ │ │ │ +0005bc70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005bcc0: 2d2d 2d0a 0a54 6865 2073 6f75 7263 6520 ---..The source │ │ │ │ -0005bcd0: 6f66 2074 6869 7320 646f 6375 6d65 6e74 of this document │ │ │ │ -0005bce0: 2069 7320 696e 0a2f 6275 696c 642f 7265 is in./build/re │ │ │ │ -0005bcf0: 7072 6f64 7563 6962 6c65 2d70 6174 682f producible-path/ │ │ │ │ -0005bd00: 6d61 6361 756c 6179 322d 312e 3236 2e30 macaulay2-1.26.0 │ │ │ │ -0005bd10: 362b 6473 2f4d 322f 4d61 6361 756c 6179 6+ds/M2/Macaulay │ │ │ │ -0005bd20: 322f 7061 636b 6167 6573 2f0a 436f 6d70 2/packages/.Comp │ │ │ │ -0005bd30: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0005bd40: 5265 736f 6c75 7469 6f6e 732e 6d32 3a33 Resolutions.m2:3 │ │ │ │ -0005bd50: 3334 393a 302e 0a1f 0a46 696c 653a 2043 349:0....File: C │ │ │ │ -0005bd60: 6f6d 706c 6574 6549 6e74 6572 7365 6374 ompleteIntersect │ │ │ │ -0005bd70: 696f 6e52 6573 6f6c 7574 696f 6e73 2e69 ionResolutions.i │ │ │ │ -0005bd80: 6e66 6f2c 204e 6f64 653a 206d 6f64 756c nfo, Node: modul │ │ │ │ -0005bd90: 6541 7345 7874 2c20 4e65 7874 3a20 6e65 eAsExt, Next: ne │ │ │ │ -0005bda0: 7745 7874 2c20 5072 6576 3a20 6d66 426f wExt, Prev: mfBo │ │ │ │ -0005bdb0: 756e 642c 2055 703a 2054 6f70 0a0a 6d6f und, Up: Top..mo │ │ │ │ -0005bdc0: 6475 6c65 4173 4578 7420 2d2d 2046 696e duleAsExt -- Fin │ │ │ │ -0005bdd0: 6420 6120 6d6f 6475 6c65 2077 6974 6820 d a module with │ │ │ │ -0005bde0: 6769 7665 6e20 6173 796d 7074 6f74 6963 given asymptotic │ │ │ │ -0005bdf0: 2072 6573 6f6c 7574 696f 6e0a 2a2a 2a2a resolution.**** │ │ │ │ +0005bcb0: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ +0005bcc0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ +0005bcd0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ +0005bce0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ +0005bcf0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ +0005bd00: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ +0005bd10: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ +0005bd20: 0a43 6f6d 706c 6574 6549 6e74 6572 7365 .CompleteInterse │ │ │ │ +0005bd30: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0005bd40: 2e6d 323a 3333 3439 3a30 2e0a 1f0a 4669 .m2:3349:0....Fi │ │ │ │ +0005bd50: 6c65 3a20 436f 6d70 6c65 7465 496e 7465 le: CompleteInte │ │ │ │ +0005bd60: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +0005bd70: 6f6e 732e 696e 666f 2c20 4e6f 6465 3a20 ons.info, Node: │ │ │ │ +0005bd80: 6d6f 6475 6c65 4173 4578 742c 204e 6578 moduleAsExt, Nex │ │ │ │ +0005bd90: 743a 206e 6577 4578 742c 2050 7265 763a t: newExt, Prev: │ │ │ │ +0005bda0: 206d 6642 6f75 6e64 2c20 5570 3a20 546f mfBound, Up: To │ │ │ │ +0005bdb0: 700a 0a6d 6f64 756c 6541 7345 7874 202d p..moduleAsExt - │ │ │ │ +0005bdc0: 2d20 4669 6e64 2061 206d 6f64 756c 6520 - Find a module │ │ │ │ +0005bdd0: 7769 7468 2067 6976 656e 2061 7379 6d70 with given asymp │ │ │ │ +0005bde0: 746f 7469 6320 7265 736f 6c75 7469 6f6e totic resolution │ │ │ │ +0005bdf0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 0005be00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005be10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005be30: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0005be40: 7361 6765 3a20 0a20 2020 2020 2020 204d sage: . M │ │ │ │ -0005be50: 203d 206d 6f64 756c 6541 7345 7874 284d = moduleAsExt(M │ │ │ │ -0005be60: 4d2c 5229 0a20 202a 2049 6e70 7574 733a M,R). * Inputs: │ │ │ │ -0005be70: 0a20 2020 2020 202a 204d 2c20 6120 2a6e . * M, a *n │ │ │ │ -0005be80: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163 ote module: (Mac │ │ │ │ -0005be90: 6175 6c61 7932 446f 6329 4d6f 6475 6c65 aulay2Doc)Module │ │ │ │ -0005bea0: 2c2c 206d 6f64 756c 6520 6f76 6572 2070 ,, module over p │ │ │ │ -0005beb0: 6f6c 796e 6f6d 6961 6c20 7269 6e67 0a20 olynomial ring. │ │ │ │ -0005bec0: 2020 2020 2020 2077 6974 6820 6320 7661 with c va │ │ │ │ -0005bed0: 7269 6162 6c65 730a 2020 2020 2020 2a20 riables. * │ │ │ │ -0005bee0: 522c 2061 202a 6e6f 7465 2072 696e 673a R, a *note ring: │ │ │ │ -0005bef0: 2028 4d61 6361 756c 6179 3244 6f63 2952 (Macaulay2Doc)R │ │ │ │ -0005bf00: 696e 672c 2c20 2867 7261 6465 6429 2063 ing,, (graded) c │ │ │ │ -0005bf10: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ -0005bf20: 7469 6f6e 0a20 2020 2020 2020 2072 696e tion. rin │ │ │ │ -0005bf30: 6720 6f66 2063 6f64 696d 656e 7369 6f6e g of codimension │ │ │ │ -0005bf40: 2063 2c20 656d 6265 6464 696e 6720 6469 c, embedding di │ │ │ │ -0005bf50: 6d65 6e73 696f 6e20 6e0a 2020 2a20 4f75 mension n. * Ou │ │ │ │ -0005bf60: 7470 7574 733a 0a20 2020 2020 202a 204e tputs:. * N │ │ │ │ -0005bf70: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ -0005bf80: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -0005bf90: 4d6f 6475 6c65 2c2c 206d 6f64 756c 6520 Module,, module │ │ │ │ -0005bfa0: 6f76 6572 2052 2073 7563 6820 7468 6174 over R such that │ │ │ │ -0005bfb0: 0a20 2020 2020 2020 2045 7874 5f52 284e . Ext_R(N │ │ │ │ -0005bfc0: 2c6b 2920 3d20 4d5c 6f74 696d 6573 205c ,k) = M\otimes \ │ │ │ │ -0005bfd0: 7765 6467 6528 6b5e 6e29 2069 6e20 6c61 wedge(k^n) in la │ │ │ │ -0005bfe0: 7267 6520 686f 6d6f 6c6f 6769 6361 6c20 rge homological │ │ │ │ -0005bff0: 6465 6772 6565 2e0a 0a44 6573 6372 6970 degree...Descrip │ │ │ │ -0005c000: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -0005c010: 0a0a 5468 6520 726f 7574 696e 6520 6060 ..The routine `` │ │ │ │ -0005c020: 6d6f 6475 6c65 4173 4578 7427 2720 6973 moduleAsExt'' is │ │ │ │ -0005c030: 2061 2070 6172 7469 616c 2069 6e76 6572 a partial inver │ │ │ │ -0005c040: 7365 2074 6f20 7468 6520 726f 7574 696e se to the routin │ │ │ │ -0005c050: 6520 4578 744d 6f64 756c 652c 0a63 6f6d e ExtModule,.com │ │ │ │ -0005c060: 7075 7465 6420 666f 6c6c 6f77 696e 6720 puted following │ │ │ │ -0005c070: 6964 6561 7320 6f66 2041 7672 616d 6f76 ideas of Avramov │ │ │ │ -0005c080: 2061 6e64 204a 6f72 6765 6e73 656e 3a20 and Jorgensen: │ │ │ │ -0005c090: 6769 7665 6e20 6120 6d6f 6475 6c65 2045 given a module E │ │ │ │ -0005c0a0: 206f 7665 7220 610a 706f 6c79 6e6f 6d69 over a.polynomi │ │ │ │ -0005c0b0: 616c 2072 696e 6720 6b5b 785f 312e 2e78 al ring k[x_1..x │ │ │ │ -0005c0c0: 5f63 5d2c 2069 7420 7072 6f76 6964 6573 _c], it provides │ │ │ │ -0005c0d0: 2061 206d 6f64 756c 6520 4e20 6f76 6572 a module N over │ │ │ │ -0005c0e0: 2061 2073 7065 6369 6669 6564 2070 6f6c a specified pol │ │ │ │ -0005c0f0: 796e 6f6d 6961 6c0a 7269 6e67 2069 6e20 ynomial.ring in │ │ │ │ -0005c100: 6e20 7661 7269 6162 6c65 7320 7375 6368 n variables such │ │ │ │ -0005c110: 2074 6861 7420 4578 7428 4e2c 6b29 2061 that Ext(N,k) a │ │ │ │ -0005c120: 6772 6565 7320 7769 7468 2024 4527 3d45 grees with $E'=E │ │ │ │ -0005c130: 5c6f 7469 6d65 7320 5c77 6564 6765 286b \otimes \wedge(k │ │ │ │ -0005c140: 5e6e 2924 0a61 6674 6572 2074 7275 6e63 ^n)$.after trunc │ │ │ │ -0005c150: 6174 696f 6e2e 2048 6572 6520 7468 6520 ation. Here the │ │ │ │ -0005c160: 6772 6164 696e 6720 6f6e 2045 2069 7320 grading on E is │ │ │ │ -0005c170: 7461 6b65 6e20 746f 2062 6520 6576 656e taken to be even │ │ │ │ -0005c180: 2c20 7768 696c 650a 245c 7765 6467 6528 , while.$\wedge( │ │ │ │ -0005c190: 6b5e 6e29 2420 6861 7320 6765 6e65 7261 k^n)$ has genera │ │ │ │ -0005c1a0: 746f 7273 2069 6e20 6465 6772 6565 2031 tors in degree 1 │ │ │ │ -0005c1b0: 2e20 5468 6520 726f 7574 696e 6520 6866 . The routine hf │ │ │ │ -0005c1c0: 4d6f 6475 6c65 4173 4578 7420 636f 6d70 ModuleAsExt comp │ │ │ │ -0005c1d0: 7574 6573 0a74 6865 2072 6573 756c 7469 utes.the resulti │ │ │ │ -0005c1e0: 6e67 2068 696c 6265 7274 2066 756e 6374 ng hilbert funct │ │ │ │ -0005c1f0: 696f 6e20 666f 7220 4527 2e20 5468 6973 ion for E'. This │ │ │ │ -0005c200: 2075 7365 7320 6964 6561 7320 6f66 2041 uses ideas of A │ │ │ │ -0005c210: 7672 616d 6f76 2061 6e64 0a4a 6f72 6765 vramov and.Jorge │ │ │ │ -0005c220: 6e73 656e 2e20 4e6f 7465 2074 6861 7420 nsen. Note that │ │ │ │ -0005c230: 7468 6520 6d6f 6475 6c65 2045 7874 284e the module Ext(N │ │ │ │ -0005c240: 2c6b 2920 2874 7275 6e63 6174 6564 2920 ,k) (truncated) │ │ │ │ -0005c250: 7769 6c6c 2061 7574 6f6d 6174 6963 616c will automatical │ │ │ │ -0005c260: 6c79 2062 6520 6672 6565 0a6f 7665 7220 ly be free.over │ │ │ │ -0005c270: 7468 6520 6578 7465 7269 6f72 2061 6c67 the exterior alg │ │ │ │ -0005c280: 6562 7261 2024 5c77 6564 6765 286b 5e6e ebra $\wedge(k^n │ │ │ │ -0005c290: 2924 2067 656e 6572 6174 6564 2062 7920 )$ generated by │ │ │ │ -0005c2a0: 4578 745e 3128 6b2c 6b29 3b20 6e6f 7420 Ext^1(k,k); not │ │ │ │ -0005c2b0: 6120 7479 7069 6361 6c0a 4578 7420 6d6f a typical.Ext mo │ │ │ │ -0005c2c0: 6475 6c65 2e0a 0a4d 6f72 6520 7072 6563 dule...More prec │ │ │ │ -0005c2d0: 6973 656c 793a 0a0a 5375 7070 6f73 6520 isely:..Suppose │ │ │ │ -0005c2e0: 7468 6174 2024 5220 3d20 6b5b 615f 312c that $R = k[a_1, │ │ │ │ -0005c2f0: 5c64 6f74 732c 2061 5f6e 5d2f 2866 5f31 \dots, a_n]/(f_1 │ │ │ │ -0005c300: 2c5c 646f 7473 2c66 5f63 2924 206c 6574 ,\dots,f_c)$ let │ │ │ │ -0005c310: 2024 4b4b 203d 0a6b 5b78 5f31 2c5c 646f $KK =.k[x_1,\do │ │ │ │ -0005c320: 7473 2c78 5f63 5d24 2c20 616e 6420 6c65 ts,x_c]$, and le │ │ │ │ -0005c330: 7420 245c 4c61 6d62 6461 203d 205c 7765 t $\Lambda = \we │ │ │ │ -0005c340: 6467 6520 6b5e 6e24 2e20 2445 203d 204b dge k^n$. $E = K │ │ │ │ -0005c350: 4b5c 6f74 696d 6573 5c4c 616d 6264 6124 K\otimes\Lambda$ │ │ │ │ -0005c360: 2c20 736f 0a74 6861 7420 7468 6520 6d69 , so.that the mi │ │ │ │ -0005c370: 6e69 6d61 6c20 2452 242d 6672 6565 2072 nimal $R$-free r │ │ │ │ -0005c380: 6573 6f6c 7574 696f 6e20 6f66 2024 6b24 esolution of $k$ │ │ │ │ -0005c390: 2068 6173 2075 6e64 6572 6c79 696e 6720 has underlying │ │ │ │ -0005c3a0: 6d6f 6475 6c65 2024 525c 6f74 696d 6573 module $R\otimes │ │ │ │ -0005c3b0: 0a45 5e2a 242c 2077 6865 7265 2024 455e .E^*$, where $E^ │ │ │ │ -0005c3c0: 2a24 2069 7320 7468 6520 6772 6164 6564 *$ is the graded │ │ │ │ -0005c3d0: 2076 6563 746f 7220 7370 6163 6520 6475 vector space du │ │ │ │ -0005c3e0: 616c 206f 6620 2445 242e 0a0a 4c65 7420 al of $E$...Let │ │ │ │ -0005c3f0: 4d4d 2062 6520 7468 6520 7265 7375 6c74 MM be the result │ │ │ │ -0005c400: 206f 6620 7472 756e 6361 7469 6e67 204d of truncating M │ │ │ │ -0005c410: 2061 7420 6974 7320 7265 6775 6c61 7269 at its regulari │ │ │ │ -0005c420: 7479 2061 6e64 2073 6869 6674 696e 6720 ty and shifting │ │ │ │ -0005c430: 6974 2073 6f20 7468 6174 0a69 7420 6973 it so that.it is │ │ │ │ -0005c440: 2067 656e 6572 6174 6564 2069 6e20 6465 generated in de │ │ │ │ -0005c450: 6772 6565 2030 2e20 4c65 7420 2446 2420 gree 0. Let $F$ │ │ │ │ -0005c460: 6265 2061 2024 4b4b 242d 6672 6565 2072 be a $KK$-free r │ │ │ │ -0005c470: 6573 6f6c 7574 696f 6e20 6f66 2024 4d4d esolution of $MM │ │ │ │ -0005c480: 242c 2061 6e64 0a77 7269 7465 2024 465f $, and.write $F_ │ │ │ │ -0005c490: 6920 3d20 4b4b 5c6f 7469 6d65 7320 565f i = KK\otimes V_ │ │ │ │ -0005c4a0: 692e 2420 5369 6e63 6520 6c69 6e65 6172 i.$ Since linear │ │ │ │ -0005c4b0: 2066 6f72 6d73 206f 7665 7220 244b 4b24 forms over $KK$ │ │ │ │ -0005c4c0: 2063 6f72 7265 7370 6f6e 6420 746f 2043 correspond to C │ │ │ │ -0005c4d0: 490a 6f70 6572 6174 6f72 7320 6f66 2064 I.operators of d │ │ │ │ -0005c4e0: 6567 7265 6520 2d32 206f 6e20 7468 6520 egree -2 on the │ │ │ │ -0005c4f0: 7265 736f 6c75 7469 6f6e 2047 206f 6620 resolution G of │ │ │ │ -0005c500: 6b20 6f76 6572 2052 2c20 7765 206d 6179 k over R, we may │ │ │ │ -0005c510: 2066 6f72 6d20 6120 6d61 7020 2424 0a64 form a map $$.d │ │ │ │ -0005c520: 5f31 2b64 5f32 3a20 5c73 756d 5f7b 693d _1+d_2: \sum_{i= │ │ │ │ -0005c530: 307d 5e6d 2047 5f7b 692b 317d 5c6f 7469 0}^m G_{i+1}\oti │ │ │ │ -0005c540: 6d65 7320 565f 7b6d 2d69 7d5e 2a20 5c74 mes V_{m-i}^* \t │ │ │ │ -0005c550: 6f20 5c73 756d 5f7b 693d 307d 5e6d 2047 o \sum_{i=0}^m G │ │ │ │ -0005c560: 5f69 5c6f 7469 6d65 730a 565f 7b6d 2d69 _i\otimes.V_{m-i │ │ │ │ -0005c570: 7d5e 2a20 2424 2077 6865 7265 2024 645f }^* $$ where $d_ │ │ │ │ -0005c580: 3124 2069 7320 7468 6520 6469 7265 6374 1$ is the direct │ │ │ │ -0005c590: 2073 756d 206f 6620 7468 6520 6469 6666 sum of the diff │ │ │ │ -0005c5a0: 6572 656e 7469 616c 7320 2428 475f 7b69 erentials $(G_{i │ │ │ │ -0005c5b0: 2b31 7d5c 746f 0a47 5f69 295c 6f74 696d +1}\to.G_i)\otim │ │ │ │ -0005c5c0: 6573 2056 5f69 5e2a 2420 616e 6420 2464 es V_i^*$ and $d │ │ │ │ -0005c5d0: 5f32 2420 6973 2074 6865 2064 6972 6563 _2$ is the direc │ │ │ │ -0005c5e0: 7420 7375 6d20 6f66 2074 6865 206d 6170 t sum of the map │ │ │ │ -0005c5f0: 7320 245c 7068 695f 6924 2064 6566 696e s $\phi_i$ defin │ │ │ │ -0005c600: 6564 0a66 726f 6d20 7468 6520 6469 6666 ed.from the diff │ │ │ │ -0005c610: 6572 656e 7469 616c 7320 6f66 2024 4624 erentials of $F$ │ │ │ │ -0005c620: 2062 7920 7375 6273 7469 7475 7469 6e67 by substituting │ │ │ │ -0005c630: 2043 4920 6f70 6572 6174 6f72 7320 666f CI operators fo │ │ │ │ -0005c640: 7220 6c69 6e65 6172 2066 6f72 6d73 2c0a r linear forms,. │ │ │ │ -0005c650: 245c 7068 695f 693a 2047 5f7b 692b 317d $\phi_i: G_{i+1} │ │ │ │ -0005c660: 5c6f 7469 6d65 7320 565f 6920 5c74 6f20 \otimes V_i \to │ │ │ │ -0005c670: 475f 7b69 2d31 7d5c 6f74 696d 6573 2056 G_{i-1}\otimes V │ │ │ │ -0005c680: 5f7b 692d 317d 242e 2054 6865 2073 6372 _{i-1}$. The scr │ │ │ │ -0005c690: 6970 7420 7265 7475 726e 7320 7468 650a ipt returns the. │ │ │ │ -0005c6a0: 6d6f 6475 6c65 204e 2074 6861 7420 6973 module N that is │ │ │ │ -0005c6b0: 2074 6865 2063 6f6b 6572 6e65 6c20 6f66 the cokernel of │ │ │ │ -0005c6c0: 2024 645f 312b 645f 3224 2e0a 0a54 6865 $d_1+d_2$...The │ │ │ │ -0005c6d0: 206d 6f64 756c 6520 2445 7874 5f52 284e module $Ext_R(N │ │ │ │ -0005c6e0: 2c6b 2924 2061 6772 6565 732c 2061 6674 ,k)$ agrees, aft │ │ │ │ -0005c6f0: 6572 2061 2066 6577 2073 7465 7073 2c20 er a few steps, │ │ │ │ -0005c700: 7769 7468 2074 6865 206d 6f64 756c 6520 with the module │ │ │ │ -0005c710: 6465 7269 7665 6420 6672 6f6d 0a24 4d4d derived from.$MM │ │ │ │ -0005c720: 2420 6279 2074 656e 736f 7269 6e67 2069 $ by tensoring i │ │ │ │ -0005c730: 7420 7769 7468 2024 5c4c 616d 6264 6124 t with $\Lambda$ │ │ │ │ -0005c740: 2c20 7468 6174 2069 732c 2077 6974 6820 , that is, with │ │ │ │ -0005c750: 7468 6520 6d6f 6475 6c65 c39f 2024 2420 the module.. $$ │ │ │ │ -0005c760: 4d4d 2720 3d20 5c73 756d 5f6a 0a28 4d4d MM' = \sum_j.(MM │ │ │ │ -0005c770: 2728 6a29 5c6f 7469 6d65 7320 5c4c 616d '(j)\otimes \Lam │ │ │ │ -0005c780: 6264 615f 6a29 2024 2420 736f 2074 6861 bda_j) $$ so tha │ │ │ │ -0005c790: 7420 244d 4d27 5f70 203d 2028 4d4d 5f70 t $MM'_p = (MM_p │ │ │ │ -0005c7a0: 5c6f 7469 6d65 7320 4c61 6d62 6461 5f30 \otimes Lambda_0 │ │ │ │ -0005c7b0: 2920 5c6f 706c 7573 0a28 4d4d 5f7b 702d ) \oplus.(MM_{p- │ │ │ │ -0005c7c0: 317d 5c6f 7469 6d65 7320 4c61 6d62 6461 1}\otimes Lambda │ │ │ │ -0005c7d0: 5f31 2920 5c6f 706c 7573 5c63 646f 7473 _1) \oplus\cdots │ │ │ │ -0005c7e0: 242e 0a0a 5468 6520 6675 6e63 7469 6f6e $...The function │ │ │ │ -0005c7f0: 2068 664d 6f64 756c 6541 7345 7874 2063 hfModuleAsExt c │ │ │ │ -0005c800: 6f6d 7075 7465 7320 7468 6520 4869 6c62 omputes the Hilb │ │ │ │ -0005c810: 6572 7420 6675 6e63 7469 6f6e 206f 6620 ert function of │ │ │ │ -0005c820: 4d4d 2720 6e75 6d65 7269 6361 6c6c 790a MM' numerically. │ │ │ │ -0005c830: 6672 6f6d 2074 6861 7420 6f66 204d 4d2e from that of MM. │ │ │ │ -0005c840: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +0005be20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0005be30: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0005be40: 2020 2020 4d20 3d20 6d6f 6475 6c65 4173 M = moduleAs │ │ │ │ +0005be50: 4578 7428 4d4d 2c52 290a 2020 2a20 496e Ext(MM,R). * In │ │ │ │ +0005be60: 7075 7473 3a0a 2020 2020 2020 2a20 4d2c puts:. * M, │ │ │ │ +0005be70: 2061 202a 6e6f 7465 206d 6f64 756c 653a a *note module: │ │ │ │ +0005be80: 2028 4d61 6361 756c 6179 3244 6f63 294d (Macaulay2Doc)M │ │ │ │ +0005be90: 6f64 756c 652c 2c20 6d6f 6475 6c65 206f odule,, module o │ │ │ │ +0005bea0: 7665 7220 706f 6c79 6e6f 6d69 616c 2072 ver polynomial r │ │ │ │ +0005beb0: 696e 670a 2020 2020 2020 2020 7769 7468 ing. with │ │ │ │ +0005bec0: 2063 2076 6172 6961 626c 6573 0a20 2020 c variables. │ │ │ │ +0005bed0: 2020 202a 2052 2c20 6120 2a6e 6f74 6520 * R, a *note │ │ │ │ +0005bee0: 7269 6e67 3a20 284d 6163 6175 6c61 7932 ring: (Macaulay2 │ │ │ │ +0005bef0: 446f 6329 5269 6e67 2c2c 2028 6772 6164 Doc)Ring,, (grad │ │ │ │ +0005bf00: 6564 2920 636f 6d70 6c65 7465 2069 6e74 ed) complete int │ │ │ │ +0005bf10: 6572 7365 6374 696f 6e0a 2020 2020 2020 ersection. │ │ │ │ +0005bf20: 2020 7269 6e67 206f 6620 636f 6469 6d65 ring of codime │ │ │ │ +0005bf30: 6e73 696f 6e20 632c 2065 6d62 6564 6469 nsion c, embeddi │ │ │ │ +0005bf40: 6e67 2064 696d 656e 7369 6f6e 206e 0a20 ng dimension n. │ │ │ │ +0005bf50: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ +0005bf60: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d * N, a *note m │ │ │ │ +0005bf70: 6f64 756c 653a 2028 4d61 6361 756c 6179 odule: (Macaulay │ │ │ │ +0005bf80: 3244 6f63 294d 6f64 756c 652c 2c20 6d6f 2Doc)Module,, mo │ │ │ │ +0005bf90: 6475 6c65 206f 7665 7220 5220 7375 6368 dule over R such │ │ │ │ +0005bfa0: 2074 6861 740a 2020 2020 2020 2020 4578 that. Ex │ │ │ │ +0005bfb0: 745f 5228 4e2c 6b29 203d 204d 5c6f 7469 t_R(N,k) = M\oti │ │ │ │ +0005bfc0: 6d65 7320 5c77 6564 6765 286b 5e6e 2920 mes \wedge(k^n) │ │ │ │ +0005bfd0: 696e 206c 6172 6765 2068 6f6d 6f6c 6f67 in large homolog │ │ │ │ +0005bfe0: 6963 616c 2064 6567 7265 652e 0a0a 4465 ical degree...De │ │ │ │ +0005bff0: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +0005c000: 3d3d 3d3d 3d0a 0a54 6865 2072 6f75 7469 =====..The routi │ │ │ │ +0005c010: 6e65 2060 606d 6f64 756c 6541 7345 7874 ne ``moduleAsExt │ │ │ │ +0005c020: 2727 2069 7320 6120 7061 7274 6961 6c20 '' is a partial │ │ │ │ +0005c030: 696e 7665 7273 6520 746f 2074 6865 2072 inverse to the r │ │ │ │ +0005c040: 6f75 7469 6e65 2045 7874 4d6f 6475 6c65 outine ExtModule │ │ │ │ +0005c050: 2c0a 636f 6d70 7574 6564 2066 6f6c 6c6f ,.computed follo │ │ │ │ +0005c060: 7769 6e67 2069 6465 6173 206f 6620 4176 wing ideas of Av │ │ │ │ +0005c070: 7261 6d6f 7620 616e 6420 4a6f 7267 656e ramov and Jorgen │ │ │ │ +0005c080: 7365 6e3a 2067 6976 656e 2061 206d 6f64 sen: given a mod │ │ │ │ +0005c090: 756c 6520 4520 6f76 6572 2061 0a70 6f6c ule E over a.pol │ │ │ │ +0005c0a0: 796e 6f6d 6961 6c20 7269 6e67 206b 5b78 ynomial ring k[x │ │ │ │ +0005c0b0: 5f31 2e2e 785f 635d 2c20 6974 2070 726f _1..x_c], it pro │ │ │ │ +0005c0c0: 7669 6465 7320 6120 6d6f 6475 6c65 204e vides a module N │ │ │ │ +0005c0d0: 206f 7665 7220 6120 7370 6563 6966 6965 over a specifie │ │ │ │ +0005c0e0: 6420 706f 6c79 6e6f 6d69 616c 0a72 696e d polynomial.rin │ │ │ │ +0005c0f0: 6720 696e 206e 2076 6172 6961 626c 6573 g in n variables │ │ │ │ +0005c100: 2073 7563 6820 7468 6174 2045 7874 284e such that Ext(N │ │ │ │ +0005c110: 2c6b 2920 6167 7265 6573 2077 6974 6820 ,k) agrees with │ │ │ │ +0005c120: 2445 273d 455c 6f74 696d 6573 205c 7765 $E'=E\otimes \we │ │ │ │ +0005c130: 6467 6528 6b5e 6e29 240a 6166 7465 7220 dge(k^n)$.after │ │ │ │ +0005c140: 7472 756e 6361 7469 6f6e 2e20 4865 7265 truncation. Here │ │ │ │ +0005c150: 2074 6865 2067 7261 6469 6e67 206f 6e20 the grading on │ │ │ │ +0005c160: 4520 6973 2074 616b 656e 2074 6f20 6265 E is taken to be │ │ │ │ +0005c170: 2065 7665 6e2c 2077 6869 6c65 0a24 5c77 even, while.$\w │ │ │ │ +0005c180: 6564 6765 286b 5e6e 2924 2068 6173 2067 edge(k^n)$ has g │ │ │ │ +0005c190: 656e 6572 6174 6f72 7320 696e 2064 6567 enerators in deg │ │ │ │ +0005c1a0: 7265 6520 312e 2054 6865 2072 6f75 7469 ree 1. The routi │ │ │ │ +0005c1b0: 6e65 2068 664d 6f64 756c 6541 7345 7874 ne hfModuleAsExt │ │ │ │ +0005c1c0: 2063 6f6d 7075 7465 730a 7468 6520 7265 computes.the re │ │ │ │ +0005c1d0: 7375 6c74 696e 6720 6869 6c62 6572 7420 sulting hilbert │ │ │ │ +0005c1e0: 6675 6e63 7469 6f6e 2066 6f72 2045 272e function for E'. │ │ │ │ +0005c1f0: 2054 6869 7320 7573 6573 2069 6465 6173 This uses ideas │ │ │ │ +0005c200: 206f 6620 4176 7261 6d6f 7620 616e 640a of Avramov and. │ │ │ │ +0005c210: 4a6f 7267 656e 7365 6e2e 204e 6f74 6520 Jorgensen. Note │ │ │ │ +0005c220: 7468 6174 2074 6865 206d 6f64 756c 6520 that the module │ │ │ │ +0005c230: 4578 7428 4e2c 6b29 2028 7472 756e 6361 Ext(N,k) (trunca │ │ │ │ +0005c240: 7465 6429 2077 696c 6c20 6175 746f 6d61 ted) will automa │ │ │ │ +0005c250: 7469 6361 6c6c 7920 6265 2066 7265 650a tically be free. │ │ │ │ +0005c260: 6f76 6572 2074 6865 2065 7874 6572 696f over the exterio │ │ │ │ +0005c270: 7220 616c 6765 6272 6120 245c 7765 6467 r algebra $\wedg │ │ │ │ +0005c280: 6528 6b5e 6e29 2420 6765 6e65 7261 7465 e(k^n)$ generate │ │ │ │ +0005c290: 6420 6279 2045 7874 5e31 286b 2c6b 293b d by Ext^1(k,k); │ │ │ │ +0005c2a0: 206e 6f74 2061 2074 7970 6963 616c 0a45 not a typical.E │ │ │ │ +0005c2b0: 7874 206d 6f64 756c 652e 0a0a 4d6f 7265 xt module...More │ │ │ │ +0005c2c0: 2070 7265 6369 7365 6c79 3a0a 0a53 7570 precisely:..Sup │ │ │ │ +0005c2d0: 706f 7365 2074 6861 7420 2452 203d 206b pose that $R = k │ │ │ │ +0005c2e0: 5b61 5f31 2c5c 646f 7473 2c20 615f 6e5d [a_1,\dots, a_n] │ │ │ │ +0005c2f0: 2f28 665f 312c 5c64 6f74 732c 665f 6329 /(f_1,\dots,f_c) │ │ │ │ +0005c300: 2420 6c65 7420 244b 4b20 3d0a 6b5b 785f $ let $KK =.k[x_ │ │ │ │ +0005c310: 312c 5c64 6f74 732c 785f 635d 242c 2061 1,\dots,x_c]$, a │ │ │ │ +0005c320: 6e64 206c 6574 2024 5c4c 616d 6264 6120 nd let $\Lambda │ │ │ │ +0005c330: 3d20 5c77 6564 6765 206b 5e6e 242e 2024 = \wedge k^n$. $ │ │ │ │ +0005c340: 4520 3d20 4b4b 5c6f 7469 6d65 735c 4c61 E = KK\otimes\La │ │ │ │ +0005c350: 6d62 6461 242c 2073 6f0a 7468 6174 2074 mbda$, so.that t │ │ │ │ +0005c360: 6865 206d 696e 696d 616c 2024 5224 2d66 he minimal $R$-f │ │ │ │ +0005c370: 7265 6520 7265 736f 6c75 7469 6f6e 206f ree resolution o │ │ │ │ +0005c380: 6620 246b 2420 6861 7320 756e 6465 726c f $k$ has underl │ │ │ │ +0005c390: 7969 6e67 206d 6f64 756c 6520 2452 5c6f ying module $R\o │ │ │ │ +0005c3a0: 7469 6d65 730a 455e 2a24 2c20 7768 6572 times.E^*$, wher │ │ │ │ +0005c3b0: 6520 2445 5e2a 2420 6973 2074 6865 2067 e $E^*$ is the g │ │ │ │ +0005c3c0: 7261 6465 6420 7665 6374 6f72 2073 7061 raded vector spa │ │ │ │ +0005c3d0: 6365 2064 7561 6c20 6f66 2024 4524 2e0a ce dual of $E$.. │ │ │ │ +0005c3e0: 0a4c 6574 204d 4d20 6265 2074 6865 2072 .Let MM be the r │ │ │ │ +0005c3f0: 6573 756c 7420 6f66 2074 7275 6e63 6174 esult of truncat │ │ │ │ +0005c400: 696e 6720 4d20 6174 2069 7473 2072 6567 ing M at its reg │ │ │ │ +0005c410: 756c 6172 6974 7920 616e 6420 7368 6966 ularity and shif │ │ │ │ +0005c420: 7469 6e67 2069 7420 736f 2074 6861 740a ting it so that. │ │ │ │ +0005c430: 6974 2069 7320 6765 6e65 7261 7465 6420 it is generated │ │ │ │ +0005c440: 696e 2064 6567 7265 6520 302e 204c 6574 in degree 0. Let │ │ │ │ +0005c450: 2024 4624 2062 6520 6120 244b 4b24 2d66 $F$ be a $KK$-f │ │ │ │ +0005c460: 7265 6520 7265 736f 6c75 7469 6f6e 206f ree resolution o │ │ │ │ +0005c470: 6620 244d 4d24 2c20 616e 640a 7772 6974 f $MM$, and.writ │ │ │ │ +0005c480: 6520 2446 5f69 203d 204b 4b5c 6f74 696d e $F_i = KK\otim │ │ │ │ +0005c490: 6573 2056 5f69 2e24 2053 696e 6365 206c es V_i.$ Since l │ │ │ │ +0005c4a0: 696e 6561 7220 666f 726d 7320 6f76 6572 inear forms over │ │ │ │ +0005c4b0: 2024 4b4b 2420 636f 7272 6573 706f 6e64 $KK$ correspond │ │ │ │ +0005c4c0: 2074 6f20 4349 0a6f 7065 7261 746f 7273 to CI.operators │ │ │ │ +0005c4d0: 206f 6620 6465 6772 6565 202d 3220 6f6e of degree -2 on │ │ │ │ +0005c4e0: 2074 6865 2072 6573 6f6c 7574 696f 6e20 the resolution │ │ │ │ +0005c4f0: 4720 6f66 206b 206f 7665 7220 522c 2077 G of k over R, w │ │ │ │ +0005c500: 6520 6d61 7920 666f 726d 2061 206d 6170 e may form a map │ │ │ │ +0005c510: 2024 240a 645f 312b 645f 323a 205c 7375 $$.d_1+d_2: \su │ │ │ │ +0005c520: 6d5f 7b69 3d30 7d5e 6d20 475f 7b69 2b31 m_{i=0}^m G_{i+1 │ │ │ │ +0005c530: 7d5c 6f74 696d 6573 2056 5f7b 6d2d 697d }\otimes V_{m-i} │ │ │ │ +0005c540: 5e2a 205c 746f 205c 7375 6d5f 7b69 3d30 ^* \to \sum_{i=0 │ │ │ │ +0005c550: 7d5e 6d20 475f 695c 6f74 696d 6573 0a56 }^m G_i\otimes.V │ │ │ │ +0005c560: 5f7b 6d2d 697d 5e2a 2024 2420 7768 6572 _{m-i}^* $$ wher │ │ │ │ +0005c570: 6520 2464 5f31 2420 6973 2074 6865 2064 e $d_1$ is the d │ │ │ │ +0005c580: 6972 6563 7420 7375 6d20 6f66 2074 6865 irect sum of the │ │ │ │ +0005c590: 2064 6966 6665 7265 6e74 6961 6c73 2024 differentials $ │ │ │ │ +0005c5a0: 2847 5f7b 692b 317d 5c74 6f0a 475f 6929 (G_{i+1}\to.G_i) │ │ │ │ +0005c5b0: 5c6f 7469 6d65 7320 565f 695e 2a24 2061 \otimes V_i^*$ a │ │ │ │ +0005c5c0: 6e64 2024 645f 3224 2069 7320 7468 6520 nd $d_2$ is the │ │ │ │ +0005c5d0: 6469 7265 6374 2073 756d 206f 6620 7468 direct sum of th │ │ │ │ +0005c5e0: 6520 6d61 7073 2024 5c70 6869 5f69 2420 e maps $\phi_i$ │ │ │ │ +0005c5f0: 6465 6669 6e65 640a 6672 6f6d 2074 6865 defined.from the │ │ │ │ +0005c600: 2064 6966 6665 7265 6e74 6961 6c73 206f differentials o │ │ │ │ +0005c610: 6620 2446 2420 6279 2073 7562 7374 6974 f $F$ by substit │ │ │ │ +0005c620: 7574 696e 6720 4349 206f 7065 7261 746f uting CI operato │ │ │ │ +0005c630: 7273 2066 6f72 206c 696e 6561 7220 666f rs for linear fo │ │ │ │ +0005c640: 726d 732c 0a24 5c70 6869 5f69 3a20 475f rms,.$\phi_i: G_ │ │ │ │ +0005c650: 7b69 2b31 7d5c 6f74 696d 6573 2056 5f69 {i+1}\otimes V_i │ │ │ │ +0005c660: 205c 746f 2047 5f7b 692d 317d 5c6f 7469 \to G_{i-1}\oti │ │ │ │ +0005c670: 6d65 7320 565f 7b69 2d31 7d24 2e20 5468 mes V_{i-1}$. Th │ │ │ │ +0005c680: 6520 7363 7269 7074 2072 6574 7572 6e73 e script returns │ │ │ │ +0005c690: 2074 6865 0a6d 6f64 756c 6520 4e20 7468 the.module N th │ │ │ │ +0005c6a0: 6174 2069 7320 7468 6520 636f 6b65 726e at is the cokern │ │ │ │ +0005c6b0: 656c 206f 6620 2464 5f31 2b64 5f32 242e el of $d_1+d_2$. │ │ │ │ +0005c6c0: 0a0a 5468 6520 6d6f 6475 6c65 2024 4578 ..The module $Ex │ │ │ │ +0005c6d0: 745f 5228 4e2c 6b29 2420 6167 7265 6573 t_R(N,k)$ agrees │ │ │ │ +0005c6e0: 2c20 6166 7465 7220 6120 6665 7720 7374 , after a few st │ │ │ │ +0005c6f0: 6570 732c 2077 6974 6820 7468 6520 6d6f eps, with the mo │ │ │ │ +0005c700: 6475 6c65 2064 6572 6976 6564 2066 726f dule derived fro │ │ │ │ +0005c710: 6d0a 244d 4d24 2062 7920 7465 6e73 6f72 m.$MM$ by tensor │ │ │ │ +0005c720: 696e 6720 6974 2077 6974 6820 245c 4c61 ing it with $\La │ │ │ │ +0005c730: 6d62 6461 242c 2074 6861 7420 6973 2c20 mbda$, that is, │ │ │ │ +0005c740: 7769 7468 2074 6865 206d 6f64 756c 65c3 with the module. │ │ │ │ +0005c750: 9f20 2424 204d 4d27 203d 205c 7375 6d5f . $$ MM' = \sum_ │ │ │ │ +0005c760: 6a0a 284d 4d27 286a 295c 6f74 696d 6573 j.(MM'(j)\otimes │ │ │ │ +0005c770: 205c 4c61 6d62 6461 5f6a 2920 2424 2073 \Lambda_j) $$ s │ │ │ │ +0005c780: 6f20 7468 6174 2024 4d4d 275f 7020 3d20 o that $MM'_p = │ │ │ │ +0005c790: 284d 4d5f 705c 6f74 696d 6573 204c 616d (MM_p\otimes Lam │ │ │ │ +0005c7a0: 6264 615f 3029 205c 6f70 6c75 730a 284d bda_0) \oplus.(M │ │ │ │ +0005c7b0: 4d5f 7b70 2d31 7d5c 6f74 696d 6573 204c M_{p-1}\otimes L │ │ │ │ +0005c7c0: 616d 6264 615f 3129 205c 6f70 6c75 735c ambda_1) \oplus\ │ │ │ │ +0005c7d0: 6364 6f74 7324 2e0a 0a54 6865 2066 756e cdots$...The fun │ │ │ │ +0005c7e0: 6374 696f 6e20 6866 4d6f 6475 6c65 4173 ction hfModuleAs │ │ │ │ +0005c7f0: 4578 7420 636f 6d70 7574 6573 2074 6865 Ext computes the │ │ │ │ +0005c800: 2048 696c 6265 7274 2066 756e 6374 696f Hilbert functio │ │ │ │ +0005c810: 6e20 6f66 204d 4d27 206e 756d 6572 6963 n of MM' numeric │ │ │ │ +0005c820: 616c 6c79 0a66 726f 6d20 7468 6174 206f ally.from that o │ │ │ │ +0005c830: 6620 4d4d 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d f MM...+-------- │ │ │ │ +0005c840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005c850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c870: 2d2d 2d2b 0a7c 6931 203a 206b 6b20 3d20 ---+.|i1 : kk = │ │ │ │ -0005c880: 5a5a 2f31 3031 3b20 2020 2020 2020 2020 ZZ/101; │ │ │ │ -0005c890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c8a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005c860: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0005c870: 6b6b 203d 205a 5a2f 3130 313b 2020 2020 kk = ZZ/101; │ │ │ │ +0005c880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005c890: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0005c8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005c8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c8d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a ---------+.|i2 : │ │ │ │ -0005c8e0: 2053 203d 206b 6b5b 612c 622c 635d 3b20 S = kk[a,b,c]; │ │ │ │ +0005c8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005c8d0: 7c69 3220 3a20 5320 3d20 6b6b 5b61 2c62 |i2 : S = kk[a,b │ │ │ │ +0005c8e0: 2c63 5d3b 2020 2020 2020 2020 2020 2020 ,c]; │ │ │ │ 0005c8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c900: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0005c900: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 0005c910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005c920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005c930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0005c940: 0a7c 6933 203a 2066 6620 3d20 6d61 7472 .|i3 : ff = matr │ │ │ │ -0005c950: 6978 7b7b 615e 342c 2062 5e34 2c63 5e34 ix{{a^4, b^4,c^4 │ │ │ │ -0005c960: 7d7d 3b20 2020 2020 2020 2020 2020 2020 }}; │ │ │ │ -0005c970: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005c930: 2d2d 2d2d 2b0a 7c69 3320 3a20 6666 203d ----+.|i3 : ff = │ │ │ │ +0005c940: 206d 6174 7269 787b 7b61 5e34 2c20 625e matrix{{a^4, b^ │ │ │ │ +0005c950: 342c 635e 347d 7d3b 2020 2020 2020 2020 4,c^4}}; │ │ │ │ +0005c960: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0005c970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005c980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c9a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0005c9b0: 2020 2020 2031 2020 2020 2020 3320 2020 1 3 │ │ │ │ -0005c9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005c9d0: 2020 2020 2020 2020 7c0a 7c6f 3320 3a20 |.|o3 : │ │ │ │ -0005c9e0: 4d61 7472 6978 2053 2020 3c2d 2d20 5320 Matrix S <-- S │ │ │ │ +0005c990: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0005c9a0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +0005c9b0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0005c9c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005c9d0: 6f33 203a 204d 6174 7269 7820 5320 203c o3 : Matrix S < │ │ │ │ +0005c9e0: 2d2d 2053 2020 2020 2020 2020 2020 2020 -- S │ │ │ │ 0005c9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ca00: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0005ca00: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0005ca10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ca20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0005ca40: 7c69 3420 3a20 5220 3d20 532f 6964 6561 |i4 : R = S/idea │ │ │ │ -0005ca50: 6c20 6666 3b20 2020 2020 2020 2020 2020 l ff; │ │ │ │ -0005ca60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ca70: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0005ca30: 2d2d 2d2b 0a7c 6934 203a 2052 203d 2053 ---+.|i4 : R = S │ │ │ │ +0005ca40: 2f69 6465 616c 2066 663b 2020 2020 2020 /ideal ff; │ │ │ │ +0005ca50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005ca60: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ca80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005caa0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4f70 7320 ----+.|i5 : Ops │ │ │ │ -0005cab0: 3d20 6b6b 5b78 5f31 2c78 5f32 2c78 5f33 = kk[x_1,x_2,x_3 │ │ │ │ -0005cac0: 5d3b 2020 2020 2020 2020 2020 2020 2020 ]; │ │ │ │ -0005cad0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0005ca90: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ +0005caa0: 204f 7073 203d 206b 6b5b 785f 312c 785f Ops = kk[x_1,x_ │ │ │ │ +0005cab0: 322c 785f 335d 3b20 2020 2020 2020 2020 2,x_3]; │ │ │ │ +0005cac0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0005cad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 ----------+.|i6 │ │ │ │ -0005cb10: 3a20 4d4d 203d 204f 7073 5e31 2f28 785f : MM = Ops^1/(x_ │ │ │ │ -0005cb20: 312a 6964 6561 6c28 785f 325e 322c 785f 1*ideal(x_2^2,x_ │ │ │ │ -0005cb30: 3329 293b 2020 2020 2020 2020 207c 0a2b 3)); |.+ │ │ │ │ +0005caf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +0005cb00: 0a7c 6936 203a 204d 4d20 3d20 4f70 735e .|i6 : MM = Ops^ │ │ │ │ +0005cb10: 312f 2878 5f31 2a69 6465 616c 2878 5f32 1/(x_1*ideal(x_2 │ │ │ │ +0005cb20: 5e32 2c78 5f33 2929 3b20 2020 2020 2020 ^2,x_3)); │ │ │ │ +0005cb30: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0005cb40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cb50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cb60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cb70: 2b0a 7c69 3720 3a20 4e20 3d20 6d6f 6475 +.|i7 : N = modu │ │ │ │ -0005cb80: 6c65 4173 4578 7428 4d4d 2c52 293b 2020 leAsExt(MM,R); │ │ │ │ -0005cb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cba0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0005cb60: 2d2d 2d2d 2d2b 0a7c 6937 203a 204e 203d -----+.|i7 : N = │ │ │ │ +0005cb70: 206d 6f64 756c 6541 7345 7874 284d 4d2c moduleAsExt(MM, │ │ │ │ +0005cb80: 5229 3b20 2020 2020 2020 2020 2020 2020 R); │ │ │ │ +0005cb90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0005cba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cbb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cbd0: 2d2d 2d2d 2d2d 2b0a 7c69 3820 3a20 6265 ------+.|i8 : be │ │ │ │ -0005cbe0: 7474 6920 6672 6565 5265 736f 6c75 7469 tti freeResoluti │ │ │ │ -0005cbf0: 6f6e 2820 4e2c 204c 656e 6774 684c 696d on( N, LengthLim │ │ │ │ -0005cc00: 6974 203d 3e20 3130 297c 0a7c 2020 2020 it => 10)|.| │ │ │ │ +0005cbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ +0005cbd0: 203a 2062 6574 7469 2066 7265 6552 6573 : betti freeRes │ │ │ │ +0005cbe0: 6f6c 7574 696f 6e28 204e 2c20 4c65 6e67 olution( N, Leng │ │ │ │ +0005cbf0: 7468 4c69 6d69 7420 3d3e 2031 3029 7c0a thLimit => 10)|. │ │ │ │ +0005cc00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005cc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005cc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cc30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0005cc40: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ -0005cc50: 2020 3220 2033 2020 3420 2035 2020 3620 2 3 4 5 6 │ │ │ │ -0005cc60: 2037 2020 3820 2039 2031 3020 2020 207c 7 8 9 10 | │ │ │ │ -0005cc70: 0a7c 6f38 203d 2074 6f74 616c 3a20 3336 .|o8 = total: 36 │ │ │ │ -0005cc80: 2032 3720 3239 2033 3120 3333 2033 3520 27 29 31 33 35 │ │ │ │ -0005cc90: 3337 2033 3920 3431 2034 3320 3435 2020 37 39 41 43 45 │ │ │ │ -0005cca0: 2020 7c0a 7c20 2020 2020 2020 202d 363a |.| -6: │ │ │ │ -0005ccb0: 2031 3820 2036 2020 2e20 202e 2020 2e20 18 6 . . . │ │ │ │ -0005ccc0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005ccd0: 2e20 2020 207c 0a7c 2020 2020 2020 2020 . |.| │ │ │ │ -0005cce0: 2d35 3a20 202e 2020 2e20 202e 2020 2e20 -5: . . . . │ │ │ │ -0005ccf0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005cd00: 2e20 202e 2020 2020 7c0a 7c20 2020 2020 . . |.| │ │ │ │ -0005cd10: 2020 202d 343a 2031 3820 3231 2032 3120 -4: 18 21 21 │ │ │ │ -0005cd20: 2037 2020 2e20 202e 2020 2e20 202e 2020 7 . . . . │ │ │ │ -0005cd30: 2e20 202e 2020 2e20 2020 207c 0a7c 2020 . . . |.| │ │ │ │ -0005cd40: 2020 2020 2020 2d33 3a20 202e 2020 2e20 -3: . . │ │ │ │ -0005cd50: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005cd60: 2e20 202e 2020 2e20 202e 2020 2020 7c0a . . . . |. │ │ │ │ -0005cd70: 7c20 2020 2020 2020 202d 323a 2020 2e20 | -2: . │ │ │ │ -0005cd80: 202e 2020 3820 3234 2032 3420 2038 2020 . 8 24 24 8 │ │ │ │ -0005cd90: 2e20 202e 2020 2e20 202e 2020 2e20 2020 . . . . . │ │ │ │ -0005cda0: 207c 0a7c 2020 2020 2020 2020 2d31 3a20 |.| -1: │ │ │ │ -0005cdb0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ -0005cdc0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005cdd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0005cde0: 303a 2020 2e20 202e 2020 2e20 202e 2020 0: . . . . │ │ │ │ -0005cdf0: 3920 3237 2032 3720 2039 2020 2e20 202e 9 27 27 9 . . │ │ │ │ -0005ce00: 2020 2e20 2020 207c 0a7c 2020 2020 2020 . |.| │ │ │ │ -0005ce10: 2020 2031 3a20 202e 2020 2e20 202e 2020 1: . . . │ │ │ │ -0005ce20: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005ce30: 2020 2e20 202e 2020 2020 7c0a 7c20 2020 . . |.| │ │ │ │ -0005ce40: 2020 2020 2020 323a 2020 2e20 202e 2020 2: . . │ │ │ │ -0005ce50: 2e20 202e 2020 2e20 202e 2031 3020 3330 . . . . 10 30 │ │ │ │ -0005ce60: 2033 3020 3130 2020 2e20 2020 207c 0a7c 30 10 . |.| │ │ │ │ -0005ce70: 2020 2020 2020 2020 2033 3a20 202e 2020 3: . │ │ │ │ -0005ce80: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005ce90: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ -0005cea0: 7c0a 7c20 2020 2020 2020 2020 343a 2020 |.| 4: │ │ │ │ -0005ceb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ -0005cec0: 2020 2e20 202e 2031 3120 3333 2033 3320 . . 11 33 33 │ │ │ │ -0005ced0: 2020 207c 0a7c 2020 2020 2020 2020 2035 |.| 5 │ │ │ │ -0005cee0: 3a20 202e 2020 2e20 202e 2020 2e20 202e : . . . . . │ │ │ │ -0005cef0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0005cf00: 202e 2020 2020 7c0a 7c20 2020 2020 2020 . |.| │ │ │ │ -0005cf10: 2020 363a 2020 2e20 202e 2020 2e20 202e 6: . . . . │ │ │ │ -0005cf20: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ -0005cf30: 202e 2031 3220 2020 207c 0a7c 2020 2020 . 12 |.| │ │ │ │ +0005cc30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0005cc40: 2030 2020 3120 2032 2020 3320 2034 2020 0 1 2 3 4 │ │ │ │ +0005cc50: 3520 2036 2020 3720 2038 2020 3920 3130 5 6 7 8 9 10 │ │ │ │ +0005cc60: 2020 2020 7c0a 7c6f 3820 3d20 746f 7461 |.|o8 = tota │ │ │ │ +0005cc70: 6c3a 2033 3620 3237 2032 3920 3331 2033 l: 36 27 29 31 3 │ │ │ │ +0005cc80: 3320 3335 2033 3720 3339 2034 3120 3433 3 35 37 39 41 43 │ │ │ │ +0005cc90: 2034 3520 2020 207c 0a7c 2020 2020 2020 45 |.| │ │ │ │ +0005cca0: 2020 2d36 3a20 3138 2020 3620 202e 2020 -6: 18 6 . │ │ │ │ +0005ccb0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0005ccc0: 2020 2e20 202e 2020 2020 7c0a 7c20 2020 . . |.| │ │ │ │ +0005ccd0: 2020 2020 202d 353a 2020 2e20 202e 2020 -5: . . │ │ │ │ +0005cce0: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0005ccf0: 2020 2e20 202e 2020 2e20 2020 207c 0a7c . . . |.| │ │ │ │ +0005cd00: 2020 2020 2020 2020 2d34 3a20 3138 2032 -4: 18 2 │ │ │ │ +0005cd10: 3120 3231 2020 3720 202e 2020 2e20 202e 1 21 7 . . . │ │ │ │ +0005cd20: 2020 2e20 202e 2020 2e20 202e 2020 2020 . . . . │ │ │ │ +0005cd30: 7c0a 7c20 2020 2020 2020 202d 333a 2020 |.| -3: │ │ │ │ +0005cd40: 2e20 202e 2020 2e20 202e 2020 2e20 202e . . . . . . │ │ │ │ +0005cd50: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005cd60: 2020 207c 0a7c 2020 2020 2020 2020 2d32 |.| -2 │ │ │ │ +0005cd70: 3a20 202e 2020 2e20 2038 2032 3420 3234 : . . 8 24 24 │ │ │ │ +0005cd80: 2020 3820 202e 2020 2e20 202e 2020 2e20 8 . . . . │ │ │ │ +0005cd90: 202e 2020 2020 7c0a 7c20 2020 2020 2020 . |.| │ │ │ │ +0005cda0: 202d 313a 2020 2e20 202e 2020 2e20 202e -1: . . . . │ │ │ │ +0005cdb0: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005cdc0: 202e 2020 2e20 2020 207c 0a7c 2020 2020 . . |.| │ │ │ │ +0005cdd0: 2020 2020 2030 3a20 202e 2020 2e20 202e 0: . . . │ │ │ │ +0005cde0: 2020 2e20 2039 2032 3720 3237 2020 3920 . 9 27 27 9 │ │ │ │ +0005cdf0: 202e 2020 2e20 202e 2020 2020 7c0a 7c20 . . . |.| │ │ │ │ +0005ce00: 2020 2020 2020 2020 313a 2020 2e20 202e 1: . . │ │ │ │ +0005ce10: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005ce20: 202e 2020 2e20 202e 2020 2e20 2020 207c . . . . | │ │ │ │ +0005ce30: 0a7c 2020 2020 2020 2020 2032 3a20 202e .| 2: . │ │ │ │ +0005ce40: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005ce50: 3130 2033 3020 3330 2031 3020 202e 2020 10 30 30 10 . │ │ │ │ +0005ce60: 2020 7c0a 7c20 2020 2020 2020 2020 333a |.| 3: │ │ │ │ +0005ce70: 2020 2e20 202e 2020 2e20 202e 2020 2e20 . . . . . │ │ │ │ +0005ce80: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0005ce90: 2e20 2020 207c 0a7c 2020 2020 2020 2020 . |.| │ │ │ │ +0005cea0: 2034 3a20 202e 2020 2e20 202e 2020 2e20 4: . . . . │ │ │ │ +0005ceb0: 202e 2020 2e20 202e 2020 2e20 3131 2033 . . . . 11 3 │ │ │ │ +0005cec0: 3320 3333 2020 2020 7c0a 7c20 2020 2020 3 33 |.| │ │ │ │ +0005ced0: 2020 2020 353a 2020 2e20 202e 2020 2e20 5: . . . │ │ │ │ +0005cee0: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0005cef0: 2e20 202e 2020 2e20 2020 207c 0a7c 2020 . . . |.| │ │ │ │ +0005cf00: 2020 2020 2020 2036 3a20 202e 2020 2e20 6: . . │ │ │ │ +0005cf10: 202e 2020 2e20 202e 2020 2e20 202e 2020 . . . . . │ │ │ │ +0005cf20: 2e20 202e 2020 2e20 3132 2020 2020 7c0a . . . 12 |. │ │ │ │ +0005cf30: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005cf40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005cf50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cf60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0005cf70: 3820 3a20 4265 7474 6954 616c 6c79 2020 8 : BettiTally │ │ │ │ +0005cf60: 207c 0a7c 6f38 203a 2042 6574 7469 5461 |.|o8 : BettiTa │ │ │ │ +0005cf70: 6c6c 7920 2020 2020 2020 2020 2020 2020 lly │ │ │ │ 0005cf80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005cf90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0005cfa0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0005cf90: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005cfa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005cfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005cfd0: 2d2d 2b0a 7c69 3920 3a20 6866 4d6f 6475 --+.|i9 : hfModu │ │ │ │ -0005cfe0: 6c65 4173 4578 7428 3132 2c4d 4d2c 3329 leAsExt(12,MM,3) │ │ │ │ -0005cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d000: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0005cfc0: 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a 2068 -------+.|i9 : h │ │ │ │ +0005cfd0: 664d 6f64 756c 6541 7345 7874 2831 322c fModuleAsExt(12, │ │ │ │ +0005cfe0: 4d4d 2c33 2920 2020 2020 2020 2020 2020 MM,3) │ │ │ │ +0005cff0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0005d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d030: 2020 2020 2020 2020 7c0a 7c6f 3920 3d20 |.|o9 = │ │ │ │ -0005d040: 2832 332c 2032 352c 2032 372c 2032 392c (23, 25, 27, 29, │ │ │ │ -0005d050: 2033 312c 2033 332c 2033 352c 2033 372c 31, 33, 35, 37, │ │ │ │ -0005d060: 2033 392c 2034 3129 2020 207c 0a7c 2020 39, 41) |.| │ │ │ │ +0005d020: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0005d030: 6f39 203d 2028 3233 2c20 3235 2c20 3237 o9 = (23, 25, 27 │ │ │ │ +0005d040: 2c20 3239 2c20 3331 2c20 3333 2c20 3335 , 29, 31, 33, 35 │ │ │ │ +0005d050: 2c20 3337 2c20 3339 2c20 3431 2920 2020 , 37, 39, 41) │ │ │ │ +0005d060: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0005d070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005d080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d090: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0005d0a0: 7c6f 3920 3a20 5365 7175 656e 6365 2020 |o9 : Sequence │ │ │ │ +0005d090: 2020 207c 0a7c 6f39 203a 2053 6571 7565 |.|o9 : Seque │ │ │ │ +0005d0a0: 6e63 6520 2020 2020 2020 2020 2020 2020 nce │ │ │ │ 0005d0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005d0d0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +0005d0c0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0005d0d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d0e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d100: 2d2d 2d2d 2b0a 0a43 6176 6561 740a 3d3d ----+..Caveat.== │ │ │ │ -0005d110: 3d3d 3d3d 0a0a 5468 6520 656c 656d 656e ====..The elemen │ │ │ │ -0005d120: 7473 2066 5f31 2e2e 665f 6320 6d75 7374 ts f_1..f_c must │ │ │ │ -0005d130: 2062 6520 686f 6d6f 6765 6e65 6f75 7320 be homogeneous │ │ │ │ -0005d140: 6f66 2074 6865 2073 616d 6520 6465 6772 of the same degr │ │ │ │ -0005d150: 6565 2e20 5468 6520 7363 7269 7074 2063 ee. The script c │ │ │ │ -0005d160: 6f75 6c64 0a62 6520 7265 7772 6974 7465 ould.be rewritte │ │ │ │ -0005d170: 6e20 746f 2061 6363 6f6d 6d6f 6461 7465 n to accommodate │ │ │ │ -0005d180: 2064 6966 6665 7265 6e74 2064 6567 7265 different degre │ │ │ │ -0005d190: 6573 2c20 6275 7420 6f6e 6c79 2062 7920 es, but only by │ │ │ │ -0005d1a0: 676f 696e 6720 746f 2074 6865 206c 6f63 going to the loc │ │ │ │ -0005d1b0: 616c 0a63 6174 6567 6f72 790a 0a53 6565 al.category..See │ │ │ │ -0005d1c0: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -0005d1d0: 2020 2a20 2a6e 6f74 6520 4578 744d 6f64 * *note ExtMod │ │ │ │ -0005d1e0: 756c 653a 2045 7874 4d6f 6475 6c65 2c20 ule: ExtModule, │ │ │ │ -0005d1f0: 2d2d 2045 7874 5e2a 284d 2c6b 2920 6f76 -- Ext^*(M,k) ov │ │ │ │ -0005d200: 6572 2061 2063 6f6d 706c 6574 6520 696e er a complete in │ │ │ │ -0005d210: 7465 7273 6563 7469 6f6e 2061 730a 2020 tersection as. │ │ │ │ -0005d220: 2020 6d6f 6475 6c65 206f 7665 7220 4349 module over CI │ │ │ │ -0005d230: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ -0005d240: 202a 202a 6e6f 7465 2065 7665 6e45 7874 * *note evenExt │ │ │ │ -0005d250: 4d6f 6475 6c65 3a20 6576 656e 4578 744d Module: evenExtM │ │ │ │ -0005d260: 6f64 756c 652c 202d 2d20 6576 656e 2070 odule, -- even p │ │ │ │ -0005d270: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ -0005d280: 2920 6f76 6572 2061 0a20 2020 2063 6f6d ) over a. com │ │ │ │ -0005d290: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ -0005d2a0: 6f6e 2061 7320 6d6f 6475 6c65 206f 7665 on as module ove │ │ │ │ -0005d2b0: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ -0005d2c0: 6e67 0a20 202a 202a 6e6f 7465 206f 6464 ng. * *note odd │ │ │ │ -0005d2d0: 4578 744d 6f64 756c 653a 206f 6464 4578 ExtModule: oddEx │ │ │ │ -0005d2e0: 744d 6f64 756c 652c 202d 2d20 6f64 6420 tModule, -- odd │ │ │ │ -0005d2f0: 7061 7274 206f 6620 4578 745e 2a28 4d2c part of Ext^*(M, │ │ │ │ -0005d300: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ -0005d310: 7465 0a20 2020 2069 6e74 6572 7365 6374 te. intersect │ │ │ │ -0005d320: 696f 6e20 6173 206d 6f64 756c 6520 6f76 ion as module ov │ │ │ │ -0005d330: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ -0005d340: 696e 670a 2020 2a20 2a6e 6f74 6520 4578 ing. * *note Ex │ │ │ │ -0005d350: 744d 6f64 756c 6544 6174 613a 2045 7874 tModuleData: Ext │ │ │ │ -0005d360: 4d6f 6475 6c65 4461 7461 2c20 2d2d 2045 ModuleData, -- E │ │ │ │ -0005d370: 7665 6e20 616e 6420 6f64 6420 4578 7420 ven and odd Ext │ │ │ │ -0005d380: 6d6f 6475 6c65 7320 616e 6420 7468 6569 modules and thei │ │ │ │ -0005d390: 720a 2020 2020 7265 6775 6c61 7269 7479 r. regularity │ │ │ │ -0005d3a0: 0a20 202a 202a 6e6f 7465 2068 664d 6f64 . * *note hfMod │ │ │ │ -0005d3b0: 756c 6541 7345 7874 3a20 6866 4d6f 6475 uleAsExt: hfModu │ │ │ │ -0005d3c0: 6c65 4173 4578 742c 202d 2d20 7072 6564 leAsExt, -- pred │ │ │ │ -0005d3d0: 6963 7420 6265 7474 6920 6e75 6d62 6572 ict betti number │ │ │ │ -0005d3e0: 7320 6f66 0a20 2020 206d 6f64 756c 6541 s of. moduleA │ │ │ │ -0005d3f0: 7345 7874 284d 2c52 290a 0a57 6179 7320 sExt(M,R)..Ways │ │ │ │ -0005d400: 746f 2075 7365 206d 6f64 756c 6541 7345 to use moduleAsE │ │ │ │ -0005d410: 7874 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d xt:.============ │ │ │ │ -0005d420: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ -0005d430: 2a20 226d 6f64 756c 6541 7345 7874 284d * "moduleAsExt(M │ │ │ │ -0005d440: 6f64 756c 652c 5269 6e67 2922 0a0a 466f odule,Ring)"..Fo │ │ │ │ -0005d450: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -0005d460: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0005d470: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -0005d480: 2a6e 6f74 6520 6d6f 6475 6c65 4173 4578 *note moduleAsEx │ │ │ │ -0005d490: 743a 206d 6f64 756c 6541 7345 7874 2c20 t: moduleAsExt, │ │ │ │ -0005d4a0: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ -0005d4b0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ -0005d4c0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ -0005d4d0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +0005d0f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 4361 7665 ---------+..Cave │ │ │ │ +0005d100: 6174 0a3d 3d3d 3d3d 3d0a 0a54 6865 2065 at.======..The e │ │ │ │ +0005d110: 6c65 6d65 6e74 7320 665f 312e 2e66 5f63 lements f_1..f_c │ │ │ │ +0005d120: 206d 7573 7420 6265 2068 6f6d 6f67 656e must be homogen │ │ │ │ +0005d130: 656f 7573 206f 6620 7468 6520 7361 6d65 eous of the same │ │ │ │ +0005d140: 2064 6567 7265 652e 2054 6865 2073 6372 degree. The scr │ │ │ │ +0005d150: 6970 7420 636f 756c 640a 6265 2072 6577 ipt could.be rew │ │ │ │ +0005d160: 7269 7474 656e 2074 6f20 6163 636f 6d6d ritten to accomm │ │ │ │ +0005d170: 6f64 6174 6520 6469 6666 6572 656e 7420 odate different │ │ │ │ +0005d180: 6465 6772 6565 732c 2062 7574 206f 6e6c degrees, but onl │ │ │ │ +0005d190: 7920 6279 2067 6f69 6e67 2074 6f20 7468 y by going to th │ │ │ │ +0005d1a0: 6520 6c6f 6361 6c0a 6361 7465 676f 7279 e local.category │ │ │ │ +0005d1b0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +0005d1c0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2045 ===.. * *note E │ │ │ │ +0005d1d0: 7874 4d6f 6475 6c65 3a20 4578 744d 6f64 xtModule: ExtMod │ │ │ │ +0005d1e0: 756c 652c 202d 2d20 4578 745e 2a28 4d2c ule, -- Ext^*(M, │ │ │ │ +0005d1f0: 6b29 206f 7665 7220 6120 636f 6d70 6c65 k) over a comple │ │ │ │ +0005d200: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ +0005d210: 6173 0a20 2020 206d 6f64 756c 6520 6f76 as. module ov │ │ │ │ +0005d220: 6572 2043 4920 6f70 6572 6174 6f72 2072 er CI operator r │ │ │ │ +0005d230: 696e 670a 2020 2a20 2a6e 6f74 6520 6576 ing. * *note ev │ │ │ │ +0005d240: 656e 4578 744d 6f64 756c 653a 2065 7665 enExtModule: eve │ │ │ │ +0005d250: 6e45 7874 4d6f 6475 6c65 2c20 2d2d 2065 nExtModule, -- e │ │ │ │ +0005d260: 7665 6e20 7061 7274 206f 6620 4578 745e ven part of Ext^ │ │ │ │ +0005d270: 2a28 4d2c 6b29 206f 7665 7220 610a 2020 *(M,k) over a. │ │ │ │ +0005d280: 2020 636f 6d70 6c65 7465 2069 6e74 6572 complete inter │ │ │ │ +0005d290: 7365 6374 696f 6e20 6173 206d 6f64 756c section as modul │ │ │ │ +0005d2a0: 6520 6f76 6572 2043 4920 6f70 6572 6174 e over CI operat │ │ │ │ +0005d2b0: 6f72 2072 696e 670a 2020 2a20 2a6e 6f74 or ring. * *not │ │ │ │ +0005d2c0: 6520 6f64 6445 7874 4d6f 6475 6c65 3a20 e oddExtModule: │ │ │ │ +0005d2d0: 6f64 6445 7874 4d6f 6475 6c65 2c20 2d2d oddExtModule, -- │ │ │ │ +0005d2e0: 206f 6464 2070 6172 7420 6f66 2045 7874 odd part of Ext │ │ │ │ +0005d2f0: 5e2a 284d 2c6b 2920 6f76 6572 2061 2063 ^*(M,k) over a c │ │ │ │ +0005d300: 6f6d 706c 6574 650a 2020 2020 696e 7465 omplete. inte │ │ │ │ +0005d310: 7273 6563 7469 6f6e 2061 7320 6d6f 6475 rsection as modu │ │ │ │ +0005d320: 6c65 206f 7665 7220 4349 206f 7065 7261 le over CI opera │ │ │ │ +0005d330: 746f 7220 7269 6e67 0a20 202a 202a 6e6f tor ring. * *no │ │ │ │ +0005d340: 7465 2045 7874 4d6f 6475 6c65 4461 7461 te ExtModuleData │ │ │ │ +0005d350: 3a20 4578 744d 6f64 756c 6544 6174 612c : ExtModuleData, │ │ │ │ +0005d360: 202d 2d20 4576 656e 2061 6e64 206f 6464 -- Even and odd │ │ │ │ +0005d370: 2045 7874 206d 6f64 756c 6573 2061 6e64 Ext modules and │ │ │ │ +0005d380: 2074 6865 6972 0a20 2020 2072 6567 756c their. regul │ │ │ │ +0005d390: 6172 6974 790a 2020 2a20 2a6e 6f74 6520 arity. * *note │ │ │ │ +0005d3a0: 6866 4d6f 6475 6c65 4173 4578 743a 2068 hfModuleAsExt: h │ │ │ │ +0005d3b0: 664d 6f64 756c 6541 7345 7874 2c20 2d2d fModuleAsExt, -- │ │ │ │ +0005d3c0: 2070 7265 6469 6374 2062 6574 7469 206e predict betti n │ │ │ │ +0005d3d0: 756d 6265 7273 206f 660a 2020 2020 6d6f umbers of. mo │ │ │ │ +0005d3e0: 6475 6c65 4173 4578 7428 4d2c 5229 0a0a duleAsExt(M,R).. │ │ │ │ +0005d3f0: 5761 7973 2074 6f20 7573 6520 6d6f 6475 Ways to use modu │ │ │ │ +0005d400: 6c65 4173 4578 743a 0a3d 3d3d 3d3d 3d3d leAsExt:.======= │ │ │ │ +0005d410: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0005d420: 3d0a 0a20 202a 2022 6d6f 6475 6c65 4173 =.. * "moduleAs │ │ │ │ +0005d430: 4578 7428 4d6f 6475 6c65 2c52 696e 6729 Ext(Module,Ring) │ │ │ │ +0005d440: 220a 0a46 6f72 2074 6865 2070 726f 6772 "..For the progr │ │ │ │ +0005d450: 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d ammer.========== │ │ │ │ +0005d460: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 ========..The ob │ │ │ │ +0005d470: 6a65 6374 202a 6e6f 7465 206d 6f64 756c ject *note modul │ │ │ │ +0005d480: 6541 7345 7874 3a20 6d6f 6475 6c65 4173 eAsExt: moduleAs │ │ │ │ +0005d490: 4578 742c 2069 7320 6120 2a6e 6f74 6520 Ext, is a *note │ │ │ │ +0005d4a0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ +0005d4b0: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ +0005d4c0: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ +0005d4d0: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ 0005d4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005d510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005d520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ -0005d530: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -0005d540: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -0005d550: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -0005d560: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -0005d570: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -0005d580: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -0005d590: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ -0005d5a0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ -0005d5b0: 6f6e 732e 6d32 3a33 3039 363a 302e 0a1f ons.m2:3096:0... │ │ │ │ -0005d5c0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ -0005d5d0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ -0005d5e0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ -0005d5f0: 653a 206e 6577 4578 742c 204e 6578 743a e: newExt, Next: │ │ │ │ -0005d600: 206f 6464 4578 744d 6f64 756c 652c 2050 oddExtModule, P │ │ │ │ -0005d610: 7265 763a 206d 6f64 756c 6541 7345 7874 rev: moduleAsExt │ │ │ │ -0005d620: 2c20 5570 3a20 546f 700a 0a6e 6577 4578 , Up: Top..newEx │ │ │ │ -0005d630: 7420 2d2d 2047 6c6f 6261 6c20 4578 7420 t -- Global Ext │ │ │ │ -0005d640: 666f 7220 6d6f 6475 6c65 7320 6f76 6572 for modules over │ │ │ │ -0005d650: 2061 2063 6f6d 706c 6574 6520 496e 7465 a complete Inte │ │ │ │ -0005d660: 7273 6563 7469 6f6e 0a2a 2a2a 2a2a 2a2a rsection.******* │ │ │ │ +0005d520: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +0005d530: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +0005d540: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +0005d550: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +0005d560: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +0005d570: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +0005d580: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ +0005d590: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ +0005d5a0: 6f6c 7574 696f 6e73 2e6d 323a 3330 3936 olutions.m2:3096 │ │ │ │ +0005d5b0: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ +0005d5c0: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ +0005d5d0: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ +0005d5e0: 2c20 4e6f 6465 3a20 6e65 7745 7874 2c20 , Node: newExt, │ │ │ │ +0005d5f0: 4e65 7874 3a20 6f64 6445 7874 4d6f 6475 Next: oddExtModu │ │ │ │ +0005d600: 6c65 2c20 5072 6576 3a20 6d6f 6475 6c65 le, Prev: module │ │ │ │ +0005d610: 4173 4578 742c 2055 703a 2054 6f70 0a0a AsExt, Up: Top.. │ │ │ │ +0005d620: 6e65 7745 7874 202d 2d20 476c 6f62 616c newExt -- Global │ │ │ │ +0005d630: 2045 7874 2066 6f72 206d 6f64 756c 6573 Ext for modules │ │ │ │ +0005d640: 206f 7665 7220 6120 636f 6d70 6c65 7465 over a complete │ │ │ │ +0005d650: 2049 6e74 6572 7365 6374 696f 6e0a 2a2a Intersection.** │ │ │ │ +0005d660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005d670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0005d680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0005d6a0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -0005d6b0: 653a 200a 2020 2020 2020 2020 4520 3d20 e: . E = │ │ │ │ -0005d6c0: 6e65 7745 7874 284d 2c4e 290a 2020 2a20 newExt(M,N). * │ │ │ │ -0005d6d0: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ -0005d6e0: 4d2c 2061 202a 6e6f 7465 206d 6f64 756c M, a *note modul │ │ │ │ -0005d6f0: 653a 2028 4d61 6361 756c 6179 3244 6f63 e: (Macaulay2Doc │ │ │ │ -0005d700: 294d 6f64 756c 652c 2c20 6f76 6572 2061 )Module,, over a │ │ │ │ -0005d710: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0005d720: 6563 7469 6f6e 0a20 2020 2020 2020 2052 ection. R │ │ │ │ -0005d730: 6261 720a 2020 2020 2020 2a20 4e2c 2061 bar. * N, a │ │ │ │ -0005d740: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ -0005d750: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ -0005d760: 756c 652c 2c20 6f76 6572 2052 6261 720a ule,, over Rbar. │ │ │ │ -0005d770: 2020 2a20 2a6e 6f74 6520 4f70 7469 6f6e * *note Option │ │ │ │ -0005d780: 616c 2069 6e70 7574 733a 2028 4d61 6361 al inputs: (Maca │ │ │ │ -0005d790: 756c 6179 3244 6f63 2975 7369 6e67 2066 ulay2Doc)using f │ │ │ │ -0005d7a0: 756e 6374 696f 6e73 2077 6974 6820 6f70 unctions with op │ │ │ │ -0005d7b0: 7469 6f6e 616c 2069 6e70 7574 732c 3a0a tional inputs,:. │ │ │ │ -0005d7c0: 2020 2020 2020 2a20 4368 6563 6b20 3d3e * Check => │ │ │ │ -0005d7d0: 202e 2e2e 2c20 6465 6661 756c 7420 7661 ..., default va │ │ │ │ -0005d7e0: 6c75 6520 6661 6c73 650a 2020 2020 2020 lue false. │ │ │ │ -0005d7f0: 2a20 4772 6164 696e 6720 3d3e 202e 2e2e * Grading => ... │ │ │ │ -0005d800: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -0005d810: 320a 2020 2020 2020 2a20 4c69 6674 203d 2. * Lift = │ │ │ │ -0005d820: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ -0005d830: 616c 7565 2066 616c 7365 0a20 2020 2020 alue false. │ │ │ │ -0005d840: 202a 2056 6172 6961 626c 6573 203d 3e20 * Variables => │ │ │ │ -0005d850: 2e2e 2e2c 2064 6566 6175 6c74 2076 616c ..., default val │ │ │ │ -0005d860: 7565 2073 0a20 202a 204f 7574 7075 7473 ue s. * Outputs │ │ │ │ -0005d870: 3a0a 2020 2020 2020 2a20 452c 2061 202a :. * E, a * │ │ │ │ -0005d880: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0005d890: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0005d8a0: 652c 2c20 6f76 6572 2061 2072 696e 6720 e,, over a ring │ │ │ │ -0005d8b0: 5320 6d61 6465 2066 726f 6d20 7269 6e67 S made from ring │ │ │ │ -0005d8c0: 0a20 2020 2020 2020 2070 7265 7365 6e74 . present │ │ │ │ -0005d8d0: 6174 696f 6e20 5262 6172 2077 6974 6820 ation Rbar with │ │ │ │ -0005d8e0: 636f 6469 6d20 5262 6172 206e 6577 2076 codim Rbar new v │ │ │ │ -0005d8f0: 6172 6961 626c 6573 0a0a 4465 7363 7269 ariables..Descri │ │ │ │ -0005d900: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -0005d910: 3d0a 0a4c 6574 2052 6261 7220 3d20 522f =..Let Rbar = R/ │ │ │ │ -0005d920: 2866 312e 2e66 6329 2c20 6120 636f 6d70 (f1..fc), a comp │ │ │ │ -0005d930: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ -0005d940: 6e20 6f66 2063 6f64 696d 656e 7369 6f6e n of codimension │ │ │ │ -0005d950: 2063 2c20 616e 6420 6c65 7420 4d2c 4e20 c, and let M,N │ │ │ │ -0005d960: 6265 0a52 6261 722d 6d6f 6475 6c65 732e be.Rbar-modules. │ │ │ │ -0005d970: 2057 6520 6173 7375 6d65 2074 6861 7420 We assume that │ │ │ │ -0005d980: 7468 6520 7075 7368 466f 7277 6172 6420 the pushForward │ │ │ │ -0005d990: 6f66 204d 2074 6f20 5220 6861 7320 6669 of M to R has fi │ │ │ │ -0005d9a0: 6e69 7465 2066 7265 650a 7265 736f 6c75 nite free.resolu │ │ │ │ -0005d9b0: 7469 6f6e 2e20 5468 6520 7363 7269 7074 tion. The script │ │ │ │ -0005d9c0: 2074 6865 6e20 636f 6d70 7574 6573 2074 then computes t │ │ │ │ -0005d9d0: 6865 2074 6f74 616c 2045 7874 284d 2c4e he total Ext(M,N │ │ │ │ -0005d9e0: 2920 6173 2061 206d 6f64 756c 6520 6f76 ) as a module ov │ │ │ │ -0005d9f0: 6572 2053 203d 0a6b 6b28 735f 312e 2e73 er S =.kk(s_1..s │ │ │ │ -0005da00: 5f63 2c67 656e 7320 5229 2c20 7573 696e _c,gens R), usin │ │ │ │ -0005da10: 6720 4569 7365 6e62 7564 5368 616d 6173 g EisenbudShamas │ │ │ │ -0005da20: 6854 6f74 616c 2e0a 0a49 6620 4368 6563 hTotal...If Chec │ │ │ │ -0005da30: 6b20 3d3e 2074 7275 652c 2074 6865 6e20 k => true, then │ │ │ │ -0005da40: 7468 6520 7265 7375 6c74 2069 7320 636f the result is co │ │ │ │ -0005da50: 6d70 6172 6564 2077 6974 6820 7468 6520 mpared with the │ │ │ │ -0005da60: 6275 696c 742d 696e 2067 6c6f 6261 6c20 built-in global │ │ │ │ -0005da70: 4578 740a 7772 6974 7465 6e20 6279 2041 Ext.written by A │ │ │ │ -0005da80: 7672 616d 6f76 2061 6e64 2047 7261 7973 vramov and Grays │ │ │ │ -0005da90: 6f6e 2028 6275 7420 6e6f 7465 2074 6865 on (but note the │ │ │ │ -0005daa0: 2064 6966 6665 7265 6e63 652c 2065 7870 difference, exp │ │ │ │ -0005dab0: 6c61 696e 6564 2062 656c 6f77 292e 0a0a lained below)... │ │ │ │ -0005dac0: 4966 204c 6966 7420 3d3e 2066 616c 7365 If Lift => false │ │ │ │ -0005dad0: 2074 6865 2072 6573 756c 7420 6973 2072 the result is r │ │ │ │ -0005dae0: 6574 7572 6e65 6420 6f76 6572 2061 6e64 eturned over and │ │ │ │ -0005daf0: 2065 7874 656e 7369 6f6e 206f 6620 5262 extension of Rb │ │ │ │ -0005db00: 6172 3b20 6966 204c 6966 7420 3d3e 0a74 ar; if Lift =>.t │ │ │ │ -0005db10: 7275 6520 7468 6520 7265 7375 6c74 2069 rue the result i │ │ │ │ -0005db20: 7320 7265 7475 726e 6564 206f 7665 7220 s returned over │ │ │ │ -0005db30: 616e 6420 6578 7465 6e73 696f 6e20 6f66 and extension of │ │ │ │ -0005db40: 2052 2e0a 0a49 6620 4772 6164 696e 6720 R...If Grading │ │ │ │ -0005db50: 3d3e 2032 2c20 7468 6520 6465 6661 756c => 2, the defaul │ │ │ │ -0005db60: 742c 2074 6865 6e20 7468 6520 7265 7375 t, then the resu │ │ │ │ -0005db70: 6c74 2069 7320 6269 6772 6164 6564 2028 lt is bigraded ( │ │ │ │ -0005db80: 7468 6973 2069 7320 6e65 6365 7373 6172 this is necessar │ │ │ │ -0005db90: 790a 7768 656e 2043 6865 636b 3d3e 7472 y.when Check=>tr │ │ │ │ -0005dba0: 7565 0a0a 5468 6520 6465 6661 756c 7420 ue..The default │ │ │ │ -0005dbb0: 5661 7269 6162 6c65 7320 3d3e 2073 796d Variables => sym │ │ │ │ -0005dbc0: 626f 6c20 2273 2220 6769 7665 7320 7468 bol "s" gives th │ │ │ │ -0005dbd0: 6520 6e65 7720 7661 7269 6162 6c65 7320 e new variables │ │ │ │ -0005dbe0: 7468 6520 6e61 6d65 2073 5f69 2c0a 693d the name s_i,.i= │ │ │ │ -0005dbf0: 302e 2e63 2d31 2e20 286e 6f74 6520 7468 0..c-1. (note th │ │ │ │ -0005dc00: 6174 2074 6865 2062 7569 6c74 696e 2045 at the builtin E │ │ │ │ -0005dc10: 7874 2075 7365 7320 585f 312e 2e58 5f63 xt uses X_1..X_c │ │ │ │ -0005dc20: 2e0a 0a4f 6e20 536f 6d65 2065 7861 6d70 ...On Some examp │ │ │ │ -0005dc30: 6c65 7320 6e65 7745 7874 2069 7320 6661 les newExt is fa │ │ │ │ -0005dc40: 7374 6572 2074 6861 6e20 4578 743b 206f ster than Ext; o │ │ │ │ -0005dc50: 6e20 6f74 6865 7273 2069 7427 7320 736c n others it's sl │ │ │ │ -0005dc60: 6f77 6572 2e0a 0a41 2073 696d 706c 6520 ower...A simple │ │ │ │ -0005dc70: 6578 616d 706c 653a 2069 6620 5220 3d20 example: if R = │ │ │ │ -0005dc80: 6b5b 785f 312e 2e78 5f6e 5d20 616e 6420 k[x_1..x_n] and │ │ │ │ -0005dc90: 4920 6973 2063 6f6e 7461 696e 6564 2069 I is contained i │ │ │ │ -0005dca0: 6e20 7468 6520 6375 6265 206f 6620 7468 n the cube of th │ │ │ │ -0005dcb0: 650a 6d61 7869 6d61 6c20 6964 6561 6c2c e.maximal ideal, │ │ │ │ -0005dcc0: 2074 6865 6e20 4578 7428 6b2c 6b29 2069 then Ext(k,k) i │ │ │ │ -0005dcd0: 7320 6120 6672 6565 2053 2f28 785f 312e s a free S/(x_1. │ │ │ │ -0005dce0: 2e78 5f6e 2920 3d20 6b5b 735f 302e 2e73 .x_n) = k[s_0..s │ │ │ │ -0005dcf0: 5f28 632d 3129 5d2d 206d 6f64 756c 650a _(c-1)]- module. │ │ │ │ -0005dd00: 7769 7468 2062 696e 6f6d 6961 6c28 6e2c with binomial(n, │ │ │ │ -0005dd10: 6929 2067 656e 6572 6174 6f72 7320 696e i) generators in │ │ │ │ -0005dd20: 2064 6567 7265 6520 690a 0a2b 2d2d 2d2d degree i..+---- │ │ │ │ +0005d690: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +0005d6a0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +0005d6b0: 2045 203d 206e 6577 4578 7428 4d2c 4e29 E = newExt(M,N) │ │ │ │ +0005d6c0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ +0005d6d0: 2020 202a 204d 2c20 6120 2a6e 6f74 6520 * M, a *note │ │ │ │ +0005d6e0: 6d6f 6475 6c65 3a20 284d 6163 6175 6c61 module: (Macaula │ │ │ │ +0005d6f0: 7932 446f 6329 4d6f 6475 6c65 2c2c 206f y2Doc)Module,, o │ │ │ │ +0005d700: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +0005d710: 6e74 6572 7365 6374 696f 6e0a 2020 2020 ntersection. │ │ │ │ +0005d720: 2020 2020 5262 6172 0a20 2020 2020 202a Rbar. * │ │ │ │ +0005d730: 204e 2c20 6120 2a6e 6f74 6520 6d6f 6475 N, a *note modu │ │ │ │ +0005d740: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ +0005d750: 6329 4d6f 6475 6c65 2c2c 206f 7665 7220 c)Module,, over │ │ │ │ +0005d760: 5262 6172 0a20 202a 202a 6e6f 7465 204f Rbar. * *note O │ │ │ │ +0005d770: 7074 696f 6e61 6c20 696e 7075 7473 3a20 ptional inputs: │ │ │ │ +0005d780: 284d 6163 6175 6c61 7932 446f 6329 7573 (Macaulay2Doc)us │ │ │ │ +0005d790: 696e 6720 6675 6e63 7469 6f6e 7320 7769 ing functions wi │ │ │ │ +0005d7a0: 7468 206f 7074 696f 6e61 6c20 696e 7075 th optional inpu │ │ │ │ +0005d7b0: 7473 2c3a 0a20 2020 2020 202a 2043 6865 ts,:. * Che │ │ │ │ +0005d7c0: 636b 203d 3e20 2e2e 2e2c 2064 6566 6175 ck => ..., defau │ │ │ │ +0005d7d0: 6c74 2076 616c 7565 2066 616c 7365 0a20 lt value false. │ │ │ │ +0005d7e0: 2020 2020 202a 2047 7261 6469 6e67 203d * Grading = │ │ │ │ +0005d7f0: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +0005d800: 616c 7565 2032 0a20 2020 2020 202a 204c alue 2. * L │ │ │ │ +0005d810: 6966 7420 3d3e 202e 2e2e 2c20 6465 6661 ift => ..., defa │ │ │ │ +0005d820: 756c 7420 7661 6c75 6520 6661 6c73 650a ult value false. │ │ │ │ +0005d830: 2020 2020 2020 2a20 5661 7269 6162 6c65 * Variable │ │ │ │ +0005d840: 7320 3d3e 202e 2e2e 2c20 6465 6661 756c s => ..., defaul │ │ │ │ +0005d850: 7420 7661 6c75 6520 730a 2020 2a20 4f75 t value s. * Ou │ │ │ │ +0005d860: 7470 7574 733a 0a20 2020 2020 202a 2045 tputs:. * E │ │ │ │ +0005d870: 2c20 6120 2a6e 6f74 6520 6d6f 6475 6c65 , a *note module │ │ │ │ +0005d880: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0005d890: 4d6f 6475 6c65 2c2c 206f 7665 7220 6120 Module,, over a │ │ │ │ +0005d8a0: 7269 6e67 2053 206d 6164 6520 6672 6f6d ring S made from │ │ │ │ +0005d8b0: 2072 696e 670a 2020 2020 2020 2020 7072 ring. pr │ │ │ │ +0005d8c0: 6573 656e 7461 7469 6f6e 2052 6261 7220 esentation Rbar │ │ │ │ +0005d8d0: 7769 7468 2063 6f64 696d 2052 6261 7220 with codim Rbar │ │ │ │ +0005d8e0: 6e65 7720 7661 7269 6162 6c65 730a 0a44 new variables..D │ │ │ │ +0005d8f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +0005d900: 3d3d 3d3d 3d3d 0a0a 4c65 7420 5262 6172 ======..Let Rbar │ │ │ │ +0005d910: 203d 2052 2f28 6631 2e2e 6663 292c 2061 = R/(f1..fc), a │ │ │ │ +0005d920: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ +0005d930: 6563 7469 6f6e 206f 6620 636f 6469 6d65 ection of codime │ │ │ │ +0005d940: 6e73 696f 6e20 632c 2061 6e64 206c 6574 nsion c, and let │ │ │ │ +0005d950: 204d 2c4e 2062 650a 5262 6172 2d6d 6f64 M,N be.Rbar-mod │ │ │ │ +0005d960: 756c 6573 2e20 5765 2061 7373 756d 6520 ules. We assume │ │ │ │ +0005d970: 7468 6174 2074 6865 2070 7573 6846 6f72 that the pushFor │ │ │ │ +0005d980: 7761 7264 206f 6620 4d20 746f 2052 2068 ward of M to R h │ │ │ │ +0005d990: 6173 2066 696e 6974 6520 6672 6565 0a72 as finite free.r │ │ │ │ +0005d9a0: 6573 6f6c 7574 696f 6e2e 2054 6865 2073 esolution. The s │ │ │ │ +0005d9b0: 6372 6970 7420 7468 656e 2063 6f6d 7075 cript then compu │ │ │ │ +0005d9c0: 7465 7320 7468 6520 746f 7461 6c20 4578 tes the total Ex │ │ │ │ +0005d9d0: 7428 4d2c 4e29 2061 7320 6120 6d6f 6475 t(M,N) as a modu │ │ │ │ +0005d9e0: 6c65 206f 7665 7220 5320 3d0a 6b6b 2873 le over S =.kk(s │ │ │ │ +0005d9f0: 5f31 2e2e 735f 632c 6765 6e73 2052 292c _1..s_c,gens R), │ │ │ │ +0005da00: 2075 7369 6e67 2045 6973 656e 6275 6453 using EisenbudS │ │ │ │ +0005da10: 6861 6d61 7368 546f 7461 6c2e 0a0a 4966 hamashTotal...If │ │ │ │ +0005da20: 2043 6865 636b 203d 3e20 7472 7565 2c20 Check => true, │ │ │ │ +0005da30: 7468 656e 2074 6865 2072 6573 756c 7420 then the result │ │ │ │ +0005da40: 6973 2063 6f6d 7061 7265 6420 7769 7468 is compared with │ │ │ │ +0005da50: 2074 6865 2062 7569 6c74 2d69 6e20 676c the built-in gl │ │ │ │ +0005da60: 6f62 616c 2045 7874 0a77 7269 7474 656e obal Ext.written │ │ │ │ +0005da70: 2062 7920 4176 7261 6d6f 7620 616e 6420 by Avramov and │ │ │ │ +0005da80: 4772 6179 736f 6e20 2862 7574 206e 6f74 Grayson (but not │ │ │ │ +0005da90: 6520 7468 6520 6469 6666 6572 656e 6365 e the difference │ │ │ │ +0005daa0: 2c20 6578 706c 6169 6e65 6420 6265 6c6f , explained belo │ │ │ │ +0005dab0: 7729 2e0a 0a49 6620 4c69 6674 203d 3e20 w)...If Lift => │ │ │ │ +0005dac0: 6661 6c73 6520 7468 6520 7265 7375 6c74 false the result │ │ │ │ +0005dad0: 2069 7320 7265 7475 726e 6564 206f 7665 is returned ove │ │ │ │ +0005dae0: 7220 616e 6420 6578 7465 6e73 696f 6e20 r and extension │ │ │ │ +0005daf0: 6f66 2052 6261 723b 2069 6620 4c69 6674 of Rbar; if Lift │ │ │ │ +0005db00: 203d 3e0a 7472 7565 2074 6865 2072 6573 =>.true the res │ │ │ │ +0005db10: 756c 7420 6973 2072 6574 7572 6e65 6420 ult is returned │ │ │ │ +0005db20: 6f76 6572 2061 6e64 2065 7874 656e 7369 over and extensi │ │ │ │ +0005db30: 6f6e 206f 6620 522e 0a0a 4966 2047 7261 on of R...If Gra │ │ │ │ +0005db40: 6469 6e67 203d 3e20 322c 2074 6865 2064 ding => 2, the d │ │ │ │ +0005db50: 6566 6175 6c74 2c20 7468 656e 2074 6865 efault, then the │ │ │ │ +0005db60: 2072 6573 756c 7420 6973 2062 6967 7261 result is bigra │ │ │ │ +0005db70: 6465 6420 2874 6869 7320 6973 206e 6563 ded (this is nec │ │ │ │ +0005db80: 6573 7361 7279 0a77 6865 6e20 4368 6563 essary.when Chec │ │ │ │ +0005db90: 6b3d 3e74 7275 650a 0a54 6865 2064 6566 k=>true..The def │ │ │ │ +0005dba0: 6175 6c74 2056 6172 6961 626c 6573 203d ault Variables = │ │ │ │ +0005dbb0: 3e20 7379 6d62 6f6c 2022 7322 2067 6976 > symbol "s" giv │ │ │ │ +0005dbc0: 6573 2074 6865 206e 6577 2076 6172 6961 es the new varia │ │ │ │ +0005dbd0: 626c 6573 2074 6865 206e 616d 6520 735f bles the name s_ │ │ │ │ +0005dbe0: 692c 0a69 3d30 2e2e 632d 312e 2028 6e6f i,.i=0..c-1. (no │ │ │ │ +0005dbf0: 7465 2074 6861 7420 7468 6520 6275 696c te that the buil │ │ │ │ +0005dc00: 7469 6e20 4578 7420 7573 6573 2058 5f31 tin Ext uses X_1 │ │ │ │ +0005dc10: 2e2e 585f 632e 0a0a 4f6e 2053 6f6d 6520 ..X_c...On Some │ │ │ │ +0005dc20: 6578 616d 706c 6573 206e 6577 4578 7420 examples newExt │ │ │ │ +0005dc30: 6973 2066 6173 7465 7220 7468 616e 2045 is faster than E │ │ │ │ +0005dc40: 7874 3b20 6f6e 206f 7468 6572 7320 6974 xt; on others it │ │ │ │ +0005dc50: 2773 2073 6c6f 7765 722e 0a0a 4120 7369 's slower...A si │ │ │ │ +0005dc60: 6d70 6c65 2065 7861 6d70 6c65 3a20 6966 mple example: if │ │ │ │ +0005dc70: 2052 203d 206b 5b78 5f31 2e2e 785f 6e5d R = k[x_1..x_n] │ │ │ │ +0005dc80: 2061 6e64 2049 2069 7320 636f 6e74 6169 and I is contai │ │ │ │ +0005dc90: 6e65 6420 696e 2074 6865 2063 7562 6520 ned in the cube │ │ │ │ +0005dca0: 6f66 2074 6865 0a6d 6178 696d 616c 2069 of the.maximal i │ │ │ │ +0005dcb0: 6465 616c 2c20 7468 656e 2045 7874 286b deal, then Ext(k │ │ │ │ +0005dcc0: 2c6b 2920 6973 2061 2066 7265 6520 532f ,k) is a free S/ │ │ │ │ +0005dcd0: 2878 5f31 2e2e 785f 6e29 203d 206b 5b73 (x_1..x_n) = k[s │ │ │ │ +0005dce0: 5f30 2e2e 735f 2863 2d31 295d 2d20 6d6f _0..s_(c-1)]- mo │ │ │ │ +0005dcf0: 6475 6c65 0a77 6974 6820 6269 6e6f 6d69 dule.with binomi │ │ │ │ +0005dd00: 616c 286e 2c69 2920 6765 6e65 7261 746f al(n,i) generato │ │ │ │ +0005dd10: 7273 2069 6e20 6465 6772 6565 2069 0a0a rs in degree i.. │ │ │ │ +0005dd20: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005dd30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dd40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dd50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dd70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a ---------+.|i1 : │ │ │ │ -0005dd80: 206e 203d 2033 3b63 3d32 3b20 2020 2020 n = 3;c=2; │ │ │ │ +0005dd60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005dd70: 7c69 3120 3a20 6e20 3d20 333b 633d 323b |i1 : n = 3;c=2; │ │ │ │ +0005dd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005dd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005dda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ddb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ddc0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005ddb0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005ddc0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005ddd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ddf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005de10: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a ---------+.|i3 : │ │ │ │ -0005de20: 2052 203d 205a 5a2f 3130 315b 785f 302e R = ZZ/101[x_0. │ │ │ │ -0005de30: 2e78 5f28 6e2d 3129 5d20 2020 2020 2020 .x_(n-1)] │ │ │ │ +0005de00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005de10: 7c69 3320 3a20 5220 3d20 5a5a 2f31 3031 |i3 : R = ZZ/101 │ │ │ │ +0005de20: 5b78 5f30 2e2e 785f 286e 2d31 295d 2020 [x_0..x_(n-1)] │ │ │ │ +0005de30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005de40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005de50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005de60: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005de50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005de60: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005de70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005de80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005de90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005dea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005deb0: 2020 2020 2020 2020 207c 0a7c 6f33 203d |.|o3 = │ │ │ │ -0005dec0: 2052 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0005dea0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005deb0: 7c6f 3320 3d20 5220 2020 2020 2020 2020 |o3 = R │ │ │ │ +0005dec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ded0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005dee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005def0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df00: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005def0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005df00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005df10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005df20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005df30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df50: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -0005df60: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +0005df40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005df50: 7c6f 3320 3a20 506f 6c79 6e6f 6d69 616c |o3 : Polynomial │ │ │ │ +0005df60: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0005df70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005df80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005df90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005dfa0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005df90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005dfa0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005dfb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dfc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005dfd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005dff0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a ---------+.|i4 : │ │ │ │ -0005e000: 2052 6261 7220 3d20 522f 2869 6465 616c Rbar = R/(ideal │ │ │ │ -0005e010: 2061 7070 6c79 2863 2c20 692d 3e20 525f apply(c, i-> R_ │ │ │ │ -0005e020: 695e 3329 2920 2020 2020 2020 2020 2020 i^3)) │ │ │ │ -0005e030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e040: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005dfe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005dff0: 7c69 3420 3a20 5262 6172 203d 2052 2f28 |i4 : Rbar = R/( │ │ │ │ +0005e000: 6964 6561 6c20 6170 706c 7928 632c 2069 ideal apply(c, i │ │ │ │ +0005e010: 2d3e 2052 5f69 5e33 2929 2020 2020 2020 -> R_i^3)) │ │ │ │ +0005e020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0005e030: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e040: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e090: 2020 2020 2020 2020 207c 0a7c 6f34 203d |.|o4 = │ │ │ │ -0005e0a0: 2052 6261 7220 2020 2020 2020 2020 2020 Rbar │ │ │ │ +0005e080: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e090: 7c6f 3420 3d20 5262 6172 2020 2020 2020 |o4 = Rbar │ │ │ │ +0005e0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e0e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e0d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e0e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e130: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ -0005e140: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +0005e120: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e130: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269 |o4 : QuotientRi │ │ │ │ +0005e140: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 0005e150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e180: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005e170: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e180: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005e190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e1d0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -0005e1e0: 204d 6261 7220 3d20 4e62 6172 203d 2063 Mbar = Nbar = c │ │ │ │ -0005e1f0: 6f6b 6572 2076 6172 7320 5262 6172 2020 oker vars Rbar │ │ │ │ +0005e1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005e1d0: 7c69 3520 3a20 4d62 6172 203d 204e 6261 |i5 : Mbar = Nba │ │ │ │ +0005e1e0: 7220 3d20 636f 6b65 7220 7661 7273 2052 r = coker vars R │ │ │ │ +0005e1f0: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 0005e200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e220: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e210: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e220: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e270: 2020 2020 2020 2020 207c 0a7c 6f35 203d |.|o5 = │ │ │ │ -0005e280: 2063 6f6b 6572 6e65 6c20 7c20 785f 3020 cokernel | x_0 │ │ │ │ -0005e290: 785f 3120 785f 3220 7c20 2020 2020 2020 x_1 x_2 | │ │ │ │ +0005e260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e270: 7c6f 3520 3d20 636f 6b65 726e 656c 207c |o5 = cokernel | │ │ │ │ +0005e280: 2078 5f30 2078 5f31 2078 5f32 207c 2020 x_0 x_1 x_2 | │ │ │ │ +0005e290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e2c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e2b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e2c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e310: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e300: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e310: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e330: 2020 2020 2020 2020 2020 2020 2020 3120 1 │ │ │ │ +0005e330: 2020 2031 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 0005e340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e360: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ -0005e370: 2052 6261 722d 6d6f 6475 6c65 2c20 7175 Rbar-module, qu │ │ │ │ -0005e380: 6f74 6965 6e74 206f 6620 5262 6172 2020 otient of Rbar │ │ │ │ +0005e350: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e360: 7c6f 3520 3a20 5262 6172 2d6d 6f64 756c |o5 : Rbar-modul │ │ │ │ +0005e370: 652c 2071 756f 7469 656e 7420 6f66 2052 e, quotient of R │ │ │ │ +0005e380: 6261 7220 2020 2020 2020 2020 2020 2020 bar │ │ │ │ 0005e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e3b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005e3a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e3b0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005e3c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e3d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005e3e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005e400: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a ---------+.|i6 : │ │ │ │ -0005e410: 2045 203d 206e 6577 4578 7428 4d62 6172 E = newExt(Mbar │ │ │ │ -0005e420: 2c4e 6261 7229 2020 2020 2020 2020 2020 ,Nbar) │ │ │ │ +0005e3f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005e400: 7c69 3620 3a20 4520 3d20 6e65 7745 7874 |i6 : E = newExt │ │ │ │ +0005e410: 284d 6261 722c 4e62 6172 2920 2020 2020 (Mbar,Nbar) │ │ │ │ +0005e420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e450: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005e440: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e450: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0005e460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005e4a0: 2020 2020 2020 2020 207c 0a7c 6f36 203d |.|o6 = │ │ │ │ -0005e4b0: 2063 6f6b 6572 6e65 6c20 7b30 2c20 307d cokernel {0, 0} │ │ │ │ -0005e4c0: 2020 207c 2078 5f32 2078 5f31 2078 5f30 | x_2 x_1 x_0 │ │ │ │ -0005e4d0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e4e0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e4f0: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e500: 2020 2020 2020 2020 2020 7b2d 322c 202d {-2, - │ │ │ │ -0005e510: 327d 207c 2030 2020 2030 2020 2030 2020 2} | 0 0 0 │ │ │ │ -0005e520: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e530: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e540: 2030 2020 2078 5f32 207c 0a7c 2020 2020 0 x_2 |.| │ │ │ │ -0005e550: 2020 2020 2020 2020 2020 7b2d 322c 202d {-2, - │ │ │ │ -0005e560: 327d 207c 2030 2020 2030 2020 2030 2020 2} | 0 0 0 │ │ │ │ -0005e570: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e580: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e590: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e5a0: 2020 2020 2020 2020 2020 7b2d 322c 202d {-2, - │ │ │ │ -0005e5b0: 327d 207c 2030 2020 2030 2020 2030 2020 2} | 0 0 0 │ │ │ │ -0005e5c0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e5d0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e5e0: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e5f0: 2020 2020 2020 2020 2020 7b2d 312c 202d {-1, - │ │ │ │ -0005e600: 317d 207c 2030 2020 2030 2020 2030 2020 1} | 0 0 0 │ │ │ │ -0005e610: 2078 5f32 2078 5f31 2078 5f30 2030 2020 x_2 x_1 x_0 0 │ │ │ │ -0005e620: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e630: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e640: 2020 2020 2020 2020 2020 7b2d 312c 202d {-1, - │ │ │ │ -0005e650: 317d 207c 2030 2020 2030 2020 2030 2020 1} | 0 0 0 │ │ │ │ -0005e660: 2030 2020 2030 2020 2030 2020 2078 5f32 0 0 0 x_2 │ │ │ │ -0005e670: 2078 5f31 2078 5f30 2030 2020 2030 2020 x_1 x_0 0 0 │ │ │ │ -0005e680: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ -0005e690: 2020 2020 2020 2020 2020 7b2d 312c 202d {-1, - │ │ │ │ -0005e6a0: 317d 207c 2030 2020 2030 2020 2030 2020 1} | 0 0 0 │ │ │ │ -0005e6b0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e6c0: 2030 2020 2030 2020 2078 5f32 2078 5f31 0 0 x_2 x_1 │ │ │ │ -0005e6d0: 2078 5f30 2030 2020 207c 0a7c 2020 2020 x_0 0 |.| │ │ │ │ -0005e6e0: 2020 2020 2020 2020 2020 7b2d 332c 202d {-3, - │ │ │ │ -0005e6f0: 337d 207c 2030 2020 2030 2020 2030 2020 3} | 0 0 0 │ │ │ │ -0005e700: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e710: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -0005e720: 2030 2020 2030 2020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +0005e490: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005e4a0: 7c6f 3620 3d20 636f 6b65 726e 656c 207b |o6 = cokernel { │ │ │ │ +0005e4b0: 302c 2030 7d20 2020 7c20 785f 3220 785f 0, 0} | x_2 x_ │ │ │ │ +0005e4c0: 3120 785f 3020 3020 2020 3020 2020 3020 1 x_0 0 0 0 │ │ │ │ +0005e4d0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e4e0: 2020 3020 2020 3020 2020 3020 2020 7c0a 0 0 0 |. │ │ │ │ +0005e4f0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e500: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020 -2, -2} | 0 0 │ │ │ │ +0005e510: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e520: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e530: 2020 3020 2020 3020 2020 785f 3220 7c0a 0 0 x_2 |. │ │ │ │ +0005e540: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e550: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020 -2, -2} | 0 0 │ │ │ │ +0005e560: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e570: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e580: 2020 3020 2020 3020 2020 3020 2020 7c0a 0 0 0 |. │ │ │ │ +0005e590: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e5a0: 2d32 2c20 2d32 7d20 7c20 3020 2020 3020 -2, -2} | 0 0 │ │ │ │ +0005e5b0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e5c0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +0005e5d0: 2020 3020 2020 3020 2020 3020 2020 7c0a 0 0 0 |. │ │ │ │ +0005e5e0: 7c20 2020 2020 2020 2020 2020 2020 207b | { │ │ │ │ +0005e5f0: 2d31 2c20 2d31 7d20 7c20 3020 2020 3020 -1, -1} | 0 0 │ │ │ │ +0005e600: 2020 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2020 2020 2020 2020 │ │ │ │ -0005f320: 2020 3020 2020 3120 2020 2020 2020 2020 0 1 │ │ │ │ +0005f2f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005f300: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0005f310: 2020 2020 2020 2030 2020 2031 2020 2020 0 1 │ │ │ │ +0005f320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f350: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0005f350: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 0005f360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f3a0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 416e 2065 ---------+..An e │ │ │ │ -0005f3b0: 7861 6d70 6c65 2077 6865 7265 2074 6865 xample where the │ │ │ │ -0005f3c0: 2062 7569 6c74 2d69 6e20 676c 6f62 616c built-in global │ │ │ │ -0005f3d0: 2045 7874 2069 7320 6861 7264 2074 6f20 Ext is hard to │ │ │ │ -0005f3e0: 636f 6d70 6172 6520 6469 7265 6374 6c79 compare directly │ │ │ │ -0005f3f0: 2077 6974 6820 6f75 720a 6d65 7468 6f64 with our.method │ │ │ │ -0005f400: 206f 6620 636f 6d70 7574 6174 696f 6e3a of computation: │ │ │ │ -0005f410: 2049 202a 6775 6573 732a 2074 6861 7420 I *guess* that │ │ │ │ -0005f420: 7468 6520 7369 676e 2063 686f 6963 6573 the sign choices │ │ │ │ -0005f430: 2069 6e20 7468 6520 6275 696c 742d 696e in the built-in │ │ │ │ -0005f440: 2061 6d6f 756e 740a 6573 7365 6e74 6961 amount.essentia │ │ │ │ -0005f450: 6c6c 7920 746f 2061 2063 6861 6e67 6520 lly to a change │ │ │ │ -0005f460: 6f66 2076 6172 6961 626c 6520 696e 2074 of variable in t │ │ │ │ -0005f470: 6865 206e 6577 2076 6172 6961 626c 6573 he new variables │ │ │ │ -0005f480: 2c20 616e 6420 7370 6f69 6c20 616e 2065 , and spoil an e │ │ │ │ -0005f490: 6173 790a 636f 6d70 6172 6973 6f6e 2e20 asy.comparison. │ │ │ │ -0005f4a0: 4275 7420 666f 7220 6578 616d 706c 6520 But for example │ │ │ │ -0005f4b0: 7468 6520 6269 2d67 7261 6465 6420 4265 the bi-graded Be │ │ │ │ -0005f4c0: 7474 6920 6e75 6d62 6572 7320 6172 6520 tti numbers are │ │ │ │ -0005f4d0: 6571 7561 6c2e 2074 6869 7320 7365 656d equal. this seem │ │ │ │ -0005f4e0: 730a 746f 2073 7461 7274 2077 6974 6820 s.to start with │ │ │ │ -0005f4f0: 633d 332e 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d c=3...+--------- │ │ │ │ +0005f390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +0005f3a0: 0a41 6e20 6578 616d 706c 6520 7768 6572 .An example wher │ │ │ │ +0005f3b0: 6520 7468 6520 6275 696c 742d 696e 2067 e the built-in g │ │ │ │ +0005f3c0: 6c6f 6261 6c20 4578 7420 6973 2068 6172 lobal Ext is har │ │ │ │ +0005f3d0: 6420 746f 2063 6f6d 7061 7265 2064 6972 d to compare dir │ │ │ │ +0005f3e0: 6563 746c 7920 7769 7468 206f 7572 0a6d ectly with our.m │ │ │ │ +0005f3f0: 6574 686f 6420 6f66 2063 6f6d 7075 7461 ethod of computa │ │ │ │ +0005f400: 7469 6f6e 3a20 4920 2a67 7565 7373 2a20 tion: I *guess* │ │ │ │ +0005f410: 7468 6174 2074 6865 2073 6967 6e20 6368 that the sign ch │ │ │ │ +0005f420: 6f69 6365 7320 696e 2074 6865 2062 7569 oices in the bui │ │ │ │ +0005f430: 6c74 2d69 6e20 616d 6f75 6e74 0a65 7373 lt-in amount.ess │ │ │ │ +0005f440: 656e 7469 616c 6c79 2074 6f20 6120 6368 entially to a ch │ │ │ │ +0005f450: 616e 6765 206f 6620 7661 7269 6162 6c65 ange of variable │ │ │ │ +0005f460: 2069 6e20 7468 6520 6e65 7720 7661 7269 in the new vari │ │ │ │ +0005f470: 6162 6c65 732c 2061 6e64 2073 706f 696c ables, and spoil │ │ │ │ +0005f480: 2061 6e20 6561 7379 0a63 6f6d 7061 7269 an easy.compari │ │ │ │ +0005f490: 736f 6e2e 2042 7574 2066 6f72 2065 7861 son. But for exa │ │ │ │ +0005f4a0: 6d70 6c65 2074 6865 2062 692d 6772 6164 mple the bi-grad │ │ │ │ +0005f4b0: 6564 2042 6574 7469 206e 756d 6265 7273 ed Betti numbers │ │ │ │ +0005f4c0: 2061 7265 2065 7175 616c 2e20 7468 6973 are equal. this │ │ │ │ +0005f4d0: 2073 6565 6d73 0a74 6f20 7374 6172 7420 seems.to start │ │ │ │ +0005f4e0: 7769 7468 2063 3d33 2e0a 0a2b 2d2d 2d2d with c=3...+---- │ │ │ │ +0005f4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f540: 2d2d 2d2d 2b0a 7c69 3920 3a20 7365 7452 ----+.|i9 : setR │ │ │ │ -0005f550: 616e 646f 6d53 6565 6420 3020 2020 2020 andomSeed 0 │ │ │ │ +0005f530: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 203a ---------+.|i9 : │ │ │ │ +0005f540: 2073 6574 5261 6e64 6f6d 5365 6564 2030 setRandomSeed 0 │ │ │ │ +0005f550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f590: 2020 2020 7c0a 7c20 2d2d 2073 6574 7469 |.| -- setti │ │ │ │ -0005f5a0: 6e67 2072 616e 646f 6d20 7365 6564 2074 ng random seed t │ │ │ │ -0005f5b0: 6f20 3020 2020 2020 2020 2020 2020 2020 o 0 │ │ │ │ +0005f580: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +0005f590: 7365 7474 696e 6720 7261 6e64 6f6d 2073 setting random s │ │ │ │ +0005f5a0: 6565 6420 746f 2030 2020 2020 2020 2020 eed to 0 │ │ │ │ +0005f5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f5e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f5d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f630: 2020 2020 7c0a 7c6f 3920 3d20 3020 2020 |.|o9 = 0 │ │ │ │ +0005f620: 2020 2020 2020 2020 207c 0a7c 6f39 203d |.|o9 = │ │ │ │ +0005f630: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0005f640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f680: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005f670: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f6a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f6b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f6d0: 2d2d 2d2d 2b0a 7c69 3130 203a 206e 203d ----+.|i10 : n = │ │ │ │ -0005f6e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0005f6c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3020 ---------+.|i10 │ │ │ │ +0005f6d0: 3a20 6e20 3d20 3320 2020 2020 2020 2020 : n = 3 │ │ │ │ +0005f6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f720: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f710: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f770: 2020 2020 7c0a 7c6f 3130 203d 2033 2020 |.|o10 = 3 │ │ │ │ +0005f760: 2020 2020 2020 2020 207c 0a7c 6f31 3020 |.|o10 │ │ │ │ +0005f770: 3d20 3320 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ 0005f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f7c0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005f7b0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f7d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f7e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f7f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f810: 2d2d 2d2d 2b0a 7c69 3131 203a 2063 203d ----+.|i11 : c = │ │ │ │ -0005f820: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0005f800: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3120 ---------+.|i11 │ │ │ │ +0005f810: 3a20 6320 3d20 3320 2020 2020 2020 2020 : c = 3 │ │ │ │ +0005f820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f860: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f850: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f8a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f8b0: 2020 2020 7c0a 7c6f 3131 203d 2033 2020 |.|o11 = 3 │ │ │ │ +0005f8a0: 2020 2020 2020 2020 207c 0a7c 6f31 3120 |.|o11 │ │ │ │ +0005f8b0: 3d20 3320 2020 2020 2020 2020 2020 2020 = 3 │ │ │ │ 0005f8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f8e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f8f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f900: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005f8f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005f900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005f930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005f950: 2d2d 2d2d 2b0a 7c69 3132 203a 206b 6b20 ----+.|i12 : kk │ │ │ │ -0005f960: 3d20 5a5a 2f31 3031 2020 2020 2020 2020 = ZZ/101 │ │ │ │ +0005f940: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3220 ---------+.|i12 │ │ │ │ +0005f950: 3a20 6b6b 203d 205a 5a2f 3130 3120 2020 : kk = ZZ/101 │ │ │ │ +0005f960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f9a0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005f990: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005f9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005f9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005f9f0: 2020 2020 7c0a 7c6f 3132 203d 206b 6b20 |.|o12 = kk │ │ │ │ +0005f9e0: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ +0005f9f0: 3d20 6b6b 2020 2020 2020 2020 2020 2020 = kk │ │ │ │ 0005fa00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa40: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fa30: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fa40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fa70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fa90: 2020 2020 7c0a 7c6f 3132 203a 2051 756f |.|o12 : Quo │ │ │ │ -0005faa0: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +0005fa80: 2020 2020 2020 2020 207c 0a7c 6f31 3220 |.|o12 │ │ │ │ +0005fa90: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +0005faa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fae0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005fad0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005fae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005faf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fb30: 2d2d 2d2d 2b0a 7c69 3133 203a 2052 203d ----+.|i13 : R = │ │ │ │ -0005fb40: 206b 6b5b 785f 302e 2e78 5f28 6e2d 3129 kk[x_0..x_(n-1) │ │ │ │ -0005fb50: 5d20 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +0005fb20: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3320 ---------+.|i13 │ │ │ │ +0005fb30: 3a20 5220 3d20 6b6b 5b78 5f30 2e2e 785f : R = kk[x_0..x_ │ │ │ │ +0005fb40: 286e 2d31 295d 2020 2020 2020 2020 2020 (n-1)] │ │ │ │ +0005fb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fb60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fb70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fb80: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fb70: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fb80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fbd0: 2020 2020 7c0a 7c6f 3133 203d 2052 2020 |.|o13 = R │ │ │ │ +0005fbc0: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0005fbd0: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0005fbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fc10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fc70: 2020 2020 7c0a 7c6f 3133 203a 2050 6f6c |.|o13 : Pol │ │ │ │ -0005fc80: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +0005fc60: 2020 2020 2020 2020 207c 0a7c 6f31 3320 |.|o13 │ │ │ │ +0005fc70: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +0005fc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fcc0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005fcb0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005fcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005fcf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005fd10: 2d2d 2d2d 2b0a 7c69 3134 203a 2049 203d ----+.|i14 : I = │ │ │ │ -0005fd20: 2069 6465 616c 2061 7070 6c79 2863 2c20 ideal apply(c, │ │ │ │ -0005fd30: 692d 3e52 5f69 5e32 2920 2020 2020 2020 i->R_i^2) │ │ │ │ +0005fd00: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3420 ---------+.|i14 │ │ │ │ +0005fd10: 3a20 4920 3d20 6964 6561 6c20 6170 706c : I = ideal appl │ │ │ │ +0005fd20: 7928 632c 2069 2d3e 525f 695e 3229 2020 y(c, i->R_i^2) │ │ │ │ +0005fd30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fd60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fd50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fd60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fd90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fdb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0005fdc0: 2020 2020 2032 2020 2032 2020 2032 2020 2 2 2 │ │ │ │ +0005fda0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fdb0: 2020 2020 2020 2020 2020 3220 2020 3220 2 2 │ │ │ │ +0005fdc0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0005fdd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fde0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe00: 2020 2020 7c0a 7c6f 3134 203d 2069 6465 |.|o14 = ide │ │ │ │ -0005fe10: 616c 2028 7820 2c20 7820 2c20 7820 2920 al (x , x , x ) │ │ │ │ +0005fdf0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +0005fe00: 3d20 6964 6561 6c20 2878 202c 2078 202c = ideal (x , x , │ │ │ │ +0005fe10: 2078 2029 2020 2020 2020 2020 2020 2020 x ) │ │ │ │ 0005fe20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fe30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -0005fe60: 2020 2020 2030 2020 2031 2020 2032 2020 0 1 2 │ │ │ │ +0005fe40: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fe50: 2020 2020 2020 2020 2020 3020 2020 3120 0 1 │ │ │ │ +0005fe60: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ 0005fe70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fe80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fe90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fea0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005fe90: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005fea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005feb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005fef0: 2020 2020 7c0a 7c6f 3134 203a 2049 6465 |.|o14 : Ide │ │ │ │ -0005ff00: 616c 206f 6620 5220 2020 2020 2020 2020 al of R │ │ │ │ +0005fee0: 2020 2020 2020 2020 207c 0a7c 6f31 3420 |.|o14 │ │ │ │ +0005fef0: 3a20 4964 6561 6c20 6f66 2052 2020 2020 : Ideal of R │ │ │ │ +0005ff00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ff10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ff20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ff30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ff40: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0005ff30: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +0005ff40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ff50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ff60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0005ff70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0005ff90: 2d2d 2d2d 2b0a 7c69 3135 203a 2066 6620 ----+.|i15 : ff │ │ │ │ -0005ffa0: 3d20 6765 6e73 2049 2020 2020 2020 2020 = gens I │ │ │ │ +0005ff80: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3520 ---------+.|i15 │ │ │ │ +0005ff90: 3a20 6666 203d 2067 656e 7320 4920 2020 : ff = gens I │ │ │ │ +0005ffa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ffb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005ffc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ffd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0005ffe0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0005ffd0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0005ffe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0005fff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060030: 2020 2020 7c0a 7c6f 3135 203d 207c 2078 |.|o15 = | x │ │ │ │ -00060040: 5f30 5e32 2078 5f31 5e32 2078 5f32 5e32 _0^2 x_1^2 x_2^2 │ │ │ │ -00060050: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00060020: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +00060030: 3d20 7c20 785f 305e 3220 785f 315e 3220 = | x_0^2 x_1^2 │ │ │ │ +00060040: 785f 325e 3220 7c20 2020 2020 2020 2020 x_2^2 | │ │ │ │ +00060050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060080: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060070: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000600a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000600b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000600c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000600d0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000600e0: 2020 2020 2031 2020 2020 2020 3320 2020 1 3 │ │ │ │ +000600c0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000600d0: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +000600e0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000600f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060120: 2020 2020 7c0a 7c6f 3135 203a 204d 6174 |.|o15 : Mat │ │ │ │ -00060130: 7269 7820 5220 203c 2d2d 2052 2020 2020 rix R <-- R │ │ │ │ +00060110: 2020 2020 2020 2020 207c 0a7c 6f31 3520 |.|o15 │ │ │ │ +00060120: 3a20 4d61 7472 6978 2052 2020 3c2d 2d20 : Matrix R <-- │ │ │ │ +00060130: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ 00060140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060170: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060160: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000601a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000601b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000601c0: 2d2d 2d2d 2b0a 7c69 3136 203a 2052 6261 ----+.|i16 : Rba │ │ │ │ -000601d0: 7220 3d20 522f 4920 2020 2020 2020 2020 r = R/I │ │ │ │ +000601b0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3620 ---------+.|i16 │ │ │ │ +000601c0: 3a20 5262 6172 203d 2052 2f49 2020 2020 : Rbar = R/I │ │ │ │ +000601d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000601e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000601f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060200: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060260: 2020 2020 7c0a 7c6f 3136 203d 2052 6261 |.|o16 = Rba │ │ │ │ -00060270: 7220 2020 2020 2020 2020 2020 2020 2020 r │ │ │ │ +00060250: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ +00060260: 3d20 5262 6172 2020 2020 2020 2020 2020 = Rbar │ │ │ │ +00060270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000602a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000602b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000602a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000602b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000602c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000602d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000602e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000602f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060300: 2020 2020 7c0a 7c6f 3136 203a 2051 756f |.|o16 : Quo │ │ │ │ -00060310: 7469 656e 7452 696e 6720 2020 2020 2020 tientRing │ │ │ │ +000602f0: 2020 2020 2020 2020 207c 0a7c 6f31 3620 |.|o16 │ │ │ │ +00060300: 3a20 5175 6f74 6965 6e74 5269 6e67 2020 : QuotientRing │ │ │ │ +00060310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060350: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060340: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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00060420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060440: 2020 2020 7c0a 7c6f 3137 203d 206d 6170 |.|o17 = map │ │ │ │ -00060450: 2028 5262 6172 2c20 522c 207b 7820 2c20 (Rbar, R, {x , │ │ │ │ -00060460: 7820 2c20 7820 7d29 2020 2020 2020 2020 x , x }) │ │ │ │ +00060430: 2020 2020 2020 2020 207c 0a7c 6f31 3720 |.|o17 │ │ │ │ +00060440: 3d20 6d61 7020 2852 6261 722c 2052 2c20 = map (Rbar, R, │ │ │ │ +00060450: 7b78 202c 2078 202c 2078 207d 2920 2020 {x , x , x }) │ │ │ │ +00060460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060490: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000604a0: 2020 2020 2020 2020 2020 2020 2030 2020 0 │ │ │ │ -000604b0: 2031 2020 2032 2020 2020 2020 2020 2020 1 2 │ │ │ │ +00060480: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000604a0: 2020 3020 2020 3120 2020 3220 2020 2020 0 1 2 │ │ │ │ +000604b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000604c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000604d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000604e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000604d0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000604e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000604f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060530: 2020 2020 7c0a 7c6f 3137 203a 2052 696e |.|o17 : Rin │ │ │ │ -00060540: 674d 6170 2052 6261 7220 3c2d 2d20 5220 gMap Rbar <-- R │ │ │ │ +00060520: 2020 2020 2020 2020 207c 0a7c 6f31 3720 |.|o17 │ │ │ │ +00060530: 3a20 5269 6e67 4d61 7020 5262 6172 203c : RingMap Rbar < │ │ │ │ +00060540: 2d2d 2052 2020 2020 2020 2020 2020 2020 -- R │ │ │ │ 00060550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060580: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060570: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000605a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000605b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000605c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000605d0: 2d2d 2d2d 2b0a 7c69 3138 203a 204b 203d ----+.|i18 : K = │ │ │ │ -000605e0: 2063 6f6b 6572 2076 6172 7320 5262 6172 coker vars Rbar │ │ │ │ +000605c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3820 ---------+.|i18 │ │ │ │ +000605d0: 3a20 4b20 3d20 636f 6b65 7220 7661 7273 : K = coker vars │ │ │ │ +000605e0: 2052 6261 7220 2020 2020 2020 2020 2020 Rbar │ │ │ │ 000605f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060620: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060610: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060670: 2020 2020 7c0a 7c6f 3138 203d 2063 6f6b |.|o18 = cok │ │ │ │ -00060680: 6572 6e65 6c20 7c20 785f 3020 785f 3120 ernel | x_0 x_1 │ │ │ │ -00060690: 785f 3220 7c20 2020 2020 2020 2020 2020 x_2 | │ │ │ │ +00060660: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +00060670: 3d20 636f 6b65 726e 656c 207c 2078 5f30 = cokernel | x_0 │ │ │ │ +00060680: 2078 5f31 2078 5f32 207c 2020 2020 2020 x_1 x_2 | │ │ │ │ +00060690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000606b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000606c0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000606b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000606c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000606f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060710: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00060720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060730: 2020 2020 2020 2020 2020 3120 2020 2020 1 │ │ │ │ +00060700: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00060720: 2020 2020 2020 2020 2020 2020 2020 2031 1 │ │ │ │ +00060730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060760: 2020 2020 7c0a 7c6f 3138 203a 2052 6261 |.|o18 : Rba │ │ │ │ -00060770: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ -00060780: 6e74 206f 6620 5262 6172 2020 2020 2020 nt of Rbar │ │ │ │ +00060750: 2020 2020 2020 2020 207c 0a7c 6f31 3820 |.|o18 │ │ │ │ +00060760: 3a20 5262 6172 2d6d 6f64 756c 652c 2071 : Rbar-module, q │ │ │ │ +00060770: 756f 7469 656e 7420 6f66 2052 6261 7220 uotient of Rbar │ │ │ │ +00060780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000607a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000607b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000607a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000607b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000607c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000607d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000607e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000607f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060800: 2d2d 2d2d 2b0a 7c69 3139 203a 204d 6261 ----+.|i19 : Mba │ │ │ │ -00060810: 7220 3d20 7072 756e 6520 636f 6b65 7220 r = prune coker │ │ │ │ -00060820: 7261 6e64 6f6d 2852 6261 725e 322c 2052 random(Rbar^2, R │ │ │ │ -00060830: 6261 725e 7b2d 322c 2d32 7d29 2020 2020 bar^{-2,-2}) │ │ │ │ -00060840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060850: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +000607f0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 3920 ---------+.|i19 │ │ │ │ +00060800: 3a20 4d62 6172 203d 2070 7275 6e65 2063 : Mbar = prune c │ │ │ │ +00060810: 6f6b 6572 2072 616e 646f 6d28 5262 6172 oker random(Rbar │ │ │ │ +00060820: 5e32 2c20 5262 6172 5e7b 2d32 2c2d 327d ^2, Rbar^{-2,-2} │ │ │ │ +00060830: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00060840: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000608a0: 2020 2020 7c0a 7c6f 3139 203d 2063 6f6b |.|o19 = cok │ │ │ │ -000608b0: 6572 6e65 6c20 7c20 785f 3078 5f31 2b31 ernel | x_0x_1+1 │ │ │ │ -000608c0: 3578 5f30 785f 322b 3338 785f 3178 5f32 5x_0x_2+38x_1x_2 │ │ │ │ -000608d0: 2034 3578 5f30 785f 322b 3239 785f 3178 45x_0x_2+29x_1x │ │ │ │ -000608e0: 5f32 2020 2020 2020 2020 7c20 2020 2020 _2 | │ │ │ │ -000608f0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00060900: 2020 2020 2020 7c20 3335 785f 3078 5f32 | 35x_0x_2 │ │ │ │ -00060910: 2d33 3078 5f31 785f 3220 2020 2020 2020 -30x_1x_2 │ │ │ │ -00060920: 2078 5f30 785f 312d 3130 785f 3078 5f32 x_0x_1-10x_0x_2 │ │ │ │ -00060930: 2d32 3278 5f31 785f 3220 7c20 2020 2020 -22x_1x_2 | │ │ │ │ -00060940: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060890: 2020 2020 2020 2020 207c 0a7c 6f31 3920 |.|o19 │ │ │ │ +000608a0: 3d20 636f 6b65 726e 656c 207c 2078 5f30 = cokernel | x_0 │ │ │ │ +000608b0: 785f 312b 3135 785f 3078 5f32 2b33 3878 x_1+15x_0x_2+38x │ │ │ │ +000608c0: 5f31 785f 3220 3435 785f 3078 5f32 2b32 _1x_2 45x_0x_2+2 │ │ │ │ +000608d0: 3978 5f31 785f 3220 2020 2020 2020 207c 9x_1x_2 | │ │ │ │ +000608e0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +000608f0: 2020 2020 2020 2020 2020 207c 2033 3578 | 35x │ │ │ │ +00060900: 5f30 785f 322d 3330 785f 3178 5f32 2020 _0x_2-30x_1x_2 │ │ │ │ +00060910: 2020 2020 2020 785f 3078 5f31 2d31 3078 x_0x_1-10x │ │ │ │ +00060920: 5f30 785f 322d 3232 785f 3178 5f32 207c _0x_2-22x_1x_2 | │ │ │ │ +00060930: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060990: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -000609a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000609b0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ +00060980: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000609a0: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +000609b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000609c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000609d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000609e0: 2020 2020 7c0a 7c6f 3139 203a 2052 6261 |.|o19 : Rba │ │ │ │ -000609f0: 722d 6d6f 6475 6c65 2c20 7175 6f74 6965 r-module, quotie │ │ │ │ -00060a00: 6e74 206f 6620 5262 6172 2020 2020 2020 nt of Rbar │ │ │ │ +000609d0: 2020 2020 2020 2020 207c 0a7c 6f31 3920 |.|o19 │ │ │ │ +000609e0: 3a20 5262 6172 2d6d 6f64 756c 652c 2071 : Rbar-module, q │ │ │ │ +000609f0: 756f 7469 656e 7420 6f66 2052 6261 7220 uotient of Rbar │ │ │ │ +00060a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00060a20: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00060a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060a80: 2d2d 2d2d 2b0a 7c69 3230 203a 2045 5320 ----+.|i20 : ES │ │ │ │ -00060a90: 3d20 6e65 7745 7874 284d 6261 722c 4b2c = newExt(Mbar,K, │ │ │ │ -00060aa0: 4c69 6674 203d 3e20 7472 7565 2920 2020 Lift => true) │ │ │ │ +00060a70: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3020 ---------+.|i20 │ │ │ │ +00060a80: 3a20 4553 203d 206e 6577 4578 7428 4d62 : ES = newExt(Mb │ │ │ │ +00060a90: 6172 2c4b 2c4c 6966 7420 3d3e 2074 7275 ar,K,Lift => tru │ │ │ │ +00060aa0: 6529 2020 2020 2020 2020 2020 2020 2020 e) │ │ │ │ 00060ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060ad0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060ac0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060b20: 2020 2020 7c0a 7c6f 3230 203d 2063 6f6b |.|o20 = cok │ │ │ │ -00060b30: 6572 6e65 6c20 7b30 2c20 307d 2020 207c ernel {0, 0} | │ │ │ │ -00060b40: 2078 5f32 2078 5f31 2078 5f30 2030 2020 x_2 x_1 x_0 0 │ │ │ │ -00060b50: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060b60: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060b70: 2073 5f32 7c0a 7c20 2020 2020 2020 2020 s_2|.| │ │ │ │ -00060b80: 2020 2020 2020 7b30 2c20 307d 2020 207c {0, 0} | │ │ │ │ -00060b90: 2030 2020 2030 2020 2030 2020 2078 5f32 0 0 0 x_2 │ │ │ │ -00060ba0: 2078 5f31 2078 5f30 2030 2020 2030 2020 x_1 x_0 0 0 │ │ │ │ -00060bb0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060bc0: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060bd0: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060be0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060bf0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c00: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c10: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060c20: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060c30: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c40: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c50: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c60: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060c70: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060c80: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060c90: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060ca0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060cb0: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060cc0: 2020 2020 2020 7b2d 322c 202d 337d 207c {-2, -3} | │ │ │ │ -00060cd0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060ce0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060cf0: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d00: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060d10: 2020 2020 2020 7b2d 312c 202d 327d 207c {-1, -2} | │ │ │ │ -00060d20: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d30: 2030 2020 2030 2020 2078 5f32 2078 5f31 0 0 x_2 x_1 │ │ │ │ -00060d40: 2078 5f30 2030 2020 2030 2020 2030 2020 x_0 0 0 0 │ │ │ │ -00060d50: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ -00060d60: 2020 2020 2020 7b2d 312c 202d 327d 207c {-1, -2} | │ │ │ │ -00060d70: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d80: 2030 2020 2030 2020 2030 2020 2030 2020 0 0 0 0 │ │ │ │ -00060d90: 2030 2020 2078 5f32 2078 5f31 2078 5f30 0 x_2 x_1 x_0 │ │ │ │ -00060da0: 2030 2020 7c0a 7c20 2020 2020 2020 2020 0 |.| │ │ │ │ +00060b10: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ +00060b20: 3d20 636f 6b65 726e 656c 207b 302c 2030 = cokernel {0, 0 │ │ │ │ +00060b30: 7d20 2020 7c20 785f 3220 785f 3120 785f } | x_2 x_1 x_ │ │ │ │ +00060b40: 3020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 0 │ │ │ │ +00060b50: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060b60: 2020 3020 2020 735f 327c 0a7c 2020 2020 0 s_2|.| │ │ │ │ +00060b70: 2020 2020 2020 2020 2020 207b 302c 2030 {0, 0 │ │ │ │ +00060b80: 7d20 2020 7c20 3020 2020 3020 2020 3020 } | 0 0 0 │ │ │ │ +00060b90: 2020 785f 3220 785f 3120 785f 3020 3020 x_2 x_1 x_0 0 │ │ │ │ +00060ba0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060bb0: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060bc0: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060bd0: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060be0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060bf0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c00: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060c10: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060c20: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060c30: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c40: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c50: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060c60: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060c70: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060c80: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060c90: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060ca0: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060cb0: 2020 2020 2020 2020 2020 207b 2d32 2c20 {-2, │ │ │ │ +00060cc0: 2d33 7d20 7c20 3020 2020 3020 2020 3020 -3} | 0 0 0 │ │ │ │ +00060cd0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060ce0: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060cf0: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060d00: 2020 2020 2020 2020 2020 207b 2d31 2c20 {-1, │ │ │ │ +00060d10: 2d32 7d20 7c20 3020 2020 3020 2020 3020 -2} | 0 0 0 │ │ │ │ +00060d20: 2020 3020 2020 3020 2020 3020 2020 785f 0 0 0 x_ │ │ │ │ +00060d30: 3220 785f 3120 785f 3020 3020 2020 3020 2 x_1 x_0 0 0 │ │ │ │ +00060d40: 2020 3020 2020 3020 207c 0a7c 2020 2020 0 0 |.| │ │ │ │ +00060d50: 2020 2020 2020 2020 2020 207b 2d31 2c20 {-1, │ │ │ │ +00060d60: 2d32 7d20 7c20 3020 2020 3020 2020 3020 -2} | 0 0 0 │ │ │ │ +00060d70: 2020 3020 2020 3020 2020 3020 2020 3020 0 0 0 0 │ │ │ │ +00060d80: 2020 3020 2020 3020 2020 785f 3220 785f 0 0 x_2 x_ │ │ │ │ +00060d90: 3120 785f 3020 3020 207c 0a7c 2020 2020 1 x_0 0 |.| │ │ │ │ +00060da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060df0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00060de0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00060e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060e30: 2020 2020 2020 2020 3820 2020 2020 2020 8 │ │ │ │ -00060e40: 2020 2020 7c0a 7c6f 3230 203a 206b 6b5b |.|o20 : kk[ │ │ │ │ -00060e50: 7320 2e2e 7320 2c20 7820 2e2e 7820 5d2d s ..s , x ..x ]- │ │ │ │ -00060e60: 6d6f 6475 6c65 2c20 7175 6f74 6965 6e74 module, quotient │ │ │ │ -00060e70: 206f 6620 286b 6b5b 7320 2e2e 7320 2c20 of (kk[s ..s , │ │ │ │ -00060e80: 7820 2e2e 7820 5d29 2020 2020 2020 2020 x ..x ]) │ │ │ │ -00060e90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ -00060ea0: 2030 2020 2032 2020 2030 2020 2032 2020 0 2 0 2 │ │ │ │ -00060eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00060ec0: 2020 2020 2020 2020 2030 2020 2032 2020 0 2 │ │ │ │ -00060ed0: 2030 2020 2032 2020 2020 2020 2020 2020 0 2 │ │ │ │ -00060ee0: 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d |.|--------- │ │ │ │ +00060e20: 2020 2020 2020 2020 2020 2020 2038 2020 8 │ │ │ │ +00060e30: 2020 2020 2020 2020 207c 0a7c 6f32 3020 |.|o20 │ │ │ │ +00060e40: 3a20 6b6b 5b73 202e 2e73 202c 2078 202e : kk[s ..s , x . │ │ │ │ +00060e50: 2e78 205d 2d6d 6f64 756c 652c 2071 756f .x ]-module, quo │ │ │ │ +00060e60: 7469 656e 7420 6f66 2028 6b6b 5b73 202e tient of (kk[s . │ │ │ │ +00060e70: 2e73 202c 2078 202e 2e78 205d 2920 2020 .s , x ..x ]) │ │ │ │ +00060e80: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00060e90: 2020 2020 2020 3020 2020 3220 2020 3020 0 2 0 │ │ │ │ +00060ea0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00060eb0: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ +00060ec0: 2020 3220 2020 3020 2020 3220 2020 2020 2 0 2 │ │ │ │ +00060ed0: 2020 2020 2020 2020 207c 0a7c 2d2d 2d2d |.|---- │ │ │ │ +00060ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00060f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00060f30: 2d2d 2d2d 7c0a 7c73 5f31 2073 5f30 2030 ----|.|s_1 s_0 0 │ │ │ │ -00060f40: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060f50: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060f60: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060f70: 2020 2030 2020 2030 2020 2020 2020 2020 0 0 │ │ │ │ -00060f80: 2020 2020 7c0a 7c30 2020 2030 2020 2073 |.|0 0 s │ │ │ │ -00060f90: 5f32 2073 5f31 2073 5f30 2030 2020 2030 _2 s_1 s_0 0 0 │ │ │ │ -00060fa0: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060fb0: 2020 2030 2020 2030 2020 2030 2020 2030 0 0 0 0 │ │ │ │ -00060fc0: 2020 2030 2020 2030 2020 2020 2020 2020 0 0 │ │ 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2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061a70: 2d2d 2d2d 2b0a 7c69 3231 203a 2053 203d ----+.|i21 : S = │ │ │ │ -00061a80: 2072 696e 6720 4553 2020 2020 2020 2020 ring ES │ │ │ │ +00061a60: 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 3120 ---------+.|i21 │ │ │ │ +00061a70: 3a20 5320 3d20 7269 6e67 2045 5320 2020 : S = ring ES │ │ │ │ +00061a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ac0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00061ab0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00061ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b10: 2020 2020 7c0a 7c6f 3231 203d 2053 2020 |.|o21 = S │ │ │ │ +00061b00: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +00061b10: 3d20 5320 2020 2020 2020 2020 2020 2020 = S │ │ │ │ 00061b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061b60: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00061b50: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00061b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061bb0: 2020 2020 7c0a 7c6f 3231 203a 2050 6f6c |.|o21 : Pol │ │ │ │ -00061bc0: 796e 6f6d 6961 6c52 696e 6720 2020 2020 ynomialRing │ │ │ │ +00061ba0: 2020 2020 2020 2020 207c 0a7c 6f32 3120 |.|o21 │ │ │ │ +00061bb0: 3a20 506f 6c79 6e6f 6d69 616c 5269 6e67 : PolynomialRing │ │ │ │ +00061bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061c00: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00061bf0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00061c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061c50: 2d2d 2d2d 2b0a 0a63 6f6d 7061 7265 2077 ----+..compare w │ │ │ │ -00061c60: 6974 6820 7468 6520 6275 696c 742d 696e ith the built-in │ │ │ │ -00061c70: 2045 7874 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d Ext..+--------- │ │ │ │ +00061c40: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 636f 6d70 ---------+..comp │ │ │ │ +00061c50: 6172 6520 7769 7468 2074 6865 2062 7569 are with the bui │ │ │ │ +00061c60: 6c74 2d69 6e20 4578 740a 0a2b 2d2d 2d2d lt-in Ext..+---- │ │ │ │ +00061c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061cb0: 2b0a 7c69 3232 203a 2045 4520 3d20 4578 +.|i22 : EE = Ex │ │ │ │ -00061cc0: 7428 4d62 6172 2c4b 293b 2020 2020 2020 t(Mbar,K); │ │ │ │ +00061ca0: 2d2d 2d2d 2d2b 0a7c 6932 3220 3a20 4545 -----+.|i22 : EE │ │ │ │ +00061cb0: 203d 2045 7874 284d 6261 722c 4b29 3b20 = Ext(Mbar,K); │ │ │ │ +00061cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061ce0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00061ce0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 00061cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061d20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3233 203a --------+.|i23 : │ │ │ │ -00061d30: 2053 2720 3d20 7269 6e67 2045 4520 2d2d S' = ring EE -- │ │ │ │ -00061d40: 206e 6f74 6520 7468 6174 2053 2720 6973 note that S' is │ │ │ │ -00061d50: 2074 6865 2070 6f6c 796e 6f6d 6961 6c20 the polynomial │ │ │ │ -00061d60: 7269 6e67 7c0a 7c20 2020 2020 2020 2020 ring|.| │ │ │ │ +00061d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ +00061d20: 6932 3320 3a20 5327 203d 2072 696e 6720 i23 : S' = ring │ │ │ │ +00061d30: 4545 202d 2d20 6e6f 7465 2074 6861 7420 EE -- note that │ │ │ │ +00061d40: 5327 2069 7320 7468 6520 706f 6c79 6e6f S' is the polyno │ │ │ │ +00061d50: 6d69 616c 2072 696e 677c 0a7c 2020 2020 mial ring|.| │ │ │ │ +00061d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061da0: 7c0a 7c6f 3233 203d 2053 2720 2020 2020 |.|o23 = S' │ │ │ │ +00061d90: 2020 2020 207c 0a7c 6f32 3320 3d20 5327 |.|o23 = S' │ │ │ │ +00061da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061dd0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00061dd0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00061de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e10: 2020 2020 2020 2020 7c0a 7c6f 3233 203a |.|o23 : │ │ │ │ -00061e20: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +00061e00: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00061e10: 6f32 3320 3a20 506f 6c79 6e6f 6d69 616c o23 : Polynomial │ │ │ │ +00061e20: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 00061e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061e50: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00061e40: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00061e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061e90: 2b0a 0a54 6865 2074 776f 2076 6572 7369 +..The two versi │ │ │ │ -00061ea0: 6f6e 7320 6f66 2045 7874 2061 7070 6561 ons of Ext appea │ │ │ │ -00061eb0: 7220 746f 2062 6520 7468 6520 7361 6d65 r to be the same │ │ │ │ -00061ec0: 2075 7020 746f 2063 6861 6e67 6520 6f66 up to change of │ │ │ │ -00061ed0: 2076 6172 6961 626c 6573 3a0a 0a2b 2d2d variables:..+-- │ │ │ │ +00061e80: 2d2d 2d2d 2d2b 0a0a 5468 6520 7477 6f20 -----+..The two │ │ │ │ +00061e90: 7665 7273 696f 6e73 206f 6620 4578 7420 versions of Ext │ │ │ │ +00061ea0: 6170 7065 6172 2074 6f20 6265 2074 6865 appear to be the │ │ │ │ +00061eb0: 2073 616d 6520 7570 2074 6f20 6368 616e same up to chan │ │ │ │ +00061ec0: 6765 206f 6620 7661 7269 6162 6c65 733a ge of variables: │ │ │ │ +00061ed0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00061ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00061f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00061f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00061f20: 6932 3420 3a20 4120 3d20 6672 6565 5265 i24 : A = freeRe │ │ │ │ -00061f30: 736f 6c75 7469 6f6e 2045 5320 2020 2020 solution ES │ │ │ │ +00061f10: 2d2d 2b0a 7c69 3234 203a 2041 203d 2066 --+.|i24 : A = f │ │ │ │ +00061f20: 7265 6552 6573 6f6c 7574 696f 6e20 4553 reeResolution ES │ │ │ │ +00061f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061f50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00061f60: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00061f50: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00061f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00061f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00061fa0: 207c 0a7c 2020 2020 2020 2038 2020 2020 |.| 8 │ │ │ │ -00061fb0: 2020 3336 2020 2020 2020 3636 2020 2020 36 66 │ │ │ │ -00061fc0: 2020 3634 2020 2020 2020 3336 2020 2020 64 36 │ │ │ │ -00061fd0: 2020 3132 2020 2020 2020 3220 2020 2020 12 2 │ │ │ │ -00061fe0: 2020 207c 0a7c 6f32 3420 3d20 5320 203c |.|o24 = S < │ │ │ │ -00061ff0: 2d2d 2053 2020 203c 2d2d 2053 2020 203c -- S <-- S < │ │ │ │ -00062000: 2d2d 2053 2020 203c 2d2d 2053 2020 203c -- S <-- S < │ │ │ │ -00062010: 2d2d 2053 2020 203c 2d2d 2053 2020 2020 -- S <-- S │ │ │ │ -00062020: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00061f90: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00061fa0: 3820 2020 2020 2033 3620 2020 2020 2036 8 36 6 │ │ │ │ +00061fb0: 3620 2020 2020 2036 3420 2020 2020 2033 6 64 3 │ │ │ │ +00061fc0: 3620 2020 2020 2031 3220 2020 2020 2032 6 12 2 │ │ │ │ +00061fd0: 2020 2020 2020 2020 7c0a 7c6f 3234 203d |.|o24 = │ │ │ │ +00061fe0: 2053 2020 3c2d 2d20 5320 2020 3c2d 2d20 S <-- S <-- │ │ │ │ +00061ff0: 5320 2020 3c2d 2d20 5320 2020 3c2d 2d20 S <-- S <-- │ │ │ │ +00062000: 5320 2020 3c2d 2d20 5320 2020 3c2d 2d20 S <-- S <-- │ │ │ │ +00062010: 5320 2020 2020 2020 2020 7c0a 7c20 2020 S |.| │ │ │ │ +00062020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062060: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00062070: 3020 2020 2020 2031 2020 2020 2020 2032 0 1 2 │ │ │ │ -00062080: 2020 2020 2020 2033 2020 2020 2020 2034 3 4 │ │ │ │ -00062090: 2020 2020 2020 2035 2020 2020 2020 2036 5 6 │ │ │ │ -000620a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +00062050: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00062060: 2020 2020 2030 2020 2020 2020 3120 2020 0 1 │ │ │ │ +00062070: 2020 2020 3220 2020 2020 2020 3320 2020 2 3 │ │ │ │ +00062080: 2020 2020 3420 2020 2020 2020 3520 2020 4 5 │ │ │ │ +00062090: 2020 2020 3620 2020 2020 2020 2020 7c0a 6 |. │ │ │ │ +000620a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000620b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000620c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000620d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000620e0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -000620f0: 3420 3a20 436f 6d70 6c65 7820 2020 2020 4 : Complex │ │ │ │ +000620e0: 7c0a 7c6f 3234 203a 2043 6f6d 706c 6578 |.|o24 : Complex │ │ │ │ +000620f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062120: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +00062120: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00062130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00062170: 0a7c 6932 3520 3a20 4220 3d20 6672 6565 .|i25 : B = free │ │ │ │ -00062180: 5265 736f 6c75 7469 6f6e 2045 4520 2020 Resolution EE │ │ │ │ +00062160: 2d2d 2d2d 2b0a 7c69 3235 203a 2042 203d ----+.|i25 : B = │ │ │ │ +00062170: 2066 7265 6552 6573 6f6c 7574 696f 6e20 freeResolution │ │ │ │ +00062180: 4545 2020 2020 2020 2020 2020 2020 2020 EE │ │ │ │ 00062190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000621a0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000621b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000621c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000621d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000621f0: 2020 207c 0a7c 2020 2020 2020 2020 3820 |.| 8 │ │ │ │ -00062200: 2020 2020 2020 3336 2020 2020 2020 2036 36 6 │ │ │ │ -00062210: 3620 2020 2020 2020 3634 2020 2020 2020 6 64 │ │ │ │ -00062220: 2033 3620 2020 2020 2020 3132 2020 2020 36 12 │ │ │ │ -00062230: 2020 2032 207c 0a7c 6f32 3520 3d20 5327 2 |.|o25 = S' │ │ │ │ -00062240: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053 <-- S' <-- S │ │ │ │ -00062250: 2720 2020 3c2d 2d20 5327 2020 203c 2d2d ' <-- S' <-- │ │ │ │ -00062260: 2053 2720 2020 3c2d 2d20 5327 2020 203c S' <-- S' < │ │ │ │ -00062270: 2d2d 2053 2720 207c 0a7c 2020 2020 2020 -- S' |.| │ │ │ │ +000621e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000621f0: 2020 2038 2020 2020 2020 2033 3620 2020 8 36 │ │ │ │ +00062200: 2020 2020 3636 2020 2020 2020 2036 3420 66 64 │ │ │ │ +00062210: 2020 2020 2020 3336 2020 2020 2020 2031 36 1 │ │ │ │ +00062220: 3220 2020 2020 2020 3220 7c0a 7c6f 3235 2 2 |.|o25 │ │ │ │ +00062230: 203d 2053 2720 203c 2d2d 2053 2720 2020 = S' <-- S' │ │ │ │ +00062240: 3c2d 2d20 5327 2020 203c 2d2d 2053 2720 <-- S' <-- S' │ │ │ │ +00062250: 2020 3c2d 2d20 5327 2020 203c 2d2d 2053 <-- S' <-- S │ │ │ │ +00062260: 2720 2020 3c2d 2d20 5327 2020 7c0a 7c20 ' <-- S' |.| │ │ │ │ +00062270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000622a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000622b0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -000622c0: 2020 3020 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ -000622d0: 2020 2032 2020 2020 2020 2020 3320 2020 2 3 │ │ │ │ -000622e0: 2020 2020 2034 2020 2020 2020 2020 3520 4 5 │ │ │ │ -000622f0: 2020 2020 2020 2036 2020 207c 0a7c 2020 6 |.| │ │ │ │ +000622a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000622b0: 7c20 2020 2020 2030 2020 2020 2020 2031 | 0 1 │ │ │ │ +000622c0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +000622d0: 2033 2020 2020 2020 2020 3420 2020 2020 3 4 │ │ │ │ +000622e0: 2020 2035 2020 2020 2020 2020 3620 2020 5 6 │ │ │ │ +000622f0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00062300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062330: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00062340: 6f32 3520 3a20 436f 6d70 6c65 7820 2020 o25 : Complex │ │ │ │ +00062330: 2020 7c0a 7c6f 3235 203a 2043 6f6d 706c |.|o25 : Compl │ │ │ │ +00062340: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ 00062350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062370: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00062380: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00062370: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00062380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000623a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000623b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000623c0: 2d2b 0a7c 6932 3620 3a20 616c 6c28 6c65 -+.|i26 : all(le │ │ │ │ -000623d0: 6e67 7468 2041 2b31 2c20 692d 3e20 736f ngth A+1, i-> so │ │ │ │ -000623e0: 7274 2064 6567 7265 6573 2041 5f69 203d rt degrees A_i = │ │ │ │ -000623f0: 3d20 736f 7274 2064 6567 7265 6573 2042 = sort degrees B │ │ │ │ -00062400: 5f69 297c 0a7c 2020 2020 2020 2020 2020 _i)|.| │ │ │ │ +000623b0: 2d2d 2d2d 2d2d 2b0a 7c69 3236 203a 2061 ------+.|i26 : a │ │ │ │ +000623c0: 6c6c 286c 656e 6774 6820 412b 312c 2069 ll(length A+1, i │ │ │ │ +000623d0: 2d3e 2073 6f72 7420 6465 6772 6565 7320 -> sort degrees │ │ │ │ +000623e0: 415f 6920 3d3d 2073 6f72 7420 6465 6772 A_i == sort degr │ │ │ │ +000623f0: 6565 7320 425f 6929 7c0a 7c20 2020 2020 ees B_i)|.| │ │ │ │ +00062400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062440: 2020 2020 207c 0a7c 6f32 3620 3d20 7472 |.|o26 = tr │ │ │ │ -00062450: 7565 2020 2020 2020 2020 2020 2020 2020 ue │ │ │ │ +00062430: 2020 2020 2020 2020 2020 7c0a 7c6f 3236 |.|o26 │ │ │ │ +00062440: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ +00062450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062480: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00062470: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00062480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000624a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000624c0: 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 6275 7420 ---------+..but │ │ │ │ -000624d0: 7468 6579 2068 6176 6520 6170 7061 7265 they have appare │ │ │ │ -000624e0: 6e74 6c79 2064 6966 6665 7265 6e74 2061 ntly different a │ │ │ │ -000624f0: 6e6e 6968 696c 6174 6f72 730a 0a2b 2d2d nnihilators..+-- │ │ │ │ +000624b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000624c0: 0a62 7574 2074 6865 7920 6861 7665 2061 .but they have a │ │ │ │ +000624d0: 7070 6172 656e 746c 7920 6469 6666 6572 pparently differ │ │ │ │ +000624e0: 656e 7420 616e 6e69 6869 6c61 746f 7273 ent annihilators │ │ │ │ +000624f0: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ 00062500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062540: 2d2d 2d2b 0a7c 6932 3720 3a20 616e 6e20 ---+.|i27 : ann │ │ │ │ -00062550: 4545 2020 2020 2020 2020 2020 2020 2020 EE │ │ │ │ +00062530: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3237 203a --------+.|i27 : │ │ │ │ +00062540: 2061 6e6e 2045 4520 2020 2020 2020 2020 ann EE │ │ │ │ +00062550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062580: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00062580: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00062590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000625a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000625b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000625c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000625d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -000625e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000625f0: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062600: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062610: 2020 2020 2020 2020 2032 207c 0a7c 6f32 2 |.|o2 │ │ │ │ -00062620: 3720 3d20 6964 6561 6c20 2878 202c 2078 7 = ideal (x , x │ │ │ │ -00062630: 202c 2078 202c 2058 2020 2b20 3431 5820 , x , X + 41X │ │ │ │ -00062640: 5820 202d 2033 3758 2020 2d20 3134 5820 X - 37X - 14X │ │ │ │ -00062650: 5820 202d 2032 3958 2058 2020 2b20 3435 X - 29X X + 45 │ │ │ │ -00062660: 5820 297c 0a7c 2020 2020 2020 2020 2020 X )|.| │ │ │ │ -00062670: 2020 2020 3220 2020 3120 2020 3020 2020 2 1 0 │ │ │ │ -00062680: 3120 2020 2020 2031 2032 2020 2020 2020 1 1 2 │ │ │ │ -00062690: 3220 2020 2020 2031 2033 2020 2020 2020 2 1 3 │ │ │ │ -000626a0: 3220 3320 2020 2020 2033 207c 0a7c 2020 2 3 3 |.| │ │ │ │ +000625c0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000625d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000625e0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +000625f0: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00062600: 2020 2020 2020 2020 2020 2020 2020 3220 2 │ │ │ │ +00062610: 7c0a 7c6f 3237 203d 2069 6465 616c 2028 |.|o27 = ideal ( │ │ │ │ +00062620: 7820 2c20 7820 2c20 7820 2c20 5820 202b x , x , x , X + │ │ │ │ +00062630: 2034 3158 2058 2020 2d20 3337 5820 202d 41X X - 37X - │ │ │ │ +00062640: 2031 3458 2058 2020 2d20 3239 5820 5820 14X X - 29X X │ │ │ │ +00062650: 202b 2034 3558 2029 7c0a 7c20 2020 2020 + 45X )|.| │ │ │ │ +00062660: 2020 2020 2020 2020 2032 2020 2031 2020 2 1 │ │ │ │ +00062670: 2030 2020 2031 2020 2020 2020 3120 3220 0 1 1 2 │ │ │ │ +00062680: 2020 2020 2032 2020 2020 2020 3120 3320 2 1 3 │ │ │ │ +00062690: 2020 2020 2032 2033 2020 2020 2020 3320 2 3 3 │ │ │ │ +000626a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000626b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000626c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000626d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000626e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000626f0: 2020 207c 0a7c 6f32 3720 3a20 4964 6561 |.|o27 : Idea │ │ │ │ -00062700: 6c20 6f66 2053 2720 2020 2020 2020 2020 l of S' │ │ │ │ +000626e0: 2020 2020 2020 2020 7c0a 7c6f 3237 203a |.|o27 : │ │ │ │ +000626f0: 2049 6465 616c 206f 6620 5327 2020 2020 Ideal of S' │ │ │ │ +00062700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062730: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00062730: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00062740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062780: 2d2d 2d2b 0a7c 6932 3820 3a20 616e 6e20 ---+.|i28 : ann │ │ │ │ -00062790: 4553 2020 2020 2020 2020 2020 2020 2020 ES │ │ │ │ +00062770: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3238 203a --------+.|i28 : │ │ │ │ +00062780: 2061 6e6e 2045 5320 2020 2020 2020 2020 ann ES │ │ │ │ +00062790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000627c0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000627c0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000627d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000627f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062810: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00062820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062830: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062840: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -00062850: 2020 2020 2020 2020 3220 207c 0a7c 6f32 2 |.|o2 │ │ │ │ -00062860: 3820 3d20 6964 6561 6c20 2878 202c 2078 8 = ideal (x , x │ │ │ │ -00062870: 202c 2078 202c 2073 2020 2b20 3432 7320 , x , s + 42s │ │ │ │ -00062880: 7320 202d 2033 3073 2020 2d20 3235 7320 s - 30s - 25s │ │ │ │ -00062890: 7320 202d 2033 3573 2073 2020 2b20 3973 s - 35s s + 9s │ │ │ │ -000628a0: 2029 207c 0a7c 2020 2020 2020 2020 2020 ) |.| │ │ │ │ -000628b0: 2020 2020 3220 2020 3120 2020 3020 2020 2 1 0 │ │ │ │ -000628c0: 3020 2020 2020 2030 2031 2020 2020 2020 0 0 1 │ │ │ │ -000628d0: 3120 2020 2020 2030 2032 2020 2020 2020 1 0 2 │ │ │ │ -000628e0: 3120 3220 2020 2020 3220 207c 0a7c 2020 1 2 2 |.| │ │ │ │ +00062800: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00062810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00062820: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00062830: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ +00062840: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00062850: 7c0a 7c6f 3238 203d 2069 6465 616c 2028 |.|o28 = ideal ( │ │ │ │ +00062860: 7820 2c20 7820 2c20 7820 2c20 7320 202b x , x , x , s + │ │ │ │ +00062870: 2034 3273 2073 2020 2d20 3330 7320 202d 42s s - 30s - │ │ │ │ +00062880: 2032 3573 2073 2020 2d20 3335 7320 7320 25s s - 35s s │ │ │ │ +00062890: 202b 2039 7320 2920 7c0a 7c20 2020 2020 + 9s ) |.| │ │ │ │ +000628a0: 2020 2020 2020 2020 2032 2020 2031 2020 2 1 │ │ │ │ +000628b0: 2030 2020 2030 2020 2020 2020 3020 3120 0 0 0 1 │ │ │ │ +000628c0: 2020 2020 2031 2020 2020 2020 3020 3220 1 0 2 │ │ │ │ +000628d0: 2020 2020 2031 2032 2020 2020 2032 2020 1 2 2 │ │ │ │ +000628e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000628f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062930: 2020 207c 0a7c 6f32 3820 3a20 4964 6561 |.|o28 : Idea │ │ │ │ -00062940: 6c20 6f66 2053 2020 2020 2020 2020 2020 l of S │ │ │ │ +00062920: 2020 2020 2020 2020 7c0a 7c6f 3238 203a |.|o28 : │ │ │ │ +00062930: 2049 6465 616c 206f 6620 5320 2020 2020 Ideal of S │ │ │ │ +00062940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062970: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00062970: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00062980: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000629a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000629b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000629c0: 2d2d 2d2b 0a0a 616e 6420 696e 2066 6163 ---+..and in fac │ │ │ │ -000629d0: 7420 7468 6579 2061 7265 206e 6f74 2069 t they are not i │ │ │ │ -000629e0: 736f 6d6f 7270 6869 633a 0a0a 2b2d 2d2d somorphic:..+--- │ │ │ │ +000629b0: 2d2d 2d2d 2d2d 2d2d 2b0a 0a61 6e64 2069 --------+..and i │ │ │ │ +000629c0: 6e20 6661 6374 2074 6865 7920 6172 6520 n fact they are │ │ │ │ +000629d0: 6e6f 7420 6973 6f6d 6f72 7068 6963 3a0a not isomorphic:. │ │ │ │ +000629e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000629f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3239 ----------+.|i29 │ │ │ │ -00062a40: 203a 2045 4574 6f45 5320 3d20 6d61 7028 : EEtoES = map( │ │ │ │ -00062a50: 7269 6e67 2045 532c 7269 6e67 2045 452c ring ES,ring EE, │ │ │ │ -00062a60: 2067 656e 7320 7269 6e67 2045 5329 2020 gens ring ES) │ │ │ │ -00062a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062a80: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00062a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00062a30: 0a7c 6932 3920 3a20 4545 746f 4553 203d .|i29 : EEtoES = │ │ │ │ +00062a40: 206d 6170 2872 696e 6720 4553 2c72 696e map(ring ES,rin │ │ │ │ +00062a50: 6720 4545 2c20 6765 6e73 2072 696e 6720 g EE, gens ring │ │ │ │ +00062a60: 4553 2920 2020 2020 2020 2020 2020 2020 ES) │ │ │ │ +00062a70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062a80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00062a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062ad0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239 |.|o29 │ │ │ │ -00062ae0: 203d 206d 6170 2028 532c 2053 272c 207b = map (S, S', { │ │ │ │ -00062af0: 7320 2c20 7320 2c20 7320 2c20 7820 2c20 s , s , s , x , │ │ │ │ -00062b00: 7820 2c20 7820 7d29 2020 2020 2020 2020 x , x }) │ │ │ │ -00062b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062b20: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00062b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062b40: 2030 2020 2031 2020 2032 2020 2030 2020 0 1 2 0 │ │ │ │ -00062b50: 2031 2020 2032 2020 2020 2020 2020 2020 1 2 │ │ │ │ -00062b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062b70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00062ac0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062ad0: 0a7c 6f32 3920 3d20 6d61 7020 2853 2c20 .|o29 = map (S, │ │ │ │ +00062ae0: 5327 2c20 7b73 202c 2073 202c 2073 202c S', {s , s , s , │ │ │ │ +00062af0: 2078 202c 2078 202c 2078 207d 2920 2020 x , x , x }) │ │ │ │ +00062b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00062b10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062b20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00062b30: 2020 2020 2020 3020 2020 3120 2020 3220 0 1 2 │ │ │ │ +00062b40: 2020 3020 2020 3120 2020 3220 2020 2020 0 1 2 │ │ │ │ +00062b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00062b60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062b70: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00062b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062bc0: 2020 2020 2020 2020 2020 7c0a 7c6f 3239 |.|o29 │ │ │ │ -00062bd0: 203a 2052 696e 674d 6170 2053 203c 2d2d : RingMap S <-- │ │ │ │ -00062be0: 2053 2720 2020 2020 2020 2020 2020 2020 S' │ │ │ │ +00062bb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062bc0: 0a7c 6f32 3920 3a20 5269 6e67 4d61 7020 .|o29 : RingMap │ │ │ │ +00062bd0: 5320 3c2d 2d20 5327 2020 2020 2020 2020 S <-- S' │ │ │ │ +00062be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00062bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00062c10: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00062c00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00062c10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00062c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00062c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00062c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3330 ----------+.|i30 │ │ │ │ -00062c70: 203a 2045 4527 203d 2063 6f6b 6572 2045 : EE' = coker E │ │ │ │ -00062c80: 4574 6f45 5320 7072 6573 656e 7461 7469 EtoES presentati │ │ │ │ -00062c90: 6f6e 2045 4520 2020 2020 2020 2020 2020 on EE │ │ │ │ -00062ca0: 2020 2020 2020 2020 2020 2020 2020 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2020 2020 .|0 │ │ │ │ 00063ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063af0: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -00063b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063b10: 2020 2020 2020 2020 2020 7c0a 7c30 2020 |.|0 │ │ │ │ +00063af0: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00063b00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063b10: 0a7c 3020 2020 2020 2020 2020 2020 2020 .|0 │ │ │ │ 00063b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063b40: 2020 2020 2020 2020 2020 2020 2020 7c20 | │ │ │ │ -00063b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063b60: 2020 2020 2020 2020 2020 7c0a 7c30 2020 |.|0 │ │ │ │ +00063b40: 2020 207c 2020 2020 2020 2020 2020 2020 | │ │ │ │ +00063b50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063b60: 0a7c 3020 2020 2020 2020 2020 2020 2020 .|0 │ │ │ │ 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2020 S); │ │ │ │ +00063bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00063c00: 0a7c 6933 3120 3a20 4820 3d20 486f 6d28 .|i31 : H = Hom( │ │ │ │ +00063c10: 4545 272c 4553 293b 2020 2020 2020 2020 EE',ES); │ │ │ │ +00063c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063c50: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00063c40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063c50: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00063c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063ca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3332 ----------+.|i32 │ │ │ │ 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d40: 2020 2020 2020 2020 2020 7c0a 7c6f 3332 |.|o32 │ │ │ │ -00063d50: 203d 207b 382c 2039 2c20 3130 2c20 3131 = {8, 9, 10, 11 │ │ │ │ -00063d60: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00063d30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063d40: 0a7c 6f33 3220 3d20 7b38 2c20 392c 2031 .|o32 = {8, 9, 1 │ │ │ │ +00063d50: 302c 2031 317d 2020 2020 2020 2020 2020 0, 11} │ │ │ │ +00063d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063d90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00063d80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063d90: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00063da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063de0: 2020 2020 2020 2020 2020 7c0a 7c6f 3332 |.|o32 │ │ │ │ -00063df0: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +00063dd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063de0: 0a7c 6f33 3220 3a20 4c69 7374 2020 2020 .|o32 : List │ │ │ │ +00063df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063e30: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00063e20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063e30: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00063e40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00063e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00063e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3333 ----------+.|i33 │ │ │ │ -00063e90: 203a 2066 203d 2073 756d 2851 2c20 702d : f = sum(Q, p- │ │ │ │ -00063ea0: 3e20 7261 6e64 6f6d 2028 535e 312c 2053 > random (S^1, S │ │ │ │ -00063eb0: 5e31 292a 2a68 6f6d 6f6d 6f72 7068 6973 ^1)**homomorphis │ │ │ │ -00063ec0: 6d20 485f 7b70 7d29 2020 2020 2020 2020 m H_{p}) │ │ │ │ -00063ed0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00063e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00063e80: 0a7c 6933 3320 3a20 6620 3d20 7375 6d28 .|i33 : f = sum( │ │ │ │ +00063e90: 512c 2070 2d3e 2072 616e 646f 6d20 2853 Q, p-> random (S │ │ │ │ +00063ea0: 5e31 2c20 535e 3129 2a2a 686f 6d6f 6d6f ^1, S^1)**homomo │ │ │ │ +00063eb0: 7270 6869 736d 2048 5f7b 707d 2920 2020 rphism H_{p}) │ │ │ │ +00063ec0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063ed0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00063ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00063f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f20: 2020 2020 2020 2020 2020 7c0a 7c6f 3333 |.|o33 │ │ │ │ -00063f30: 203d 207b 302c 2030 7d20 2020 7c20 2d33 = {0, 0} | -3 │ │ │ │ -00063f40: 3820 3339 2030 2030 2030 2030 2030 2030 8 39 0 0 0 0 0 0 │ │ │ │ -00063f50: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00063f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063f70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00063f80: 2020 207b 302c 2030 7d20 2020 7c20 2d31 {0, 0} | -1 │ │ │ │ -00063f90: 3620 3231 2030 2030 2030 2030 2030 2030 6 21 0 0 0 0 0 0 │ │ │ │ -00063fa0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00063fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00063fc0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00063fd0: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00063fe0: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00063ff0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064010: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00064020: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00064030: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00064040: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064060: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00064070: 2020 207b 2d32 2c20 2d33 7d20 7c20 3020 {-2, -3} | 0 │ │ │ │ -00064080: 2020 3020 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 │ │ │ │ -00064090: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000640a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000640b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -000640c0: 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2020 2020 2020 2020 207c | │ │ │ │ +00063f20: 0a7c 6f33 3320 3d20 7b30 2c20 307d 2020 .|o33 = {0, 0} │ │ │ │ +00063f30: 207c 202d 3338 2033 3920 3020 3020 3020 | -38 39 0 0 0 │ │ │ │ +00063f40: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00063f60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063f70: 0a7c 2020 2020 2020 7b30 2c20 307d 2020 .| {0, 0} │ │ │ │ +00063f80: 207c 202d 3136 2032 3120 3020 3020 3020 | -16 21 0 0 0 │ │ │ │ +00063f90: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00063fb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00063fc0: 0a7c 2020 2020 2020 7b2d 322c 202d 337d .| {-2, -3} │ │ │ │ +00063fd0: 207c 2030 2020 2030 2020 3020 3020 3020 | 0 0 0 0 0 │ │ │ │ +00063fe0: 3020 3020 3020 7c20 2020 2020 2020 2020 0 0 0 | │ │ │ │ +00063ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064000: 2020 2020 2020 2020 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2020 2020 7c0a 7c6f 3333 |.|o33 │ │ │ │ -00064200: 203a 204d 6174 7269 7820 4553 203c 2d2d : Matrix ES <-- │ │ │ │ -00064210: 2045 4527 2020 2020 2020 2020 2020 2020 EE' │ │ │ │ +000641e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000641f0: 0a7c 6f33 3320 3a20 4d61 7472 6978 2045 .|o33 : Matrix E │ │ │ │ +00064200: 5320 3c2d 2d20 4545 2720 2020 2020 2020 S <-- EE' │ │ │ │ +00064210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064240: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +00064230: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00064240: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00064250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064270: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064290: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a49 6620 ----------+..If │ │ │ │ -000642a0: 4545 2061 6e64 2045 5320 7765 7265 2069 EE and ES were i │ │ │ │ -000642b0: 736f 6d6f 7270 6869 632c 2077 6520 776f somorphic, we wo │ │ │ │ -000642c0: 756c 6420 6578 7065 6374 2063 6f6b 6572 uld expect coker │ │ │ │ -000642d0: 2066 2074 6f20 6265 2030 2c20 616e 6420 f to be 0, and │ │ │ │ -000642e0: 6974 2773 206e 6f74 2e0a 7072 756e 6520 it's not..prune │ │ │ │ -000642f0: 636f 6b65 7220 660a 0a53 6565 2061 6c73 coker f..See als │ │ │ │ -00064300: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ -00064310: 2a6e 6f74 6520 4578 743a 2028 4d61 6361 *note Ext: (Maca │ │ │ │ -00064320: 756c 6179 3244 6f63 2945 7874 2c20 2d2d ulay2Doc)Ext, -- │ │ │ │ -00064330: 2063 6f6d 7075 7465 2061 6e20 4578 7420 compute an Ext │ │ │ │ -00064340: 6d6f 6475 6c65 0a20 202a 202a 6e6f 7465 module. * *note │ │ │ │ -00064350: 2045 6973 656e 6275 6453 6861 6d61 7368 EisenbudShamash │ │ │ │ -00064360: 546f 7461 6c3a 2045 6973 656e 6275 6453 Total: EisenbudS │ │ │ │ -00064370: 6861 6d61 7368 546f 7461 6c2c 202d 2d20 hamashTotal, -- │ │ │ │ -00064380: 5072 6563 7572 736f 7220 636f 6d70 6c65 Precursor comple │ │ │ │ -00064390: 7820 6f66 0a20 2020 2074 6f74 616c 2045 x of. total E │ │ │ │ -000643a0: 7874 0a0a 5761 7973 2074 6f20 7573 6520 xt..Ways to use │ │ │ │ -000643b0: 6e65 7745 7874 3a0a 3d3d 3d3d 3d3d 3d3d newExt:.======== │ │ │ │ -000643c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -000643d0: 2022 6e65 7745 7874 284d 6f64 756c 652c "newExt(Module, │ │ │ │ -000643e0: 4d6f 6475 6c65 2922 0a0a 466f 7220 7468 Module)"..For th │ │ │ │ -000643f0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -00064400: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00064410: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -00064420: 6520 6e65 7745 7874 3a20 6e65 7745 7874 e newExt: newExt │ │ │ │ -00064430: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -00064440: 686f 6420 6675 6e63 7469 6f6e 2077 6974 hod function wit │ │ │ │ -00064450: 6820 6f70 7469 6f6e 733a 0a28 4d61 6361 h options:.(Maca │ │ │ │ -00064460: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ -00064470: 756e 6374 696f 6e57 6974 684f 7074 696f unctionWithOptio │ │ │ │ -00064480: 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ns,...---------- │ │ │ │ +00064280: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00064290: 0a0a 4966 2045 4520 616e 6420 4553 2077 ..If EE and ES w │ │ │ │ +000642a0: 6572 6520 6973 6f6d 6f72 7068 6963 2c20 ere isomorphic, │ │ │ │ +000642b0: 7765 2077 6f75 6c64 2065 7870 6563 7420 we would expect │ │ │ │ +000642c0: 636f 6b65 7220 6620 746f 2062 6520 302c coker f to be 0, │ │ │ │ +000642d0: 2061 6e64 2069 7427 7320 6e6f 742e 0a70 and it's not..p │ │ │ │ +000642e0: 7275 6e65 2063 6f6b 6572 2066 0a0a 5365 rune coker f..Se │ │ │ │ +000642f0: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ +00064300: 0a20 202a 202a 6e6f 7465 2045 7874 3a20 . * *note Ext: │ │ │ │ +00064310: 284d 6163 6175 6c61 7932 446f 6329 4578 (Macaulay2Doc)Ex │ │ │ │ +00064320: 742c 202d 2d20 636f 6d70 7574 6520 616e t, -- compute an │ │ │ │ +00064330: 2045 7874 206d 6f64 756c 650a 2020 2a20 Ext module. * │ │ │ │ +00064340: 2a6e 6f74 6520 4569 7365 6e62 7564 5368 *note EisenbudSh │ │ │ │ +00064350: 616d 6173 6854 6f74 616c 3a20 4569 7365 amashTotal: Eise │ │ │ │ +00064360: 6e62 7564 5368 616d 6173 6854 6f74 616c nbudShamashTotal │ │ │ │ +00064370: 2c20 2d2d 2050 7265 6375 7273 6f72 2063 , -- Precursor c │ │ │ │ +00064380: 6f6d 706c 6578 206f 660a 2020 2020 746f omplex of. to │ │ │ │ +00064390: 7461 6c20 4578 740a 0a57 6179 7320 746f tal Ext..Ways to │ │ │ │ +000643a0: 2075 7365 206e 6577 4578 743a 0a3d 3d3d use newExt:.=== │ │ │ │ +000643b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000643c0: 0a0a 2020 2a20 226e 6577 4578 7428 4d6f .. * "newExt(Mo │ │ │ │ +000643d0: 6475 6c65 2c4d 6f64 756c 6529 220a 0a46 dule,Module)"..F │ │ │ │ +000643e0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +000643f0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00064400: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00064410: 202a 6e6f 7465 206e 6577 4578 743a 206e *note newExt: n │ │ │ │ +00064420: 6577 4578 742c 2069 7320 6120 2a6e 6f74 ewExt, is a *not │ │ │ │ +00064430: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +00064440: 6e20 7769 7468 206f 7074 696f 6e73 3a0a n with options:. │ │ │ │ +00064450: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ +00064460: 7468 6f64 4675 6e63 7469 6f6e 5769 7468 thodFunctionWith │ │ │ │ +00064470: 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d Options,...----- │ │ │ │ +00064480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000644a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000644b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000644d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -000644e0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -000644f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00064500: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00064510: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00064520: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -00064530: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ -00064540: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -00064550: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ -00064560: 3a32 3536 343a 302e 0a1f 0a46 696c 653a :2564:0....File: │ │ │ │ -00064570: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -00064580: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -00064590: 2e69 6e66 6f2c 204e 6f64 653a 206f 6464 .info, Node: odd │ │ │ │ -000645a0: 4578 744d 6f64 756c 652c 204e 6578 743a ExtModule, Next: │ │ │ │ -000645b0: 204f 7074 696d 6973 6d2c 2050 7265 763a Optimism, Prev: │ │ │ │ -000645c0: 206e 6577 4578 742c 2055 703a 2054 6f70 newExt, Up: Top │ │ │ │ -000645d0: 0a0a 6f64 6445 7874 4d6f 6475 6c65 202d ..oddExtModule - │ │ │ │ -000645e0: 2d20 6f64 6420 7061 7274 206f 6620 4578 - odd part of Ex │ │ │ │ -000645f0: 745e 2a28 4d2c 6b29 206f 7665 7220 6120 t^*(M,k) over a │ │ │ │ -00064600: 636f 6d70 6c65 7465 2069 6e74 6572 7365 complete interse │ │ │ │ -00064610: 6374 696f 6e20 6173 206d 6f64 756c 6520 ction as module │ │ │ │ -00064620: 6f76 6572 2043 4920 6f70 6572 6174 6f72 over CI operator │ │ │ │ -00064630: 2072 696e 670a 2a2a 2a2a 2a2a 2a2a 2a2a ring.********** │ │ │ │ +000644c0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +000644d0: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +000644e0: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +000644f0: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +00064500: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +00064510: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +00064520: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +00064530: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +00064540: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +00064550: 6e73 2e6d 323a 3235 3634 3a30 2e0a 1f0a ns.m2:2564:0.... │ │ │ │ +00064560: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +00064570: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +00064580: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +00064590: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ +000645a0: 4e65 7874 3a20 4f70 7469 6d69 736d 2c20 Next: Optimism, │ │ │ │ +000645b0: 5072 6576 3a20 6e65 7745 7874 2c20 5570 Prev: newExt, Up │ │ │ │ +000645c0: 3a20 546f 700a 0a6f 6464 4578 744d 6f64 : Top..oddExtMod │ │ │ │ +000645d0: 756c 6520 2d2d 206f 6464 2070 6172 7420 ule -- odd part │ │ │ │ +000645e0: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +000645f0: 6572 2061 2063 6f6d 706c 6574 6520 696e er a complete in │ │ │ │ +00064600: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ +00064610: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ +00064620: 7261 746f 7220 7269 6e67 0a2a 2a2a 2a2a rator ring.***** │ │ │ │ +00064630: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064640: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064650: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064660: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00064670: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00064690: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -000646a0: 7361 6765 3a20 0a20 2020 2020 2020 2045 sage: . E │ │ │ │ -000646b0: 203d 206f 6464 4578 744d 6f64 756c 6520 = oddExtModule │ │ │ │ -000646c0: 4d0a 2020 2a20 496e 7075 7473 3a0a 2020 M. * Inputs:. │ │ │ │ -000646d0: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ -000646e0: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ -000646f0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ -00064700: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ -00064710: 696e 7465 7273 6563 7469 6f6e 0a20 2020 intersection. │ │ │ │ -00064720: 2020 2020 2072 696e 670a 2020 2a20 2a6e ring. * *n │ │ │ │ -00064730: 6f74 6520 4f70 7469 6f6e 616c 2069 6e70 ote Optional inp │ │ │ │ -00064740: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ -00064750: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ -00064760: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ -00064770: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ -00064780: 2a20 4f75 7452 696e 6720 3d3e 202e 2e2e * OutRing => ... │ │ │ │ -00064790: 2c20 6465 6661 756c 7420 7661 6c75 6520 , default value │ │ │ │ -000647a0: 300a 2020 2a20 4f75 7470 7574 733a 0a20 0. * Outputs:. │ │ │ │ -000647b0: 2020 2020 202a 2045 2c20 6120 2a6e 6f74 * E, a *not │ │ │ │ -000647c0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -000647d0: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -000647e0: 206f 7665 7220 6120 706f 6c79 6e6f 6d69 over a polynomi │ │ │ │ -000647f0: 616c 2072 696e 6720 7769 7468 0a20 2020 al ring with. │ │ │ │ -00064800: 2020 2020 2067 656e 7320 696e 2064 6567 gens in deg │ │ │ │ -00064810: 7265 6520 310a 0a44 6573 6372 6970 7469 ree 1..Descripti │ │ │ │ -00064820: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00064830: 4578 7472 6163 7473 2074 6865 206f 6464 Extracts the odd │ │ │ │ -00064840: 2064 6567 7265 6520 7061 7274 2066 726f degree part fro │ │ │ │ -00064850: 6d20 4578 744d 6f64 756c 6520 4d2e 2049 m ExtModule M. I │ │ │ │ -00064860: 6620 7468 6520 6f70 7469 6f6e 616c 2061 f the optional a │ │ │ │ -00064870: 7267 756d 656e 7420 4f75 7452 696e 670a rgument OutRing. │ │ │ │ -00064880: 3d3e 2054 2069 7320 6769 7665 6e2c 2061 => T is given, a │ │ │ │ -00064890: 6e64 2063 6c61 7373 2054 203d 3d3d 2050 nd class T === P │ │ │ │ -000648a0: 6f6c 796e 6f6d 6961 6c52 696e 672c 2074 olynomialRing, t │ │ │ │ -000648b0: 6865 6e20 7468 6520 6f75 7470 7574 2077 hen the output w │ │ │ │ -000648c0: 696c 6c20 6265 2061 206d 6f64 756c 650a ill be a module. │ │ │ │ -000648d0: 6f76 6572 2054 2e0a 0a2b 2d2d 2d2d 2d2d over T...+------ │ │ │ │ +00064680: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +00064690: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +000646a0: 2020 2020 4520 3d20 6f64 6445 7874 4d6f E = oddExtMo │ │ │ │ +000646b0: 6475 6c65 204d 0a20 202a 2049 6e70 7574 dule M. * Input │ │ │ │ +000646c0: 733a 0a20 2020 2020 202a 204d 2c20 6120 s:. * M, a │ │ │ │ +000646d0: 2a6e 6f74 6520 6d6f 6475 6c65 3a20 284d *note module: (M │ │ │ │ +000646e0: 6163 6175 6c61 7932 446f 6329 4d6f 6475 acaulay2Doc)Modu │ │ │ │ +000646f0: 6c65 2c2c 206f 7665 7220 6120 636f 6d70 le,, over a comp │ │ │ │ +00064700: 6c65 7465 2069 6e74 6572 7365 6374 696f lete intersectio │ │ │ │ +00064710: 6e0a 2020 2020 2020 2020 7269 6e67 0a20 n. ring. │ │ │ │ +00064720: 202a 202a 6e6f 7465 204f 7074 696f 6e61 * *note Optiona │ │ │ │ +00064730: 6c20 696e 7075 7473 3a20 284d 6163 6175 l inputs: (Macau │ │ │ │ +00064740: 6c61 7932 446f 6329 7573 696e 6720 6675 lay2Doc)using fu │ │ │ │ +00064750: 6e63 7469 6f6e 7320 7769 7468 206f 7074 nctions with opt │ │ │ │ +00064760: 696f 6e61 6c20 696e 7075 7473 2c3a 0a20 ional inputs,:. │ │ │ │ +00064770: 2020 2020 202a 204f 7574 5269 6e67 203d * OutRing = │ │ │ │ +00064780: 3e20 2e2e 2e2c 2064 6566 6175 6c74 2076 > ..., default v │ │ │ │ +00064790: 616c 7565 2030 0a20 202a 204f 7574 7075 alue 0. * Outpu │ │ │ │ +000647a0: 7473 3a0a 2020 2020 2020 2a20 452c 2061 ts:. * E, a │ │ │ │ +000647b0: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +000647c0: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +000647d0: 756c 652c 2c20 6f76 6572 2061 2070 6f6c ule,, over a pol │ │ │ │ +000647e0: 796e 6f6d 6961 6c20 7269 6e67 2077 6974 ynomial ring wit │ │ │ │ +000647f0: 680a 2020 2020 2020 2020 6765 6e73 2069 h. gens i │ │ │ │ +00064800: 6e20 6465 6772 6565 2031 0a0a 4465 7363 n degree 1..Desc │ │ │ │ +00064810: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00064820: 3d3d 3d0a 0a45 7874 7261 6374 7320 7468 ===..Extracts th │ │ │ │ +00064830: 6520 6f64 6420 6465 6772 6565 2070 6172 e odd degree par │ │ │ │ +00064840: 7420 6672 6f6d 2045 7874 4d6f 6475 6c65 t from ExtModule │ │ │ │ +00064850: 204d 2e20 4966 2074 6865 206f 7074 696f M. If the optio │ │ │ │ +00064860: 6e61 6c20 6172 6775 6d65 6e74 204f 7574 nal argument Out │ │ │ │ +00064870: 5269 6e67 0a3d 3e20 5420 6973 2067 6976 Ring.=> T is giv │ │ │ │ +00064880: 656e 2c20 616e 6420 636c 6173 7320 5420 en, and class T │ │ │ │ +00064890: 3d3d 3d20 506f 6c79 6e6f 6d69 616c 5269 === PolynomialRi │ │ │ │ +000648a0: 6e67 2c20 7468 656e 2074 6865 206f 7574 ng, then the out │ │ │ │ +000648b0: 7075 7420 7769 6c6c 2062 6520 6120 6d6f put will be a mo │ │ │ │ +000648c0: 6475 6c65 0a6f 7665 7220 542e 0a0a 2b2d dule.over T...+- │ │ │ │ +000648d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000648e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000648f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064910: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a ---+.|i1 : kk= Z │ │ │ │ -00064920: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +00064900: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +00064910: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020 kk= ZZ/101 │ │ │ │ +00064920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064940: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064950: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00064940: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064980: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -00064990: 203d 206b 6b20 2020 2020 2020 2020 2020 = kk │ │ │ │ +00064980: 7c0a 7c6f 3120 3d20 6b6b 2020 2020 2020 |.|o1 = kk │ │ │ │ +00064990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000649a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000649b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000649c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000649b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000649c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000649d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000649e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000649f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064a00: 2020 207c 0a7c 6f31 203a 2051 756f 7469 |.|o1 : Quoti │ │ │ │ -00064a10: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +000649f0: 2020 2020 2020 2020 7c0a 7c6f 3120 3a20 |.|o1 : │ │ │ │ +00064a00: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +00064a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064a30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064a40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00064a30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00064a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064a50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00064a80: 203a 2053 203d 206b 6b5b 782c 792c 7a5d : S = kk[x,y,z] │ │ │ │ +00064a70: 2b0a 7c69 3220 3a20 5320 3d20 6b6b 5b78 +.|i2 : S = kk[x │ │ │ │ +00064a80: 2c79 2c7a 5d20 2020 2020 2020 2020 2020 ,y,z] │ │ │ │ 00064a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ab0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00064aa0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00064ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064af0: 2020 207c 0a7c 6f32 203d 2053 2020 2020 |.|o2 = S │ │ │ │ +00064ae0: 2020 2020 2020 2020 7c0a 7c6f 3220 3d20 |.|o2 = │ │ │ │ +00064af0: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ 00064b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064b20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064b30: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00064b20: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064b60: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -00064b70: 203a 2050 6f6c 796e 6f6d 6961 6c52 696e : PolynomialRin │ │ │ │ -00064b80: 6720 2020 2020 2020 2020 2020 2020 2020 g │ │ │ │ -00064b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ba0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00064b60: 7c0a 7c6f 3220 3a20 506f 6c79 6e6f 6d69 |.|o2 : Polynomi │ │ │ │ +00064b70: 616c 5269 6e67 2020 2020 2020 2020 2020 alRing │ │ │ │ +00064b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064b90: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00064ba0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064bb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064be0: 2d2d 2d2b 0a7c 6933 203a 2049 3220 3d20 ---+.|i3 : I2 = │ │ │ │ -00064bf0: 6964 6561 6c22 7833 2c79 7a22 2020 2020 ideal"x3,yz" │ │ │ │ +00064bd0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00064be0: 4932 203d 2069 6465 616c 2278 332c 797a I2 = ideal"x3,yz │ │ │ │ +00064bf0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ 00064c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c10: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064c20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00064c10: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c50: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00064c60: 2020 2020 2020 2020 2020 2033 2020 2020 3 │ │ │ │ +00064c50: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064c60: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 00064c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064c90: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -00064ca0: 6465 616c 2028 7820 2c20 792a 7a29 2020 deal (x , y*z) │ │ │ │ +00064c80: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00064c90: 3320 3d20 6964 6561 6c20 2878 202c 2079 3 = ideal (x , y │ │ │ │ +00064ca0: 2a7a 2920 2020 2020 2020 2020 2020 2020 *z) │ │ │ │ 00064cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064cd0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064cc0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00064cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064d00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064d10: 0a7c 6f33 203a 2049 6465 616c 206f 6620 .|o3 : Ideal of │ │ │ │ -00064d20: 5320 2020 2020 2020 2020 2020 2020 2020 S │ │ │ │ +00064d00: 2020 2020 7c0a 7c6f 3320 3a20 4964 6561 |.|o3 : Idea │ │ │ │ +00064d10: 6c20 6f66 2053 2020 2020 2020 2020 2020 l of S │ │ │ │ +00064d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064d40: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00064d40: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00064d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064d80: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 -------+.|i4 : R │ │ │ │ -00064d90: 3220 3d20 532f 4932 2020 2020 2020 2020 2 = S/I2 │ │ │ │ +00064d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00064d80: 3420 3a20 5232 203d 2053 2f49 3220 2020 4 : R2 = S/I2 │ │ │ │ +00064d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064dc0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064db0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00064dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064e00: 0a7c 6f34 203d 2052 3220 2020 2020 2020 .|o4 = R2 │ │ │ │ +00064df0: 2020 2020 7c0a 7c6f 3420 3d20 5232 2020 |.|o4 = R2 │ │ │ │ +00064e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064e30: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00064e30: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00064e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064e70: 2020 2020 2020 207c 0a7c 6f34 203a 2051 |.|o4 : Q │ │ │ │ -00064e80: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +00064e60: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00064e70: 3420 3a20 5175 6f74 6965 6e74 5269 6e67 4 : QuotientRing │ │ │ │ +00064e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064e90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064eb0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00064ea0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00064eb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00064ed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00064ee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00064ef0: 0a7c 6935 203a 204d 3220 3d20 5232 5e31 .|i5 : M2 = R2^1 │ │ │ │ -00064f00: 2f69 6465 616c 2278 322c 792c 7a22 2020 /ideal"x2,y,z" │ │ │ │ +00064ee0: 2d2d 2d2d 2b0a 7c69 3520 3a20 4d32 203d ----+.|i5 : M2 = │ │ │ │ +00064ef0: 2052 325e 312f 6964 6561 6c22 7832 2c79 R2^1/ideal"x2,y │ │ │ │ +00064f00: 2c7a 2220 2020 2020 2020 2020 2020 2020 ,z" │ │ │ │ 00064f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064f20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00064f20: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00064f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064f60: 2020 2020 2020 207c 0a7c 6f35 203d 2063 |.|o5 = c │ │ │ │ -00064f70: 6f6b 6572 6e65 6c20 7c20 7832 2079 207a okernel | x2 y z │ │ │ │ -00064f80: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -00064f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064fa0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00064f50: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00064f60: 3520 3d20 636f 6b65 726e 656c 207c 2078 5 = cokernel | x │ │ │ │ +00064f70: 3220 7920 7a20 7c20 2020 2020 2020 2020 2 y z | │ │ │ │ +00064f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064f90: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00064fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00064fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00064fd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00064fe0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ -00064ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065000: 3120 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ -00065010: 2020 2020 2020 2020 2020 207c 0a7c 6f35 |.|o5 │ │ │ │ -00065020: 203a 2052 322d 6d6f 6475 6c65 2c20 7175 : R2-module, qu │ │ │ │ -00065030: 6f74 6965 6e74 206f 6620 5232 2020 2020 otient of R2 │ │ │ │ -00065040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065050: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00064fd0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00064fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00064ff0: 2020 2020 2031 2020 2020 2020 2020 2020 1 │ │ │ │ +00065000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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|.|o │ │ │ │ +00065320: 3620 3a20 4265 7474 6954 616c 6c79 2020 6 : BettiTally │ │ │ │ +00065330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065360: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +00065350: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00065360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -000653a0: 0a7c 6937 203a 2045 203d 2045 7874 4d6f .|i7 : E = ExtMo │ │ │ │ -000653b0: 6475 6c65 204d 3220 2020 2020 2020 2020 dule M2 │ │ │ │ +00065390: 2d2d 2d2d 2b0a 7c69 3720 3a20 4520 3d20 ----+.|i7 : E = │ │ │ │ +000653a0: 4578 744d 6f64 756c 6520 4d32 2020 2020 ExtModule M2 │ │ │ │ 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+000657a0: 2020 2020 3420 2020 2020 2020 2020 2020 4 │ │ │ │ 000657b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000657c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000657d0: 2020 2020 2020 207c 0a7c 6f39 203d 2028 |.|o9 = ( │ │ │ │ -000657e0: 6b6b 5b58 202e 2e58 205d 2920 2020 2020 kk[X ..X ]) │ │ │ │ +000657c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000657d0: 3920 3d20 286b 6b5b 5820 2e2e 5820 5d29 9 = (kk[X ..X ]) │ │ │ │ +000657e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000657f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065810: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ -00065820: 3020 2020 3120 2020 2020 2020 2020 2020 0 1 │ │ │ │ +00065800: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00065810: 2020 2020 2030 2020 2031 2020 2020 2020 0 1 │ │ │ │ +00065820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065840: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00065850: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00065840: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00065850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065880: 2020 2020 2020 2020 2020 207c 0a7c 6f39 |.|o9 │ │ │ │ -00065890: 203a 206b 6b5b 5820 2e2e 5820 5d2d 6d6f : kk[X ..X ]-mo │ │ │ │ -000658a0: 6475 6c65 2c20 6672 6565 2c20 6465 6772 dule, free, degr │ │ │ │ -000658b0: 6565 7320 7b33 3a30 2c20 317d 2020 2020 ees {3:0, 1} │ │ │ │ -000658c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000658d0: 2020 2030 2020 2031 2020 2020 2020 2020 0 1 │ │ │ │ +00065880: 7c0a 7c6f 3920 3a20 6b6b 5b58 202e 2e58 |.|o9 : kk[X ..X │ │ │ │ +00065890: 205d 2d6d 6f64 756c 652c 2066 7265 652c ]-module, free, │ │ │ │ +000658a0: 2064 6567 7265 6573 207b 333a 302c 2031 degrees {3:0, 1 │ │ │ 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6464 2929 2020 207c 0a7c 2020 , Eodd)) |.| │ │ │ │ +00065930: 2d2d 2d2d 2b0a 7c69 3130 203a 2061 7070 ----+.|i10 : app │ │ │ │ +00065940: 6c79 2874 6f4c 6973 7428 302e 2e35 292c ly(toList(0..5), │ │ │ │ +00065950: 2069 2d3e 6869 6c62 6572 7446 756e 6374 i->hilbertFunct │ │ │ │ +00065960: 696f 6e28 692c 2045 6f64 6429 2920 2020 ion(i, Eodd)) │ │ │ │ +00065970: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00065980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000659a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000659b0: 2020 2020 2020 207c 0a7c 6f31 3020 3d20 |.|o10 = │ │ │ │ -000659c0: 7b33 2c20 372c 2031 312c 2031 352c 2031 {3, 7, 11, 15, 1 │ │ │ │ -000659d0: 392c 2032 337d 2020 2020 2020 2020 2020 9, 23} │ │ │ │ -000659e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000659f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000659a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000659b0: 3130 203d 207b 332c 2037 2c20 3131 2c20 10 = {3, 7, 11, │ │ │ │ +000659c0: 3135 2c20 3139 2c20 3233 7d20 2020 2020 15, 19, 23} │ │ │ │ +000659d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000659e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +000659f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065a20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00065a30: 0a7c 6f31 3020 3a20 4c69 7374 2020 2020 .|o10 : List │ │ │ │ +00065a20: 2020 2020 7c0a 7c6f 3130 203a 204c 6973 |.|o10 : Lis │ │ │ │ +00065a30: 7420 2020 2020 2020 2020 2020 2020 2020 t │ │ │ │ 00065a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00065a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00065a60: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +00065a60: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 00065a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065aa0: 2d2d 2d2d 2d2d 2d2b 0a0a 5365 6520 616c -------+..See al │ │ │ │ -00065ab0: 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a so.========.. * │ │ │ │ -00065ac0: 202a 6e6f 7465 2045 7874 4d6f 6475 6c65 *note ExtModule │ │ │ │ -00065ad0: 3a20 4578 744d 6f64 756c 652c 202d 2d20 : ExtModule, -- │ │ │ │ -00065ae0: 4578 745e 2a28 4d2c 6b29 206f 7665 7220 Ext^*(M,k) over │ │ │ │ -00065af0: 6120 636f 6d70 6c65 7465 2069 6e74 6572 a complete inter │ │ │ │ -00065b00: 7365 6374 696f 6e20 6173 0a20 2020 206d section as. m │ │ │ │ -00065b10: 6f64 756c 6520 6f76 6572 2043 4920 6f70 odule over CI op │ │ │ │ -00065b20: 6572 6174 6f72 2072 696e 670a 2020 2a20 erator ring. * │ │ │ │ -00065b30: 2a6e 6f74 6520 6576 656e 4578 744d 6f64 *note evenExtMod │ │ │ │ -00065b40: 756c 653a 2065 7665 6e45 7874 4d6f 6475 ule: evenExtModu │ │ │ │ -00065b50: 6c65 2c20 2d2d 2065 7665 6e20 7061 7274 le, -- even part │ │ │ │ -00065b60: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ -00065b70: 7665 7220 610a 2020 2020 636f 6d70 6c65 ver a. comple │ │ │ │ -00065b80: 7465 2069 6e74 6572 7365 6374 696f 6e20 te intersection │ │ │ │ -00065b90: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ -00065ba0: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ -00065bb0: 2020 2a20 2a6e 6f74 6520 4f75 7452 696e * *note OutRin │ │ │ │ -00065bc0: 673a 204f 7574 5269 6e67 2c20 2d2d 204f g: OutRing, -- O │ │ │ │ -00065bd0: 7074 696f 6e20 616c 6c6f 7769 6e67 2073 ption allowing s │ │ │ │ -00065be0: 7065 6369 6669 6361 7469 6f6e 206f 6620 pecification of │ │ │ │ -00065bf0: 7468 6520 7269 6e67 206f 7665 720a 2020 the ring over. │ │ │ │ -00065c00: 2020 7768 6963 6820 7468 6520 6f75 7470 which the outp │ │ │ │ -00065c10: 7574 2069 7320 6465 6669 6e65 640a 0a57 ut is defined..W │ │ │ │ -00065c20: 6179 7320 746f 2075 7365 206f 6464 4578 ays to use oddEx │ │ │ │ -00065c30: 744d 6f64 756c 653a 0a3d 3d3d 3d3d 3d3d tModule:.======= │ │ │ │ -00065c40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00065c50: 3d3d 0a0a 2020 2a20 226f 6464 4578 744d ==.. * "oddExtM │ │ │ │ -00065c60: 6f64 756c 6528 4d6f 6475 6c65 2922 0a0a odule(Module)".. │ │ │ │ -00065c70: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -00065c80: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -00065c90: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -00065ca0: 7420 2a6e 6f74 6520 6f64 6445 7874 4d6f t *note oddExtMo │ │ │ │ -00065cb0: 6475 6c65 3a20 6f64 6445 7874 4d6f 6475 dule: oddExtModu │ │ │ │ -00065cc0: 6c65 2c20 6973 2061 202a 6e6f 7465 206d le, is a *note m │ │ │ │ -00065cd0: 6574 686f 6420 6675 6e63 7469 6f6e 2077 ethod function w │ │ │ │ -00065ce0: 6974 680a 6f70 7469 6f6e 733a 2028 4d61 ith.options: (Ma │ │ │ │ -00065cf0: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ -00065d00: 6446 756e 6374 696f 6e57 6974 684f 7074 dFunctionWithOpt │ │ │ │ -00065d10: 696f 6e73 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d ions,...-------- │ │ │ │ +00065a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +00065aa0: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +00065ab0: 0a0a 2020 2a20 2a6e 6f74 6520 4578 744d .. * *note ExtM │ │ │ │ +00065ac0: 6f64 756c 653a 2045 7874 4d6f 6475 6c65 odule: ExtModule │ │ │ │ +00065ad0: 2c20 2d2d 2045 7874 5e2a 284d 2c6b 2920 , -- Ext^*(M,k) │ │ │ │ +00065ae0: 6f76 6572 2061 2063 6f6d 706c 6574 6520 over a complete │ │ │ │ +00065af0: 696e 7465 7273 6563 7469 6f6e 2061 730a intersection as. │ │ │ │ +00065b00: 2020 2020 6d6f 6475 6c65 206f 7665 7220 module over │ │ │ │ +00065b10: 4349 206f 7065 7261 746f 7220 7269 6e67 CI operator ring │ │ │ │ +00065b20: 0a20 202a 202a 6e6f 7465 2065 7665 6e45 . * *note evenE │ │ │ │ +00065b30: 7874 4d6f 6475 6c65 3a20 6576 656e 4578 xtModule: evenEx │ │ │ │ +00065b40: 744d 6f64 756c 652c 202d 2d20 6576 656e tModule, -- even │ │ │ │ +00065b50: 2070 6172 7420 6f66 2045 7874 5e2a 284d part of Ext^*(M │ │ │ │ +00065b60: 2c6b 2920 6f76 6572 2061 0a20 2020 2063 ,k) over a. c │ │ │ │ +00065b70: 6f6d 706c 6574 6520 696e 7465 7273 6563 omplete intersec │ │ │ │ +00065b80: 7469 6f6e 2061 7320 6d6f 6475 6c65 206f tion as module o │ │ │ │ +00065b90: 7665 7220 4349 206f 7065 7261 746f 7220 ver CI operator │ │ │ │ +00065ba0: 7269 6e67 0a20 202a 202a 6e6f 7465 204f ring. * *note O │ │ │ │ +00065bb0: 7574 5269 6e67 3a20 4f75 7452 696e 672c utRing: OutRing, │ │ │ │ +00065bc0: 202d 2d20 4f70 7469 6f6e 2061 6c6c 6f77 -- Option allow │ │ │ │ +00065bd0: 696e 6720 7370 6563 6966 6963 6174 696f ing specificatio │ │ │ │ +00065be0: 6e20 6f66 2074 6865 2072 696e 6720 6f76 n of the ring ov │ │ │ │ +00065bf0: 6572 0a20 2020 2077 6869 6368 2074 6865 er. which the │ │ │ │ +00065c00: 206f 7574 7075 7420 6973 2064 6566 696e output is defin │ │ │ │ +00065c10: 6564 0a0a 5761 7973 2074 6f20 7573 6520 ed..Ways to use │ │ │ │ +00065c20: 6f64 6445 7874 4d6f 6475 6c65 3a0a 3d3d oddExtModule:.== │ │ │ │ +00065c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00065c40: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 6f64 =======.. * "od │ │ │ │ +00065c50: 6445 7874 4d6f 6475 6c65 284d 6f64 756c dExtModule(Modul │ │ │ │ +00065c60: 6529 220a 0a46 6f72 2074 6865 2070 726f e)"..For the pro │ │ │ │ +00065c70: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ +00065c80: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ +00065c90: 6f62 6a65 6374 202a 6e6f 7465 206f 6464 object *note odd │ │ │ │ +00065ca0: 4578 744d 6f64 756c 653a 206f 6464 4578 ExtModule: oddEx │ │ │ │ +00065cb0: 744d 6f64 756c 652c 2069 7320 6120 2a6e tModule, is a *n │ │ │ │ +00065cc0: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ +00065cd0: 696f 6e20 7769 7468 0a6f 7074 696f 6e73 ion with.options │ │ │ │ +00065ce0: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00065cf0: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ +00065d00: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ +00065d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065d30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00065d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00065d60: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00065d70: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00065d80: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00065d90: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00065da0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00065db0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -00065dc0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00065dd0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00065de0: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00065df0: 6d32 3a33 3637 393a 302e 0a1f 0a46 696c m2:3679:0....Fil │ │ │ │ -00065e00: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -00065e10: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00065e20: 6e73 2e69 6e66 6f2c 204e 6f64 653a 204f ns.info, Node: O │ │ │ │ -00065e30: 7074 696d 6973 6d2c 204e 6578 743a 204f ptimism, Next: O │ │ │ │ -00065e40: 7574 5269 6e67 2c20 5072 6576 3a20 6f64 utRing, Prev: od │ │ │ │ -00065e50: 6445 7874 4d6f 6475 6c65 2c20 5570 3a20 dExtModule, Up: │ │ │ │ -00065e60: 546f 700a 0a4f 7074 696d 6973 6d20 2d2d Top..Optimism -- │ │ │ │ -00065e70: 204f 7074 696f 6e20 746f 2068 6967 6853 Option to highS │ │ │ │ -00065e80: 797a 7967 790a 2a2a 2a2a 2a2a 2a2a 2a2a yzygy.********** │ │ │ │ -00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00065ea0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00065eb0: 653a 200a 2020 2020 2020 2020 6869 6768 e: . high │ │ │ │ -00065ec0: 5379 7a79 6779 284d 2c20 4f70 7469 6d69 Syzygy(M, Optimi │ │ │ │ -00065ed0: 736d 203d 3e20 3129 0a20 202a 2049 6e70 sm => 1). * Inp │ │ │ │ -00065ee0: 7574 733a 0a20 2020 2020 202a 204f 7074 uts:. * Opt │ │ │ │ -00065ef0: 696d 6973 6d2c 2061 6e20 2a6e 6f74 6520 imism, an *note │ │ │ │ -00065f00: 696e 7465 6765 723a 2028 4d61 6361 756c integer: (Macaul │ │ │ │ -00065f10: 6179 3244 6f63 295a 5a2c 2c20 0a0a 4465 ay2Doc)ZZ,, ..De │ │ │ │ -00065f20: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -00065f30: 3d3d 3d3d 3d0a 0a49 6620 6869 6768 5379 =====..If highSy │ │ │ │ -00065f40: 7a79 6779 284d 2920 6368 6f6f 7365 7320 zygy(M) chooses │ │ │ │ -00065f50: 7468 6520 702d 7468 2073 797a 7967 792c the p-th syzygy, │ │ │ │ -00065f60: 2074 6865 6e20 6869 6768 5379 7a79 6779 then highSyzygy │ │ │ │ -00065f70: 284d 2c4f 7074 696d 6973 6d3d 3e72 290a (M,Optimism=>r). │ │ │ │ -00065f80: 6368 6f6f 7365 7320 7468 6520 2870 2d72 chooses the (p-r │ │ │ │ -00065f90: 292d 7468 2073 797a 7967 792e 2028 506f )-th syzygy. (Po │ │ │ │ -00065fa0: 7369 7469 7665 204f 7074 696d 6973 6d20 sitive Optimism │ │ │ │ -00065fb0: 6368 6f6f 7365 7320 6120 6c6f 7765 7220 chooses a lower │ │ │ │ -00065fc0: 2268 6967 6822 2073 797a 7967 792c 0a6e "high" syzygy,.n │ │ │ │ -00065fd0: 6567 6174 6976 6520 4f70 7469 6d69 736d egative Optimism │ │ │ │ -00065fe0: 2061 2068 6967 6865 7220 2268 6967 6822 a higher "high" │ │ │ │ -00065ff0: 2073 797a 7967 792e 0a0a 4361 7665 6174 syzygy...Caveat │ │ │ │ -00066000: 0a3d 3d3d 3d3d 3d0a 0a41 7265 2074 6865 .======..Are the │ │ │ │ -00066010: 7265 2063 6173 6573 2077 6865 6e20 706f re cases when po │ │ │ │ -00066020: 7369 7469 7665 204f 7074 696d 6973 6d20 sitive Optimism │ │ │ │ -00066030: 6973 206a 7573 7469 6669 6564 3f0a 0a53 is justified?..S │ │ │ │ -00066040: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ -00066050: 0a0a 2020 2a20 2a6e 6f74 6520 6d66 426f .. * *note mfBo │ │ │ │ -00066060: 756e 643a 206d 6642 6f75 6e64 2c20 2d2d und: mfBound, -- │ │ │ │ -00066070: 2064 6574 6572 6d69 6e65 7320 686f 7720 determines how │ │ │ │ -00066080: 6869 6768 2061 2073 797a 7967 7920 746f high a syzygy to │ │ │ │ -00066090: 2074 616b 6520 666f 720a 2020 2020 226d take for. "m │ │ │ │ -000660a0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -000660b0: 6f6e 220a 2020 2a20 2a6e 6f74 6520 6869 on". * *note hi │ │ │ │ -000660c0: 6768 5379 7a79 6779 3a20 6869 6768 5379 ghSyzygy: highSy │ │ │ │ -000660d0: 7a79 6779 2c20 2d2d 2052 6574 7572 6e73 zygy, -- Returns │ │ │ │ -000660e0: 2061 2073 797a 7967 7920 6d6f 6475 6c65 a syzygy module │ │ │ │ -000660f0: 206f 6e65 2062 6579 6f6e 6420 7468 650a one beyond the. │ │ │ │ -00066100: 2020 2020 7265 6775 6c61 7269 7479 206f regularity o │ │ │ │ -00066110: 6620 4578 7428 4d2c 6b29 0a0a 4675 6e63 f Ext(M,k)..Func │ │ │ │ -00066120: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -00066130: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -00066140: 6564 204f 7074 696d 6973 6d3a 0a3d 3d3d ed Optimism:.=== │ │ │ │ +00065d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00065d60: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00065d70: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00065d80: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00065d90: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00065da0: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00065db0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00065dc0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00065dd0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00065de0: 696f 6e73 2e6d 323a 3336 3739 3a30 2e0a ions.m2:3679:0.. │ │ │ │ +00065df0: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00065e00: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00065e10: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00065e20: 6465 3a20 4f70 7469 6d69 736d 2c20 4e65 de: Optimism, Ne │ │ │ │ +00065e30: 7874 3a20 4f75 7452 696e 672c 2050 7265 xt: OutRing, Pre │ │ │ │ +00065e40: 763a 206f 6464 4578 744d 6f64 756c 652c v: oddExtModule, │ │ │ │ +00065e50: 2055 703a 2054 6f70 0a0a 4f70 7469 6d69 Up: Top..Optimi │ │ │ │ +00065e60: 736d 202d 2d20 4f70 7469 6f6e 2074 6f20 sm -- Option to │ │ │ │ +00065e70: 6869 6768 5379 7a79 6779 0a2a 2a2a 2a2a highSyzygy.***** │ │ │ │ +00065e80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00065e90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ +00065ea0: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ +00065eb0: 2068 6967 6853 797a 7967 7928 4d2c 204f highSyzygy(M, O │ │ │ │ +00065ec0: 7074 696d 6973 6d20 3d3e 2031 290a 2020 ptimism => 1). │ │ │ │ +00065ed0: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ +00065ee0: 2a20 4f70 7469 6d69 736d 2c20 616e 202a * Optimism, an * │ │ │ │ +00065ef0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +00065f00: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +00065f10: 200a 0a44 6573 6372 6970 7469 6f6e 0a3d ..Description.= │ │ │ │ +00065f20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 2068 ==========..If h │ │ │ │ +00065f30: 6967 6853 797a 7967 7928 4d29 2063 686f ighSyzygy(M) cho │ │ │ │ +00065f40: 6f73 6573 2074 6865 2070 2d74 6820 7379 oses the p-th sy │ │ │ │ +00065f50: 7a79 6779 2c20 7468 656e 2068 6967 6853 zygy, then highS │ │ │ │ +00065f60: 797a 7967 7928 4d2c 4f70 7469 6d69 736d yzygy(M,Optimism │ │ │ │ +00065f70: 3d3e 7229 0a63 686f 6f73 6573 2074 6865 =>r).chooses the │ │ │ │ +00065f80: 2028 702d 7229 2d74 6820 7379 7a79 6779 (p-r)-th syzygy │ │ │ │ +00065f90: 2e20 2850 6f73 6974 6976 6520 4f70 7469 . (Positive Opti │ │ │ │ +00065fa0: 6d69 736d 2063 686f 6f73 6573 2061 206c mism chooses a l │ │ │ │ +00065fb0: 6f77 6572 2022 6869 6768 2220 7379 7a79 ower "high" syzy │ │ │ │ +00065fc0: 6779 2c0a 6e65 6761 7469 7665 204f 7074 gy,.negative Opt │ │ │ │ +00065fd0: 696d 6973 6d20 6120 6869 6768 6572 2022 imism a higher " │ │ │ │ +00065fe0: 6869 6768 2220 7379 7a79 6779 2e0a 0a43 high" syzygy...C │ │ │ │ +00065ff0: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4172 aveat.======..Ar │ │ │ │ +00066000: 6520 7468 6572 6520 6361 7365 7320 7768 e there cases wh │ │ │ │ +00066010: 656e 2070 6f73 6974 6976 6520 4f70 7469 en positive Opti │ │ │ │ +00066020: 6d69 736d 2069 7320 6a75 7374 6966 6965 mism is justifie │ │ │ │ +00066030: 643f 0a0a 5365 6520 616c 736f 0a3d 3d3d d?..See also.=== │ │ │ │ +00066040: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +00066050: 206d 6642 6f75 6e64 3a20 6d66 426f 756e mfBound: mfBoun │ │ │ │ +00066060: 642c 202d 2d20 6465 7465 726d 696e 6573 d, -- determines │ │ │ │ +00066070: 2068 6f77 2068 6967 6820 6120 7379 7a79 how high a syzy │ │ │ │ +00066080: 6779 2074 6f20 7461 6b65 2066 6f72 0a20 gy to take for. │ │ │ │ +00066090: 2020 2022 6d61 7472 6978 4661 6374 6f72 "matrixFactor │ │ │ │ +000660a0: 697a 6174 696f 6e22 0a20 202a 202a 6e6f ization". * *no │ │ │ │ +000660b0: 7465 2068 6967 6853 797a 7967 793a 2068 te highSyzygy: h │ │ │ │ +000660c0: 6967 6853 797a 7967 792c 202d 2d20 5265 ighSyzygy, -- Re │ │ │ │ +000660d0: 7475 726e 7320 6120 7379 7a79 6779 206d turns a syzygy m │ │ │ │ +000660e0: 6f64 756c 6520 6f6e 6520 6265 796f 6e64 odule one beyond │ │ │ │ +000660f0: 2074 6865 0a20 2020 2072 6567 756c 6172 the. regular │ │ │ │ +00066100: 6974 7920 6f66 2045 7874 284d 2c6b 290a ity of Ext(M,k). │ │ │ │ +00066110: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ +00066120: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +00066130: 7420 6e61 6d65 6420 4f70 7469 6d69 736d t named Optimism │ │ │ │ +00066140: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ 00066150: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00066160: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00066170: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ -00066180: 202a 2022 6869 6768 5379 7a79 6779 282e * "highSyzygy(. │ │ │ │ -00066190: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e ..,Optimism=>... │ │ │ │ -000661a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -000661b0: 6869 6768 5379 7a79 6779 3a20 6869 6768 highSyzygy: high │ │ │ │ -000661c0: 5379 7a79 6779 2c20 2d2d 0a20 2020 2052 Syzygy, --. R │ │ │ │ -000661d0: 6574 7572 6e73 2061 2073 797a 7967 7920 eturns a syzygy │ │ │ │ -000661e0: 6d6f 6475 6c65 206f 6e65 2062 6579 6f6e module one beyon │ │ │ │ -000661f0: 6420 7468 6520 7265 6775 6c61 7269 7479 d the regularity │ │ │ │ -00066200: 206f 6620 4578 7428 4d2c 6b29 0a20 202a of Ext(M,k). * │ │ │ │ -00066210: 2022 7477 6f4d 6f6e 6f6d 6961 6c73 282e "twoMonomials(. │ │ │ │ -00066220: 2e2e 2c4f 7074 696d 6973 6d3d 3e2e 2e2e ..,Optimism=>... │ │ │ │ -00066230: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00066240: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 twoMonomials: tw │ │ │ │ -00066250: 6f4d 6f6e 6f6d 6961 6c73 2c0a 2020 2020 oMonomials,. │ │ │ │ -00066260: 2d2d 2074 616c 6c79 2074 6865 2073 6571 -- tally the seq │ │ │ │ -00066270: 7565 6e63 6573 206f 6620 4252 616e 6b73 uences of BRanks │ │ │ │ -00066280: 2066 6f72 2063 6572 7461 696e 2065 7861 for certain exa │ │ │ │ -00066290: 6d70 6c65 730a 0a46 6f72 2074 6865 2070 mples..For the p │ │ │ │ -000662a0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ -000662b0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ -000662c0: 6520 6f62 6a65 6374 202a 6e6f 7465 204f e object *note O │ │ │ │ -000662d0: 7074 696d 6973 6d3a 204f 7074 696d 6973 ptimism: Optimis │ │ │ │ -000662e0: 6d2c 2069 7320 6120 2a6e 6f74 6520 7379 m, is a *note sy │ │ │ │ -000662f0: 6d62 6f6c 3a20 284d 6163 6175 6c61 7932 mbol: (Macaulay2 │ │ │ │ -00066300: 446f 6329 5379 6d62 6f6c 2c2e 0a0a 2d2d Doc)Symbol,...-- │ │ │ │ +00066170: 3d3d 0a0a 2020 2a20 2268 6967 6853 797a ==.. * "highSyz │ │ │ │ +00066180: 7967 7928 2e2e 2e2c 4f70 7469 6d69 736d ygy(...,Optimism │ │ │ │ +00066190: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +000661a0: 6e6f 7465 2068 6967 6853 797a 7967 793a note highSyzygy: │ │ │ │ +000661b0: 2068 6967 6853 797a 7967 792c 202d 2d0a highSyzygy, --. │ │ │ │ +000661c0: 2020 2020 5265 7475 726e 7320 6120 7379 Returns a sy │ │ │ │ +000661d0: 7a79 6779 206d 6f64 756c 6520 6f6e 6520 zygy module one │ │ │ │ +000661e0: 6265 796f 6e64 2074 6865 2072 6567 756c beyond the regul │ │ │ │ +000661f0: 6172 6974 7920 6f66 2045 7874 284d 2c6b arity of Ext(M,k │ │ │ │ +00066200: 290a 2020 2a20 2274 776f 4d6f 6e6f 6d69 ). * "twoMonomi │ │ │ │ +00066210: 616c 7328 2e2e 2e2c 4f70 7469 6d69 736d als(...,Optimism │ │ │ │ +00066220: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +00066230: 6e6f 7465 2074 776f 4d6f 6e6f 6d69 616c note twoMonomial │ │ │ │ +00066240: 733a 2074 776f 4d6f 6e6f 6d69 616c 732c s: twoMonomials, │ │ │ │ +00066250: 0a20 2020 202d 2d20 7461 6c6c 7920 7468 . -- tally th │ │ │ │ +00066260: 6520 7365 7175 656e 6365 7320 6f66 2042 e sequences of B │ │ │ │ +00066270: 5261 6e6b 7320 666f 7220 6365 7274 6169 Ranks for certai │ │ │ │ +00066280: 6e20 6578 616d 706c 6573 0a0a 466f 7220 n examples..For │ │ │ │ +00066290: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +000662a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +000662b0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +000662c0: 6f74 6520 4f70 7469 6d69 736d 3a20 4f70 ote Optimism: Op │ │ │ │ +000662d0: 7469 6d69 736d 2c20 6973 2061 202a 6e6f timism, is a *no │ │ │ │ +000662e0: 7465 2073 796d 626f 6c3a 2028 4d61 6361 te symbol: (Maca │ │ │ │ +000662f0: 756c 6179 3244 6f63 2953 796d 626f 6c2c ulay2Doc)Symbol, │ │ │ │ +00066300: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 00066310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -00066360: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -00066370: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -00066380: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -00066390: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -000663a0: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -000663b0: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -000663c0: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -000663d0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -000663e0: 7469 6f6e 732e 6d32 3a33 3136 353a 302e tions.m2:3165:0. │ │ │ │ -000663f0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -00066400: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -00066410: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -00066420: 6f64 653a 204f 7574 5269 6e67 2c20 4e65 ode: OutRing, Ne │ │ │ │ -00066430: 7874 3a20 7073 694d 6170 732c 2050 7265 xt: psiMaps, Pre │ │ │ │ -00066440: 763a 204f 7074 696d 6973 6d2c 2055 703a v: Optimism, Up: │ │ │ │ -00066450: 2054 6f70 0a0a 4f75 7452 696e 6720 2d2d Top..OutRing -- │ │ │ │ -00066460: 204f 7074 696f 6e20 616c 6c6f 7769 6e67 Option allowing │ │ │ │ -00066470: 2073 7065 6369 6669 6361 7469 6f6e 206f specification o │ │ │ │ -00066480: 6620 7468 6520 7269 6e67 206f 7665 7220 f the ring over │ │ │ │ -00066490: 7768 6963 6820 7468 6520 6f75 7470 7574 which the output │ │ │ │ -000664a0: 2069 7320 6465 6669 6e65 640a 2a2a 2a2a is defined.**** │ │ │ │ +00066350: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +00066360: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +00066370: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +00066380: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +00066390: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +000663a0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +000663b0: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +000663c0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +000663d0: 6573 6f6c 7574 696f 6e73 2e6d 323a 3331 esolutions.m2:31 │ │ │ │ +000663e0: 3635 3a30 2e0a 1f0a 4669 6c65 3a20 436f 65:0....File: Co │ │ │ │ +000663f0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +00066400: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +00066410: 666f 2c20 4e6f 6465 3a20 4f75 7452 696e fo, Node: OutRin │ │ │ │ +00066420: 672c 204e 6578 743a 2070 7369 4d61 7073 g, Next: psiMaps │ │ │ │ +00066430: 2c20 5072 6576 3a20 4f70 7469 6d69 736d , Prev: Optimism │ │ │ │ +00066440: 2c20 5570 3a20 546f 700a 0a4f 7574 5269 , Up: Top..OutRi │ │ │ │ +00066450: 6e67 202d 2d20 4f70 7469 6f6e 2061 6c6c ng -- Option all │ │ │ │ +00066460: 6f77 696e 6720 7370 6563 6966 6963 6174 owing specificat │ │ │ │ +00066470: 696f 6e20 6f66 2074 6865 2072 696e 6720 ion of the ring │ │ │ │ +00066480: 6f76 6572 2077 6869 6368 2074 6865 206f over which the o │ │ │ │ +00066490: 7574 7075 7420 6973 2064 6566 696e 6564 utput is defined │ │ │ │ +000664a0: 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a .*************** │ │ │ │ 000664b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000664c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000664d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000664e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000664f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00066500: 2a0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d *..See also.==== │ │ │ │ -00066510: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00066520: 6576 656e 4578 744d 6f64 756c 653a 2065 evenExtModule: e │ │ │ │ -00066530: 7665 6e45 7874 4d6f 6475 6c65 2c20 2d2d venExtModule, -- │ │ │ │ -00066540: 2065 7665 6e20 7061 7274 206f 6620 4578 even part of Ex │ │ │ │ -00066550: 745e 2a28 4d2c 6b29 206f 7665 7220 610a t^*(M,k) over a. │ │ │ │ -00066560: 2020 2020 636f 6d70 6c65 7465 2069 6e74 complete int │ │ │ │ -00066570: 6572 7365 6374 696f 6e20 6173 206d 6f64 ersection as mod │ │ │ │ -00066580: 756c 6520 6f76 6572 2043 4920 6f70 6572 ule over CI oper │ │ │ │ -00066590: 6174 6f72 2072 696e 670a 2020 2a20 2a6e ator ring. * *n │ │ │ │ -000665a0: 6f74 6520 6f64 6445 7874 4d6f 6475 6c65 ote oddExtModule │ │ │ │ -000665b0: 3a20 6f64 6445 7874 4d6f 6475 6c65 2c20 : oddExtModule, │ │ │ │ -000665c0: 2d2d 206f 6464 2070 6172 7420 6f66 2045 -- odd part of E │ │ │ │ -000665d0: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -000665e0: 2063 6f6d 706c 6574 650a 2020 2020 696e complete. in │ │ │ │ -000665f0: 7465 7273 6563 7469 6f6e 2061 7320 6d6f tersection as mo │ │ │ │ -00066600: 6475 6c65 206f 7665 7220 4349 206f 7065 dule over CI ope │ │ │ │ -00066610: 7261 746f 7220 7269 6e67 0a0a 4675 6e63 rator ring..Func │ │ │ │ -00066620: 7469 6f6e 7320 7769 7468 206f 7074 696f tions with optio │ │ │ │ -00066630: 6e61 6c20 6172 6775 6d65 6e74 206e 616d nal argument nam │ │ │ │ -00066640: 6564 204f 7574 5269 6e67 3a0a 3d3d 3d3d ed OutRing:.==== │ │ │ │ +000664f0: 2a2a 2a2a 2a2a 0a0a 5365 6520 616c 736f ******..See also │ │ │ │ +00066500: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +00066510: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ +00066520: 6c65 3a20 6576 656e 4578 744d 6f64 756c le: evenExtModul │ │ │ │ +00066530: 652c 202d 2d20 6576 656e 2070 6172 7420 e, -- even part │ │ │ │ +00066540: 6f66 2045 7874 5e2a 284d 2c6b 2920 6f76 of Ext^*(M,k) ov │ │ │ │ +00066550: 6572 2061 0a20 2020 2063 6f6d 706c 6574 er a. complet │ │ │ │ +00066560: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ +00066570: 7320 6d6f 6475 6c65 206f 7665 7220 4349 s module over CI │ │ │ │ +00066580: 206f 7065 7261 746f 7220 7269 6e67 0a20 operator ring. │ │ │ │ +00066590: 202a 202a 6e6f 7465 206f 6464 4578 744d * *note oddExtM │ │ │ │ +000665a0: 6f64 756c 653a 206f 6464 4578 744d 6f64 odule: oddExtMod │ │ │ │ +000665b0: 756c 652c 202d 2d20 6f64 6420 7061 7274 ule, -- odd part │ │ │ │ +000665c0: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +000665d0: 7665 7220 6120 636f 6d70 6c65 7465 0a20 ver a complete. │ │ │ │ +000665e0: 2020 2069 6e74 6572 7365 6374 696f 6e20 intersection │ │ │ │ +000665f0: 6173 206d 6f64 756c 6520 6f76 6572 2043 as module over C │ │ │ │ +00066600: 4920 6f70 6572 6174 6f72 2072 696e 670a I operator ring. │ │ │ │ +00066610: 0a46 756e 6374 696f 6e73 2077 6974 6820 .Functions with │ │ │ │ +00066620: 6f70 7469 6f6e 616c 2061 7267 756d 656e optional argumen │ │ │ │ +00066630: 7420 6e61 6d65 6420 4f75 7452 696e 673a t named OutRing: │ │ │ │ +00066640: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ 00066650: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00066660: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00066670: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00066680: 2022 6576 656e 4578 744d 6f64 756c 6528 "evenExtModule( │ │ │ │ -00066690: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e ...,OutRing=>... │ │ │ │ -000666a0: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -000666b0: 6576 656e 4578 744d 6f64 756c 653a 0a20 evenExtModule:. │ │ │ │ -000666c0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65 evenExtModule │ │ │ │ -000666d0: 2c20 2d2d 2065 7665 6e20 7061 7274 206f , -- even part o │ │ │ │ -000666e0: 6620 4578 745e 2a28 4d2c 6b29 206f 7665 f Ext^*(M,k) ove │ │ │ │ -000666f0: 7220 6120 636f 6d70 6c65 7465 2069 6e74 r a complete int │ │ │ │ -00066700: 6572 7365 6374 696f 6e20 6173 0a20 2020 ersection as. │ │ │ │ -00066710: 206d 6f64 756c 6520 6f76 6572 2043 4920 module over CI │ │ │ │ -00066720: 6f70 6572 6174 6f72 2072 696e 670a 2020 operator ring. │ │ │ │ -00066730: 2a20 226f 6464 4578 744d 6f64 756c 6528 * "oddExtModule( │ │ │ │ -00066740: 2e2e 2e2c 4f75 7452 696e 673d 3e2e 2e2e ...,OutRing=>... │ │ │ │ -00066750: 2922 202d 2d20 7365 6520 2a6e 6f74 6520 )" -- see *note │ │ │ │ -00066760: 6f64 6445 7874 4d6f 6475 6c65 3a20 6f64 oddExtModule: od │ │ │ │ -00066770: 6445 7874 4d6f 6475 6c65 2c0a 2020 2020 dExtModule,. │ │ │ │ -00066780: 2d2d 206f 6464 2070 6172 7420 6f66 2045 -- odd part of E │ │ │ │ -00066790: 7874 5e2a 284d 2c6b 2920 6f76 6572 2061 xt^*(M,k) over a │ │ │ │ -000667a0: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -000667b0: 6563 7469 6f6e 2061 7320 6d6f 6475 6c65 ection as module │ │ │ │ -000667c0: 206f 7665 7220 4349 0a20 2020 206f 7065 over CI. ope │ │ │ │ -000667d0: 7261 746f 7220 7269 6e67 0a0a 466f 7220 rator ring..For │ │ │ │ -000667e0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -000667f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00066800: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00066810: 6f74 6520 4f75 7452 696e 673a 204f 7574 ote OutRing: Out │ │ │ │ -00066820: 5269 6e67 2c20 6973 2061 202a 6e6f 7465 Ring, is a *note │ │ │ │ -00066830: 2073 796d 626f 6c3a 2028 4d61 6361 756c symbol: (Macaul │ │ │ │ -00066840: 6179 3244 6f63 2953 796d 626f 6c2c 2e0a ay2Doc)Symbol,.. │ │ │ │ -00066850: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +00066670: 0a0a 2020 2a20 2265 7665 6e45 7874 4d6f .. * "evenExtMo │ │ │ │ +00066680: 6475 6c65 282e 2e2e 2c4f 7574 5269 6e67 dule(...,OutRing │ │ │ │ +00066690: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +000666a0: 6e6f 7465 2065 7665 6e45 7874 4d6f 6475 note evenExtModu │ │ │ │ +000666b0: 6c65 3a0a 2020 2020 6576 656e 4578 744d le:. evenExtM │ │ │ │ +000666c0: 6f64 756c 652c 202d 2d20 6576 656e 2070 odule, -- even p │ │ │ │ +000666d0: 6172 7420 6f66 2045 7874 5e2a 284d 2c6b art of Ext^*(M,k │ │ │ │ +000666e0: 2920 6f76 6572 2061 2063 6f6d 706c 6574 ) over a complet │ │ │ │ +000666f0: 6520 696e 7465 7273 6563 7469 6f6e 2061 e intersection a │ │ │ │ +00066700: 730a 2020 2020 6d6f 6475 6c65 206f 7665 s. module ove │ │ │ │ +00066710: 7220 4349 206f 7065 7261 746f 7220 7269 r CI operator ri │ │ │ │ +00066720: 6e67 0a20 202a 2022 6f64 6445 7874 4d6f ng. * "oddExtMo │ │ │ │ +00066730: 6475 6c65 282e 2e2e 2c4f 7574 5269 6e67 dule(...,OutRing │ │ │ │ +00066740: 3d3e 2e2e 2e29 2220 2d2d 2073 6565 202a =>...)" -- see * │ │ │ │ +00066750: 6e6f 7465 206f 6464 4578 744d 6f64 756c note oddExtModul │ │ │ │ +00066760: 653a 206f 6464 4578 744d 6f64 756c 652c e: oddExtModule, │ │ │ │ +00066770: 0a20 2020 202d 2d20 6f64 6420 7061 7274 . -- odd part │ │ │ │ +00066780: 206f 6620 4578 745e 2a28 4d2c 6b29 206f of Ext^*(M,k) o │ │ │ │ +00066790: 7665 7220 6120 636f 6d70 6c65 7465 2069 ver a complete i │ │ │ │ +000667a0: 6e74 6572 7365 6374 696f 6e20 6173 206d ntersection as m │ │ │ │ +000667b0: 6f64 756c 6520 6f76 6572 2043 490a 2020 odule over CI. │ │ │ │ +000667c0: 2020 6f70 6572 6174 6f72 2072 696e 670a operator ring. │ │ │ │ +000667d0: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +000667e0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +000667f0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00066800: 6374 202a 6e6f 7465 204f 7574 5269 6e67 ct *note OutRing │ │ │ │ +00066810: 3a20 4f75 7452 696e 672c 2069 7320 6120 : OutRing, is a │ │ │ │ +00066820: 2a6e 6f74 6520 7379 6d62 6f6c 3a20 284d *note symbol: (M │ │ │ │ +00066830: 6163 6175 6c61 7932 446f 6329 5379 6d62 acaulay2Doc)Symb │ │ │ │ +00066840: 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d ol,...---------- │ │ │ │ +00066850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000668a0: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -000668b0: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -000668c0: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -000668d0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -000668e0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -000668f0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -00066900: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ -00066910: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -00066920: 6f6c 7574 696f 6e73 2e6d 323a 3336 3035 olutions.m2:3605 │ │ │ │ -00066930: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ -00066940: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -00066950: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ -00066960: 2c20 4e6f 6465 3a20 7073 694d 6170 732c , Node: psiMaps, │ │ │ │ -00066970: 204e 6578 743a 2072 6567 756c 6172 6974 Next: regularit │ │ │ │ -00066980: 7953 6571 7565 6e63 652c 2050 7265 763a ySequence, Prev: │ │ │ │ -00066990: 204f 7574 5269 6e67 2c20 5570 3a20 546f OutRing, Up: To │ │ │ │ -000669a0: 700a 0a70 7369 4d61 7073 202d 2d20 6c69 p..psiMaps -- li │ │ │ │ -000669b0: 7374 2074 6865 206d 6170 7320 2070 7369 st the maps psi │ │ │ │ -000669c0: 2870 293a 2042 5f31 2870 2920 2d2d 3e20 (p): B_1(p) --> │ │ │ │ -000669d0: 415f 3028 702d 3129 2069 6e20 6120 6d61 A_0(p-1) in a ma │ │ │ │ -000669e0: 7472 6978 4661 6374 6f72 697a 6174 696f trixFactorizatio │ │ │ │ -000669f0: 6e0a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a n.************** │ │ │ │ +00066890: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +000668a0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +000668b0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +000668c0: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +000668d0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +000668e0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +000668f0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ +00066900: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +00066910: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ +00066920: 3a33 3630 353a 302e 0a1f 0a46 696c 653a :3605:0....File: │ │ │ │ +00066930: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ +00066940: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +00066950: 2e69 6e66 6f2c 204e 6f64 653a 2070 7369 .info, Node: psi │ │ │ │ +00066960: 4d61 7073 2c20 4e65 7874 3a20 7265 6775 Maps, Next: regu │ │ │ │ +00066970: 6c61 7269 7479 5365 7175 656e 6365 2c20 laritySequence, │ │ │ │ +00066980: 5072 6576 3a20 4f75 7452 696e 672c 2055 Prev: OutRing, U │ │ │ │ +00066990: 703a 2054 6f70 0a0a 7073 694d 6170 7320 p: Top..psiMaps │ │ │ │ +000669a0: 2d2d 206c 6973 7420 7468 6520 6d61 7073 -- list the maps │ │ │ │ +000669b0: 2020 7073 6928 7029 3a20 425f 3128 7029 psi(p): B_1(p) │ │ │ │ +000669c0: 202d 2d3e 2041 5f30 2870 2d31 2920 696e --> A_0(p-1) in │ │ │ │ +000669d0: 2061 206d 6174 7269 7846 6163 746f 7269 a matrixFactori │ │ │ │ +000669e0: 7a61 7469 6f6e 0a2a 2a2a 2a2a 2a2a 2a2a zation.********* │ │ │ │ +000669f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00066a00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00066a10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00066a20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00066a30: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00066a40: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ -00066a50: 2020 2020 2020 7073 6d61 7073 203d 2070 psmaps = p │ │ │ │ -00066a60: 7369 4d61 7073 206d 660a 2020 2a20 496e siMaps mf. * In │ │ │ │ -00066a70: 7075 7473 3a0a 2020 2020 2020 2a20 6d66 puts:. * mf │ │ │ │ -00066a80: 2c20 6120 2a6e 6f74 6520 6c69 7374 3a20 , a *note list: │ │ │ │ -00066a90: 284d 6163 6175 6c61 7932 446f 6329 4c69 (Macaulay2Doc)Li │ │ │ │ -00066aa0: 7374 2c2c 206f 7574 7075 7420 6f66 2061 st,, output of a │ │ │ │ -00066ab0: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ -00066ac0: 7469 6f6e 0a20 2020 2020 2020 2063 6f6d tion. com │ │ │ │ -00066ad0: 7075 7461 7469 6f6e 0a20 202a 204f 7574 putation. * Out │ │ │ │ -00066ae0: 7075 7473 3a0a 2020 2020 2020 2a20 7073 puts:. * ps │ │ │ │ -00066af0: 6d61 7073 2c20 6120 2a6e 6f74 6520 6c69 maps, a *note li │ │ │ │ -00066b00: 7374 3a20 284d 6163 6175 6c61 7932 446f st: (Macaulay2Do │ │ │ │ -00066b10: 6329 4c69 7374 2c2c 206c 6973 7420 6d61 c)List,, list ma │ │ │ │ -00066b20: 7472 6963 6573 2024 645f 703a 0a20 2020 trices $d_p:. │ │ │ │ -00066b30: 2020 2020 2042 5f31 2870 295c 746f 2041 B_1(p)\to A │ │ │ │ -00066b40: 5f30 2870 2d31 2924 0a0a 4465 7363 7269 _0(p-1)$..Descri │ │ │ │ -00066b50: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ -00066b60: 3d0a 0a53 6565 2074 6865 2064 6f63 756d =..See the docum │ │ │ │ -00066b70: 656e 7461 7469 6f6e 2066 6f72 206d 6174 entation for mat │ │ │ │ -00066b80: 7269 7846 6163 746f 7269 7a61 7469 6f6e rixFactorization │ │ │ │ -00066b90: 2066 6f72 2061 6e20 6578 616d 706c 652e for an example. │ │ │ │ -00066ba0: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00066bb0: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -00066bc0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -00066bd0: 6f6e 3a20 6d61 7472 6978 4661 6374 6f72 on: matrixFactor │ │ │ │ -00066be0: 697a 6174 696f 6e2c 202d 2d20 4d61 7073 ization, -- Maps │ │ │ │ -00066bf0: 2069 6e20 6120 6869 6768 6572 0a20 2020 in a higher. │ │ │ │ -00066c00: 2063 6f64 696d 656e 7369 6f6e 206d 6174 codimension mat │ │ │ │ -00066c10: 7269 7820 6661 6374 6f72 697a 6174 696f rix factorizatio │ │ │ │ -00066c20: 6e0a 2020 2a20 2a6e 6f74 6520 4252 616e n. * *note BRan │ │ │ │ -00066c30: 6b73 3a20 4252 616e 6b73 2c20 2d2d 2072 ks: BRanks, -- r │ │ │ │ -00066c40: 616e 6b73 206f 6620 7468 6520 6d6f 6475 anks of the modu │ │ │ │ -00066c50: 6c65 7320 425f 6928 6429 2069 6e20 610a les B_i(d) in a. │ │ │ │ -00066c60: 2020 2020 6d61 7472 6978 4661 6374 6f72 matrixFactor │ │ │ │ -00066c70: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74 ization. * *not │ │ │ │ -00066c80: 6520 624d 6170 733a 2062 4d61 7073 2c20 e bMaps: bMaps, │ │ │ │ -00066c90: 2d2d 206c 6973 7420 7468 6520 6d61 7073 -- list the maps │ │ │ │ -00066ca0: 2020 645f 703a 425f 3128 7029 2d2d 3e42 d_p:B_1(p)-->B │ │ │ │ -00066cb0: 5f30 2870 2920 696e 2061 0a20 2020 206d _0(p) in a. m │ │ │ │ -00066cc0: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ -00066cd0: 6f6e 0a20 202a 202a 6e6f 7465 2064 4d61 on. * *note dMa │ │ │ │ -00066ce0: 7073 3a20 644d 6170 732c 202d 2d20 6c69 ps: dMaps, -- li │ │ │ │ -00066cf0: 7374 2074 6865 206d 6170 7320 2064 2870 st the maps d(p │ │ │ │ -00066d00: 293a 415f 3128 7029 2d2d 3e20 415f 3028 ):A_1(p)--> A_0( │ │ │ │ -00066d10: 7029 2069 6e20 610a 2020 2020 6d61 7472 p) in a. matr │ │ │ │ -00066d20: 6978 4661 6374 6f72 697a 6174 696f 6e0a ixFactorization. │ │ │ │ -00066d30: 2020 2a20 2a6e 6f74 6520 684d 6170 733a * *note hMaps: │ │ │ │ -00066d40: 2068 4d61 7073 2c20 2d2d 206c 6973 7420 hMaps, -- list │ │ │ │ -00066d50: 7468 6520 6d61 7073 2020 6828 7029 3a20 the maps h(p): │ │ │ │ -00066d60: 415f 3028 7029 2d2d 3e20 415f 3128 7029 A_0(p)--> A_1(p) │ │ │ │ -00066d70: 2069 6e20 610a 2020 2020 6d61 7472 6978 in a. matrix │ │ │ │ -00066d80: 4661 6374 6f72 697a 6174 696f 6e0a 0a57 Factorization..W │ │ │ │ -00066d90: 6179 7320 746f 2075 7365 2070 7369 4d61 ays to use psiMa │ │ │ │ -00066da0: 7073 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ps:.============ │ │ │ │ -00066db0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2270 ========.. * "p │ │ │ │ -00066dc0: 7369 4d61 7073 284c 6973 7429 220a 0a46 siMaps(List)"..F │ │ │ │ -00066dd0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ -00066de0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ -00066df0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ -00066e00: 202a 6e6f 7465 2070 7369 4d61 7073 3a20 *note psiMaps: │ │ │ │ -00066e10: 7073 694d 6170 732c 2069 7320 6120 2a6e psiMaps, is a *n │ │ │ │ -00066e20: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00066e30: 696f 6e3a 0a28 4d61 6361 756c 6179 3244 ion:.(Macaulay2D │ │ │ │ -00066e40: 6f63 294d 6574 686f 6446 756e 6374 696f oc)MethodFunctio │ │ │ │ -00066e50: 6e2c 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d n,...----------- │ │ │ │ +00066a30: 2a2a 2a2a 2a0a 0a20 202a 2055 7361 6765 *****.. * Usage │ │ │ │ +00066a40: 3a20 0a20 2020 2020 2020 2070 736d 6170 : . psmap │ │ │ │ +00066a50: 7320 3d20 7073 694d 6170 7320 6d66 0a20 s = psiMaps mf. │ │ │ │ +00066a60: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ +00066a70: 202a 206d 662c 2061 202a 6e6f 7465 206c * mf, a *note l │ │ │ │ +00066a80: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ +00066a90: 6f63 294c 6973 742c 2c20 6f75 7470 7574 oc)List,, output │ │ │ │ +00066aa0: 206f 6620 6120 6d61 7472 6978 4661 6374 of a matrixFact │ │ │ │ +00066ab0: 6f72 697a 6174 696f 6e0a 2020 2020 2020 orization. │ │ │ │ +00066ac0: 2020 636f 6d70 7574 6174 696f 6e0a 2020 computation. │ │ │ │ +00066ad0: 2a20 4f75 7470 7574 733a 0a20 2020 2020 * Outputs:. │ │ │ │ +00066ae0: 202a 2070 736d 6170 732c 2061 202a 6e6f * psmaps, a *no │ │ │ │ +00066af0: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ +00066b00: 6179 3244 6f63 294c 6973 742c 2c20 6c69 ay2Doc)List,, li │ │ │ │ +00066b10: 7374 206d 6174 7269 6365 7320 2464 5f70 st matrices $d_p │ │ │ │ +00066b20: 3a0a 2020 2020 2020 2020 425f 3128 7029 :. B_1(p) │ │ │ │ +00066b30: 5c74 6f20 415f 3028 702d 3129 240a 0a44 \to A_0(p-1)$..D │ │ │ │ +00066b40: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00066b50: 3d3d 3d3d 3d3d 0a0a 5365 6520 7468 6520 ======..See the │ │ │ │ +00066b60: 646f 6375 6d65 6e74 6174 696f 6e20 666f documentation fo │ │ │ │ +00066b70: 7220 6d61 7472 6978 4661 6374 6f72 697a r matrixFactoriz │ │ │ │ +00066b80: 6174 696f 6e20 666f 7220 616e 2065 7861 ation for an exa │ │ │ │ +00066b90: 6d70 6c65 2e0a 0a53 6565 2061 6c73 6f0a mple...See also. │ │ │ │ +00066ba0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e ========.. * *n │ │ │ │ +00066bb0: 6f74 6520 6d61 7472 6978 4661 6374 6f72 ote matrixFactor │ │ │ │ +00066bc0: 697a 6174 696f 6e3a 206d 6174 7269 7846 ization: matrixF │ │ │ │ +00066bd0: 6163 746f 7269 7a61 7469 6f6e 2c20 2d2d actorization, -- │ │ │ │ +00066be0: 204d 6170 7320 696e 2061 2068 6967 6865 Maps in a highe │ │ │ │ +00066bf0: 720a 2020 2020 636f 6469 6d65 6e73 696f r. codimensio │ │ │ │ +00066c00: 6e20 6d61 7472 6978 2066 6163 746f 7269 n matrix factori │ │ │ │ +00066c10: 7a61 7469 6f6e 0a20 202a 202a 6e6f 7465 zation. * *note │ │ │ │ +00066c20: 2042 5261 6e6b 733a 2042 5261 6e6b 732c BRanks: BRanks, │ │ │ │ +00066c30: 202d 2d20 7261 6e6b 7320 6f66 2074 6865 -- ranks of the │ │ │ │ +00066c40: 206d 6f64 756c 6573 2042 5f69 2864 2920 modules B_i(d) │ │ │ │ +00066c50: 696e 2061 0a20 2020 206d 6174 7269 7846 in a. matrixF │ │ │ │ +00066c60: 6163 746f 7269 7a61 7469 6f6e 0a20 202a actorization. * │ │ │ │ +00066c70: 202a 6e6f 7465 2062 4d61 7073 3a20 624d *note bMaps: bM │ │ │ │ +00066c80: 6170 732c 202d 2d20 6c69 7374 2074 6865 aps, -- list the │ │ │ │ +00066c90: 206d 6170 7320 2064 5f70 3a42 5f31 2870 maps d_p:B_1(p │ │ │ │ +00066ca0: 292d 2d3e 425f 3028 7029 2069 6e20 610a )-->B_0(p) in a. │ │ │ │ +00066cb0: 2020 2020 6d61 7472 6978 4661 6374 6f72 matrixFactor │ │ │ │ +00066cc0: 697a 6174 696f 6e0a 2020 2a20 2a6e 6f74 ization. * *not │ │ │ │ +00066cd0: 6520 644d 6170 733a 2064 4d61 7073 2c20 e dMaps: dMaps, │ │ │ │ +00066ce0: 2d2d 206c 6973 7420 7468 6520 6d61 7073 -- list the maps │ │ │ │ +00066cf0: 2020 6428 7029 3a41 5f31 2870 292d 2d3e d(p):A_1(p)--> │ │ │ │ +00066d00: 2041 5f30 2870 2920 696e 2061 0a20 2020 A_0(p) in a. │ │ │ │ +00066d10: 206d 6174 7269 7846 6163 746f 7269 7a61 matrixFactoriza │ │ │ │ +00066d20: 7469 6f6e 0a20 202a 202a 6e6f 7465 2068 tion. * *note h │ │ │ │ +00066d30: 4d61 7073 3a20 684d 6170 732c 202d 2d20 Maps: hMaps, -- │ │ │ │ +00066d40: 6c69 7374 2074 6865 206d 6170 7320 2068 list the maps h │ │ │ │ +00066d50: 2870 293a 2041 5f30 2870 292d 2d3e 2041 (p): A_0(p)--> A │ │ │ │ +00066d60: 5f31 2870 2920 696e 2061 0a20 2020 206d _1(p) in a. m │ │ │ │ +00066d70: 6174 7269 7846 6163 746f 7269 7a61 7469 atrixFactorizati │ │ │ │ +00066d80: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520 on..Ways to use │ │ │ │ +00066d90: 7073 694d 6170 733a 0a3d 3d3d 3d3d 3d3d psiMaps:.======= │ │ │ │ +00066da0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00066db0: 202a 2022 7073 694d 6170 7328 4c69 7374 * "psiMaps(List │ │ │ │ +00066dc0: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00066dd0: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00066de0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00066df0: 626a 6563 7420 2a6e 6f74 6520 7073 694d bject *note psiM │ │ │ │ +00066e00: 6170 733a 2070 7369 4d61 7073 2c20 6973 aps: psiMaps, is │ │ │ │ +00066e10: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ +00066e20: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ +00066e30: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ +00066e40: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00066e50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066e60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00066e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066e90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00066ea0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ -00066eb0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ -00066ec0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ -00066ed0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ -00066ee0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ -00066ef0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ -00066f00: 7932 2f70 6163 6b61 6765 732f 0a43 6f6d y2/packages/.Com │ │ │ │ -00066f10: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ -00066f20: 6e52 6573 6f6c 7574 696f 6e73 2e6d 323a nResolutions.m2: │ │ │ │ -00066f30: 3434 3832 3a30 2e0a 1f0a 4669 6c65 3a20 4482:0....File: │ │ │ │ -00066f40: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00066f50: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00066f60: 696e 666f 2c20 4e6f 6465 3a20 7265 6775 info, Node: regu │ │ │ │ -00066f70: 6c61 7269 7479 5365 7175 656e 6365 2c20 laritySequence, │ │ │ │ -00066f80: 4e65 7874 3a20 5332 2c20 5072 6576 3a20 Next: S2, Prev: │ │ │ │ -00066f90: 7073 694d 6170 732c 2055 703a 2054 6f70 psiMaps, Up: Top │ │ │ │ -00066fa0: 0a0a 7265 6775 6c61 7269 7479 5365 7175 ..regularitySequ │ │ │ │ -00066fb0: 656e 6365 202d 2d20 7265 6775 6c61 7269 ence -- regulari │ │ │ │ -00066fc0: 7479 206f 6620 4578 7420 6d6f 6475 6c65 ty of Ext module │ │ │ │ -00066fd0: 7320 666f 7220 6120 7365 7175 656e 6365 s for a sequence │ │ │ │ -00066fe0: 206f 6620 4d43 4d20 6170 7072 6f78 696d of MCM approxim │ │ │ │ -00066ff0: 6174 696f 6e73 0a2a 2a2a 2a2a 2a2a 2a2a ations.********* │ │ │ │ +00066e90: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ +00066ea0: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ +00066eb0: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ +00066ec0: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ +00066ed0: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ +00066ee0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ +00066ef0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ +00066f00: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ +00066f10: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ +00066f20: 732e 6d32 3a34 3438 323a 302e 0a1f 0a46 s.m2:4482:0....F │ │ │ │ +00066f30: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ +00066f40: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00066f50: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ +00066f60: 2072 6567 756c 6172 6974 7953 6571 7565 regularitySeque │ │ │ │ +00066f70: 6e63 652c 204e 6578 743a 2053 322c 2050 nce, Next: S2, P │ │ │ │ +00066f80: 7265 763a 2070 7369 4d61 7073 2c20 5570 rev: psiMaps, Up │ │ │ │ +00066f90: 3a20 546f 700a 0a72 6567 756c 6172 6974 : Top..regularit │ │ │ │ +00066fa0: 7953 6571 7565 6e63 6520 2d2d 2072 6567 ySequence -- reg │ │ │ │ +00066fb0: 756c 6172 6974 7920 6f66 2045 7874 206d ularity of Ext m │ │ │ │ +00066fc0: 6f64 756c 6573 2066 6f72 2061 2073 6571 odules for a seq │ │ │ │ +00066fd0: 7565 6e63 6520 6f66 204d 434d 2061 7070 uence of MCM app │ │ │ │ +00066fe0: 726f 7869 6d61 7469 6f6e 730a 2a2a 2a2a roximations.**** │ │ │ │ +00066ff0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067000: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067010: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067030: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067040: 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a ***********.. * │ │ │ │ -00067050: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -00067060: 204c 203d 2072 6567 756c 6172 6974 7953 L = regularityS │ │ │ │ -00067070: 6571 7565 6e63 6520 2852 2c4d 290a 2020 equence (R,M). │ │ │ │ -00067080: 2a20 496e 7075 7473 3a0a 2020 2020 2020 * Inputs:. │ │ │ │ -00067090: 2a20 522c 2061 202a 6e6f 7465 206c 6973 * R, a *note lis │ │ │ │ -000670a0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ -000670b0: 294c 6973 742c 2c20 6c69 7374 206f 6620 )List,, list of │ │ │ │ -000670c0: 7269 6e67 7320 525f 6920 3d0a 2020 2020 rings R_i =. │ │ │ │ -000670d0: 2020 2020 532f 2866 5f30 2e2e 665f 7b28 S/(f_0..f_{( │ │ │ │ -000670e0: 692d 3129 7d29 2c20 636f 6d70 6c65 7465 i-1)}), complete │ │ │ │ -000670f0: 2069 6e74 6572 7365 6374 696f 6e73 0a20 intersections. │ │ │ │ -00067100: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ -00067110: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ -00067120: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ -00067130: 206d 6f64 756c 6520 6f76 6572 2052 5f63 module over R_c │ │ │ │ -00067140: 2077 6865 7265 2063 203d 0a20 2020 2020 where c =. │ │ │ │ -00067150: 2020 206c 656e 6774 6820 5220 2d20 312e length R - 1. │ │ │ │ -00067160: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ -00067170: 2020 2020 2a20 4c2c 2061 202a 6e6f 7465 * L, a *note │ │ │ │ -00067180: 206c 6973 743a 2028 4d61 6361 756c 6179 list: (Macaulay │ │ │ │ -00067190: 3244 6f63 294c 6973 742c 2c20 4c69 7374 2Doc)List,, List │ │ │ │ -000671a0: 206f 6620 7061 6972 7320 7b72 6567 756c of pairs {regul │ │ │ │ -000671b0: 6172 6974 790a 2020 2020 2020 2020 6576 arity. ev │ │ │ │ -000671c0: 656e 4578 744d 6f64 756c 6520 4d5f 692c enExtModule M_i, │ │ │ │ -000671d0: 2072 6567 756c 6172 6974 7920 6f64 6445 regularity oddE │ │ │ │ -000671e0: 7874 4d6f 6475 6c65 204d 5f69 290a 0a44 xtModule M_i)..D │ │ │ │ -000671f0: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ -00067200: 3d3d 3d3d 3d3d 0a0a 436f 6d70 7574 6573 ======..Computes │ │ │ │ -00067210: 2074 6865 206e 6f6e 2d66 7265 6520 7061 the non-free pa │ │ │ │ -00067220: 7274 7320 4d5f 6920 6f66 2074 6865 204d rts M_i of the M │ │ │ │ -00067230: 434d 2061 7070 726f 7869 6d61 7469 6f6e CM approximation │ │ │ │ -00067240: 2074 6f20 4d20 6f76 6572 2052 5f69 2c0a to M over R_i,. │ │ │ │ -00067250: 7374 6f70 7069 6e67 2077 6865 6e20 4d5f stopping when M_ │ │ │ │ -00067260: 6920 6265 636f 6d65 7320 6672 6565 2c20 i becomes free, │ │ │ │ -00067270: 616e 6420 7265 7475 726e 7320 7468 6520 and returns the │ │ │ │ -00067280: 6c69 7374 2077 686f 7365 2065 6c65 6d65 list whose eleme │ │ │ │ -00067290: 6e74 7320 6172 6520 7468 650a 7061 6972 nts are the.pair │ │ │ │ -000672a0: 7320 6f66 2072 6567 756c 6172 6974 6965 s of regularitie │ │ │ │ -000672b0: 732c 2073 7461 7274 696e 6720 7769 7468 s, starting with │ │ │ │ -000672c0: 204d 5f7b 2863 2d31 297d 204e 6f74 6520 M_{(c-1)} Note │ │ │ │ -000672d0: 7468 6174 2074 6865 2066 6972 7374 2070 that the first p │ │ │ │ -000672e0: 6169 7220 6973 2066 6f72 0a74 6865 0a0a air is for.the.. │ │ │ │ -000672f0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00067040: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +00067050: 2020 2020 2020 4c20 3d20 7265 6775 6c61 L = regula │ │ │ │ +00067060: 7269 7479 5365 7175 656e 6365 2028 522c ritySequence (R, │ │ │ │ +00067070: 4d29 0a20 202a 2049 6e70 7574 733a 0a20 M). * Inputs:. │ │ │ │ +00067080: 2020 2020 202a 2052 2c20 6120 2a6e 6f74 * R, a *not │ │ │ │ +00067090: 6520 6c69 7374 3a20 284d 6163 6175 6c61 e list: (Macaula │ │ │ │ +000670a0: 7932 446f 6329 4c69 7374 2c2c 206c 6973 y2Doc)List,, lis │ │ │ │ +000670b0: 7420 6f66 2072 696e 6773 2052 5f69 203d t of rings R_i = │ │ │ │ +000670c0: 0a20 2020 2020 2020 2053 2f28 665f 302e . S/(f_0. │ │ │ │ +000670d0: 2e66 5f7b 2869 2d31 297d 292c 2063 6f6d .f_{(i-1)}), com │ │ │ │ +000670e0: 706c 6574 6520 696e 7465 7273 6563 7469 plete intersecti │ │ │ │ +000670f0: 6f6e 730a 2020 2020 2020 2a20 4d2c 2061 ons. * M, a │ │ │ │ +00067100: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +00067110: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +00067120: 756c 652c 2c20 6d6f 6475 6c65 206f 7665 ule,, module ove │ │ │ │ +00067130: 7220 525f 6320 7768 6572 6520 6320 3d0a r R_c where c =. │ │ │ │ +00067140: 2020 2020 2020 2020 6c65 6e67 7468 2052 length R │ │ │ │ +00067150: 202d 2031 2e0a 2020 2a20 4f75 7470 7574 - 1.. * Output │ │ │ │ +00067160: 733a 0a20 2020 2020 202a 204c 2c20 6120 s:. * L, a │ │ │ │ +00067170: 2a6e 6f74 6520 6c69 7374 3a20 284d 6163 *note list: (Mac │ │ │ │ +00067180: 6175 6c61 7932 446f 6329 4c69 7374 2c2c aulay2Doc)List,, │ │ │ │ +00067190: 204c 6973 7420 6f66 2070 6169 7273 207b List of pairs { │ │ │ │ +000671a0: 7265 6775 6c61 7269 7479 0a20 2020 2020 regularity. │ │ │ │ +000671b0: 2020 2065 7665 6e45 7874 4d6f 6475 6c65 evenExtModule │ │ │ │ +000671c0: 204d 5f69 2c20 7265 6775 6c61 7269 7479 M_i, regularity │ │ │ │ +000671d0: 206f 6464 4578 744d 6f64 756c 6520 4d5f oddExtModule M_ │ │ │ │ +000671e0: 6929 0a0a 4465 7363 7269 7074 696f 6e0a i)..Description. │ │ │ │ +000671f0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a43 6f6d ===========..Com │ │ │ │ +00067200: 7075 7465 7320 7468 6520 6e6f 6e2d 6672 putes the non-fr │ │ │ │ +00067210: 6565 2070 6172 7473 204d 5f69 206f 6620 ee parts M_i of │ │ │ │ +00067220: 7468 6520 4d43 4d20 6170 7072 6f78 696d the MCM approxim │ │ │ │ +00067230: 6174 696f 6e20 746f 204d 206f 7665 7220 ation to M over │ │ │ │ +00067240: 525f 692c 0a73 746f 7070 696e 6720 7768 R_i,.stopping wh │ │ │ │ +00067250: 656e 204d 5f69 2062 6563 6f6d 6573 2066 en M_i becomes f │ │ │ │ +00067260: 7265 652c 2061 6e64 2072 6574 7572 6e73 ree, and returns │ │ │ │ +00067270: 2074 6865 206c 6973 7420 7768 6f73 6520 the list whose │ │ │ │ +00067280: 656c 656d 656e 7473 2061 7265 2074 6865 elements are the │ │ │ │ +00067290: 0a70 6169 7273 206f 6620 7265 6775 6c61 .pairs of regula │ │ │ │ +000672a0: 7269 7469 6573 2c20 7374 6172 7469 6e67 rities, starting │ │ │ │ +000672b0: 2077 6974 6820 4d5f 7b28 632d 3129 7d20 with M_{(c-1)} │ │ │ │ +000672c0: 4e6f 7465 2074 6861 7420 7468 6520 6669 Note that the fi │ │ │ │ +000672d0: 7273 7420 7061 6972 2069 7320 666f 720a rst pair is for. │ │ │ │ +000672e0: 7468 650a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d the..+---------- │ │ │ │ +000672f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -00067330: 0a7c 6931 203a 2063 203d 2033 3b64 3d32 .|i1 : c = 3;d=2 │ │ │ │ +00067320: 2d2d 2d2d 2b0a 7c69 3120 3a20 6320 3d20 ----+.|i1 : c = │ │ │ │ +00067330: 333b 643d 3220 2020 2020 2020 2020 2020 3;d=2 │ │ │ │ 00067340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067370: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00067360: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00067370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673b0: 207c 0a7c 6f32 203d 2032 2020 2020 2020 |.|o2 = 2 │ │ │ │ +000673a0: 2020 2020 2020 7c0a 7c6f 3220 3d20 3220 |.|o2 = 2 │ │ │ │ +000673b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000673c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000673d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000673f0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +000673e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000673f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067430: 2d2d 2d2b 0a7c 6933 203a 2052 203d 2073 ---+.|i3 : R = s │ │ │ │ -00067440: 6574 7570 5269 6e67 7328 632c 6429 3b20 etupRings(c,d); │ │ │ │ +00067420: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 --------+.|i3 : │ │ │ │ +00067430: 5220 3d20 7365 7475 7052 696e 6773 2863 R = setupRings(c │ │ │ │ +00067440: 2c64 293b 2020 2020 2020 2020 2020 2020 ,d); │ │ │ │ 00067450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067470: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00067460: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +00067470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067480: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067490: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000674b0: 2d2d 2d2d 2d2b 0a7c 6934 203a 2052 6320 -----+.|i4 : Rc │ │ │ │ -000674c0: 3d20 525f 6320 2020 2020 2020 2020 2020 = R_c │ │ │ │ +000674a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3420 ----------+.|i4 │ │ │ │ +000674b0: 3a20 5263 203d 2052 5f63 2020 2020 2020 : Rc = R_c │ │ │ │ +000674c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000674d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000674e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000674f0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +000674e0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000674f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067530: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ -00067540: 6320 2020 2020 2020 2020 2020 2020 2020 c │ │ │ │ +00067520: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00067530: 3420 3d20 5263 2020 2020 2020 2020 2020 4 = Rc │ │ │ │ +00067540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067570: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00067560: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00067570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675b0: 2020 2020 2020 2020 207c 0a7c 6f34 203a |.|o4 : │ │ │ │ -000675c0: 2051 756f 7469 656e 7452 696e 6720 2020 QuotientRing │ │ │ │ +000675a0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000675b0: 7c6f 3420 3a20 5175 6f74 6965 6e74 5269 |o4 : QuotientRi │ │ │ │ +000675c0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 000675d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000675f0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +000675e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000675f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 00067600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ -00067640: 203a 204d 203d 2063 6f6b 6572 206d 6174 : M = coker mat │ │ │ │ -00067650: 7269 787b 7b52 635f 302c 5263 5f31 2c52 rix{{Rc_0,Rc_1,R │ │ │ │ -00067660: 635f 327d 2c7b 5263 5f31 2c52 635f 322c c_2},{Rc_1,Rc_2, │ │ │ │ -00067670: 5263 5f30 7d7d 2020 2020 2020 7c0a 7c20 Rc_0}} |.| │ │ │ │ +00067630: 2b0a 7c69 3520 3a20 4d20 3d20 636f 6b65 +.|i5 : M = coke │ │ │ │ +00067640: 7220 6d61 7472 6978 7b7b 5263 5f30 2c52 r matrix{{Rc_0,R │ │ │ │ +00067650: 635f 312c 5263 5f32 7d2c 7b52 635f 312c c_1,Rc_2},{Rc_1, │ │ │ │ +00067660: 5263 5f32 2c52 635f 307d 7d20 2020 2020 Rc_2,Rc_0}} │ │ │ │ +00067670: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00067680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000676a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000676b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000676c0: 6f35 203d 2063 6f6b 6572 6e65 6c20 7c20 o5 = cokernel | │ │ │ │ -000676d0: 785f 3020 785f 3120 785f 3220 7c20 2020 x_0 x_1 x_2 | │ │ │ │ +000676b0: 2020 7c0a 7c6f 3520 3d20 636f 6b65 726e |.|o5 = cokern │ │ │ │ +000676c0: 656c 207c 2078 5f30 2078 5f31 2078 5f32 el | x_0 x_1 x_2 │ │ │ │ +000676d0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000676e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000676f0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00067700: 7c20 2020 2020 2020 2020 2020 2020 207c | | │ │ │ │ -00067710: 2078 5f31 2078 5f32 2078 5f30 207c 2020 x_1 x_2 x_0 | │ │ │ │ +000676f0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00067700: 2020 2020 7c20 785f 3120 785f 3220 785f | x_1 x_2 x_ │ │ │ │ +00067710: 3020 7c20 2020 2020 2020 2020 2020 2020 0 | │ │ │ │ 00067720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067730: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00067740: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00067730: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00067740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00067760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067780: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -00067790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000677a0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ -000677b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000677c0: 207c 0a7c 6f35 203a 2052 632d 6d6f 6475 |.|o5 : Rc-modu │ │ │ │ -000677d0: 6c65 2c20 7175 6f74 6965 6e74 206f 6620 le, quotient of │ │ │ │ -000677e0: 5263 2020 2020 2020 2020 2020 2020 2020 Rc │ │ │ │ -000677f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067800: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ +00067770: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00067780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067790: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ +000677a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000677b0: 2020 2020 2020 7c0a 7c6f 3520 3a20 5263 |.|o5 : Rc │ │ │ │ +000677c0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ +000677d0: 7420 6f66 2052 6320 2020 2020 2020 2020 t of Rc │ │ │ │ +000677e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000677f0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00067800: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067840: 2d2d 2d2b 0a7c 6936 203a 2072 6567 756c ---+.|i6 : regul │ │ │ │ -00067850: 6172 6974 7953 6571 7565 6e63 6528 522c aritySequence(R, │ │ │ │ -00067860: 4d29 2020 2020 2020 2020 2020 2020 2020 M) │ │ │ │ -00067870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067880: 2020 2020 7c0a 7c72 6567 2065 7665 6e20 |.|reg even │ │ │ │ -00067890: 6578 742c 2073 6f63 2064 6567 7320 6576 ext, soc degs ev │ │ │ │ -000678a0: 656e 2065 7874 2c20 7265 6720 6f64 6420 en ext, reg odd │ │ │ │ -000678b0: 6578 742c 2073 6f63 2064 6567 7320 6f64 ext, soc degs od │ │ │ │ -000678c0: 6420 6578 747c 0a7c 2020 2020 2020 2020 d ext|.| │ │ │ │ +00067830: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3620 3a20 --------+.|i6 : │ │ │ │ +00067840: 7265 6775 6c61 7269 7479 5365 7175 656e regularitySequen │ │ │ │ +00067850: 6365 2852 2c4d 2920 2020 2020 2020 2020 ce(R,M) │ │ │ │ +00067860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067870: 2020 2020 2020 2020 207c 0a7c 7265 6720 |.|reg │ │ │ │ +00067880: 6576 656e 2065 7874 2c20 736f 6320 6465 even ext, soc de │ │ │ │ +00067890: 6773 2065 7665 6e20 6578 742c 2072 6567 gs even ext, reg │ │ │ │ +000678a0: 206f 6464 2065 7874 2c20 736f 6320 6465 odd ext, soc de │ │ │ │ +000678b0: 6773 206f 6464 2065 7874 7c0a 7c20 2020 gs odd ext|.| │ │ │ │ +000678c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000678d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000678e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000678f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067900: 2020 2020 2020 7c0a 7c7b 332c 207b 312c |.|{3, {1, │ │ │ │ -00067910: 2031 2c20 317d 2c20 322c 207b 312c 2031 1, 1}, 2, {1, 1 │ │ │ │ -00067920: 7d7d 2020 2020 2020 2020 2020 2020 2020 }} │ │ │ │ -00067930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067940: 2020 2020 2020 207c 0a7c 7b32 2c20 7b30 |.|{2, {0 │ │ │ │ -00067950: 2c20 302c 2030 2c20 317d 2c20 322c 207b , 0, 0, 1}, 2, { │ │ │ │ -00067960: 302c 2030 2c20 307d 7d20 2020 2020 2020 0, 0, 0}} │ │ │ │ -00067970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00067980: 2020 2020 2020 2020 7c0a 7c7b 302c 207b |.|{0, { │ │ │ │ -00067990: 7d2c 2030 2c20 7b7d 7d20 2020 2020 2020 }, 0, {}} │ │ │ │ +000678f0: 2020 2020 2020 2020 2020 207c 0a7c 7b33 |.|{3 │ │ │ │ +00067900: 2c20 7b31 2c20 312c 2031 7d2c 2032 2c20 , {1, 1, 1}, 2, │ │ │ │ +00067910: 7b31 2c20 317d 7d20 2020 2020 2020 2020 {1, 1}} │ │ │ │ +00067920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067930: 2020 2020 2020 2020 2020 2020 7c0a 7c7b |.|{ │ │ │ │ +00067940: 322c 207b 302c 2030 2c20 302c 2031 7d2c 2, {0, 0, 0, 1}, │ │ │ │ +00067950: 2032 2c20 7b30 2c20 302c 2030 7d7d 2020 2, {0, 0, 0}} │ │ │ │ +00067960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00067970: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00067980: 7b30 2c20 7b7d 2c20 302c 207b 7d7d 2020 {0, {}, 0, {}} │ │ │ │ +00067990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000679a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000679b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000679c0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ +000679b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000679c0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000679d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000679e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -00067a10: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -00067a20: 2020 2a20 2a6e 6f74 6520 6170 7072 6f78 * *note approx │ │ │ │ -00067a30: 696d 6174 696f 6e3a 2028 4d43 4d41 7070 imation: (MCMApp │ │ │ │ -00067a40: 726f 7869 6d61 7469 6f6e 7329 6170 7072 roximations)appr │ │ │ │ -00067a50: 6f78 696d 6174 696f 6e2c 202d 2d20 7265 oximation, -- re │ │ │ │ -00067a60: 7475 726e 7320 7061 6972 206f 660a 2020 turns pair of. │ │ │ │ -00067a70: 2020 636f 6d70 6f6e 656e 7473 206f 6620 components of │ │ │ │ -00067a80: 7468 6520 6d61 7020 6672 6f6d 2074 6865 the map from the │ │ │ │ -00067a90: 204d 434d 2061 7070 726f 7869 6d61 7469 MCM approximati │ │ │ │ -00067aa0: 6f6e 0a20 202a 202a 6e6f 7465 2061 7573 on. * *note aus │ │ │ │ -00067ab0: 6c61 6e64 6572 496e 7661 7269 616e 743a landerInvariant: │ │ │ │ -00067ac0: 2028 4d43 4d41 7070 726f 7869 6d61 7469 (MCMApproximati │ │ │ │ -00067ad0: 6f6e 7329 6175 736c 616e 6465 7249 6e76 ons)auslanderInv │ │ │ │ -00067ae0: 6172 6961 6e74 2c20 2d2d 0a20 2020 206d ariant, --. m │ │ │ │ -00067af0: 6561 7375 7265 7320 6661 696c 7572 6520 easures failure │ │ │ │ -00067b00: 6f66 2073 7572 6a65 6374 6976 6974 7920 of surjectivity │ │ │ │ -00067b10: 6f66 2074 6865 2065 7373 656e 7469 616c of the essential │ │ │ │ -00067b20: 204d 434d 2061 7070 726f 7869 6d61 7469 MCM approximati │ │ │ │ -00067b30: 6f6e 0a0a 5761 7973 2074 6f20 7573 6520 on..Ways to use │ │ │ │ -00067b40: 7265 6775 6c61 7269 7479 5365 7175 656e regularitySequen │ │ │ │ -00067b50: 6365 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ce:.============ │ │ │ │ -00067b60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00067b70: 3d3d 3d0a 0a20 202a 2022 7265 6775 6c61 ===.. * "regula │ │ │ │ -00067b80: 7269 7479 5365 7175 656e 6365 284c 6973 ritySequence(Lis │ │ │ │ -00067b90: 742c 4d6f 6475 6c65 2922 0a0a 466f 7220 t,Module)"..For │ │ │ │ -00067ba0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -00067bb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00067bc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -00067bd0: 6f74 6520 7265 6775 6c61 7269 7479 5365 ote regularitySe │ │ │ │ -00067be0: 7175 656e 6365 3a20 7265 6775 6c61 7269 quence: regulari │ │ │ │ -00067bf0: 7479 5365 7175 656e 6365 2c20 6973 2061 tySequence, is a │ │ │ │ -00067c00: 202a 6e6f 7465 206d 6574 686f 640a 6675 *note method.fu │ │ │ │ -00067c10: 6e63 7469 6f6e 3a20 284d 6163 6175 6c61 nction: (Macaula │ │ │ │ -00067c20: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -00067c30: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +000679f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +00067a00: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ +00067a10: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 2061 ===.. * *note a │ │ │ │ +00067a20: 7070 726f 7869 6d61 7469 6f6e 3a20 284d pproximation: (M │ │ │ │ +00067a30: 434d 4170 7072 6f78 696d 6174 696f 6e73 CMApproximations │ │ │ │ +00067a40: 2961 7070 726f 7869 6d61 7469 6f6e 2c20 )approximation, │ │ │ │ +00067a50: 2d2d 2072 6574 7572 6e73 2070 6169 7220 -- returns pair │ │ │ │ +00067a60: 6f66 0a20 2020 2063 6f6d 706f 6e65 6e74 of. component │ │ │ │ +00067a70: 7320 6f66 2074 6865 206d 6170 2066 726f s of the map fro │ │ │ │ +00067a80: 6d20 7468 6520 4d43 4d20 6170 7072 6f78 m the MCM approx │ │ │ │ +00067a90: 696d 6174 696f 6e0a 2020 2a20 2a6e 6f74 imation. * *not │ │ │ │ +00067aa0: 6520 6175 736c 616e 6465 7249 6e76 6172 e auslanderInvar │ │ │ │ +00067ab0: 6961 6e74 3a20 284d 434d 4170 7072 6f78 iant: (MCMApprox │ │ │ │ +00067ac0: 696d 6174 696f 6e73 2961 7573 6c61 6e64 imations)ausland │ │ │ │ +00067ad0: 6572 496e 7661 7269 616e 742c 202d 2d0a erInvariant, --. │ │ │ │ +00067ae0: 2020 2020 6d65 6173 7572 6573 2066 6169 measures fai │ │ │ │ +00067af0: 6c75 7265 206f 6620 7375 726a 6563 7469 lure of surjecti │ │ │ │ +00067b00: 7669 7479 206f 6620 7468 6520 6573 7365 vity of the esse │ │ │ │ +00067b10: 6e74 6961 6c20 4d43 4d20 6170 7072 6f78 ntial MCM approx │ │ │ │ +00067b20: 696d 6174 696f 6e0a 0a57 6179 7320 746f imation..Ways to │ │ │ │ +00067b30: 2075 7365 2072 6567 756c 6172 6974 7953 use regularityS │ │ │ │ +00067b40: 6571 7565 6e63 653a 0a3d 3d3d 3d3d 3d3d equence:.======= │ │ │ │ +00067b50: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00067b60: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2272 ========.. * "r │ │ │ │ +00067b70: 6567 756c 6172 6974 7953 6571 7565 6e63 egularitySequenc │ │ │ │ +00067b80: 6528 4c69 7374 2c4d 6f64 756c 6529 220a e(List,Module)". │ │ │ │ +00067b90: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +00067ba0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +00067bb0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +00067bc0: 6374 202a 6e6f 7465 2072 6567 756c 6172 ct *note regular │ │ │ │ +00067bd0: 6974 7953 6571 7565 6e63 653a 2072 6567 itySequence: reg │ │ │ │ +00067be0: 756c 6172 6974 7953 6571 7565 6e63 652c ularitySequence, │ │ │ │ +00067bf0: 2069 7320 6120 2a6e 6f74 6520 6d65 7468 is a *note meth │ │ │ │ +00067c00: 6f64 0a66 756e 6374 696f 6e3a 2028 4d61 od.function: (Ma │ │ │ │ +00067c10: 6361 756c 6179 3244 6f63 294d 6574 686f caulay2Doc)Metho │ │ │ │ +00067c20: 6446 756e 6374 696f 6e2c 2e0a 0a2d 2d2d dFunction,...--- │ │ │ │ +00067c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067c40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067c50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00067c60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00067c80: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00067c90: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00067ca0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00067cb0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00067cc0: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00067cd0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -00067ce0: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ -00067cf0: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ -00067d00: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ -00067d10: 6d32 3a32 3630 303a 302e 0a1f 0a46 696c m2:2600:0....Fil │ │ │ │ -00067d20: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ -00067d30: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ -00067d40: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053 ns.info, Node: S │ │ │ │ -00067d50: 322c 204e 6578 743a 2053 6861 6d61 7368 2, Next: Shamash │ │ │ │ -00067d60: 2c20 5072 6576 3a20 7265 6775 6c61 7269 , Prev: regulari │ │ │ │ -00067d70: 7479 5365 7175 656e 6365 2c20 5570 3a20 tySequence, Up: │ │ │ │ -00067d80: 546f 700a 0a53 3220 2d2d 2055 6e69 7665 Top..S2 -- Unive │ │ │ │ -00067d90: 7273 616c 206d 6170 2074 6f20 6120 6d6f rsal map to a mo │ │ │ │ -00067da0: 6475 6c65 2073 6174 6973 6679 696e 6720 dule satisfying │ │ │ │ -00067db0: 5365 7272 6527 7320 636f 6e64 6974 696f Serre's conditio │ │ │ │ -00067dc0: 6e20 5332 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a n S2.*********** │ │ │ │ +00067c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 ------------..Th │ │ │ │ +00067c80: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ +00067c90: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ +00067ca0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ +00067cb0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ +00067cc0: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ +00067cd0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ +00067ce0: 6765 732f 0a43 6f6d 706c 6574 6549 6e74 ges/.CompleteInt │ │ │ │ +00067cf0: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ +00067d00: 696f 6e73 2e6d 323a 3236 3030 3a30 2e0a ions.m2:2600:0.. │ │ │ │ +00067d10: 1f0a 4669 6c65 3a20 436f 6d70 6c65 7465 ..File: Complete │ │ │ │ +00067d20: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ +00067d30: 6c75 7469 6f6e 732e 696e 666f 2c20 4e6f lutions.info, No │ │ │ │ +00067d40: 6465 3a20 5332 2c20 4e65 7874 3a20 5368 de: S2, Next: Sh │ │ │ │ +00067d50: 616d 6173 682c 2050 7265 763a 2072 6567 amash, Prev: reg │ │ │ │ +00067d60: 756c 6172 6974 7953 6571 7565 6e63 652c ularitySequence, │ │ │ │ +00067d70: 2055 703a 2054 6f70 0a0a 5332 202d 2d20 Up: Top..S2 -- │ │ │ │ +00067d80: 556e 6976 6572 7361 6c20 6d61 7020 746f Universal map to │ │ │ │ +00067d90: 2061 206d 6f64 756c 6520 7361 7469 7366 a module satisf │ │ │ │ +00067da0: 7969 6e67 2053 6572 7265 2773 2063 6f6e ying Serre's con │ │ │ │ +00067db0: 6469 7469 6f6e 2053 320a 2a2a 2a2a 2a2a dition S2.****** │ │ │ │ +00067dc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067dd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00067de0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067df0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00067e00: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -00067e10: 200a 2020 2020 2020 2020 6620 3d20 5332 . f = S2 │ │ │ │ -00067e20: 2862 2c4d 290a 2020 2a20 496e 7075 7473 (b,M). * Inputs │ │ │ │ -00067e30: 3a0a 2020 2020 2020 2a20 622c 2061 6e20 :. * b, an │ │ │ │ -00067e40: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ -00067e50: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ -00067e60: 2c20 6465 6772 6565 2062 6f75 6e64 2074 , degree bound t │ │ │ │ -00067e70: 6f20 7768 6963 6820 746f 2063 6172 7279 o which to carry │ │ │ │ -00067e80: 0a20 2020 2020 2020 2074 6865 2063 6f6d . the com │ │ │ │ -00067e90: 7075 7461 7469 6f6e 0a20 2020 2020 202a putation. * │ │ │ │ -00067ea0: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -00067eb0: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -00067ec0: 6329 4d6f 6475 6c65 2c2c 200a 2020 2a20 c)Module,, . * │ │ │ │ -00067ed0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -00067ee0: 2066 2c20 6120 2a6e 6f74 6520 6d61 7472 f, a *note matr │ │ │ │ -00067ef0: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -00067f00: 6329 4d61 7472 6978 2c2c 2064 6566 696e c)Matrix,, defin │ │ │ │ -00067f10: 696e 6720 6120 6d61 7020 4d2d 2d3e 4d27 ing a map M-->M' │ │ │ │ -00067f20: 2074 6861 740a 2020 2020 2020 2020 6167 that. ag │ │ │ │ -00067f30: 7265 6573 2077 6974 6820 7468 6520 5332 rees with the S2 │ │ │ │ -00067f40: 2d69 6669 6361 7469 6f6e 206f 6620 4d20 -ification of M │ │ │ │ -00067f50: 696e 2064 6567 7265 6573 2024 5c67 6571 in degrees $\geq │ │ │ │ -00067f60: 2062 240a 0a44 6573 6372 6970 7469 6f6e b$..Description │ │ │ │ -00067f70: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4966 .===========..If │ │ │ │ -00067f80: 204d 2069 7320 6120 6772 6164 6564 206d M is a graded m │ │ │ │ -00067f90: 6f64 756c 6520 6f76 6572 2061 2072 696e odule over a rin │ │ │ │ -00067fa0: 6720 532c 2074 6865 6e20 7468 6520 5332 g S, then the S2 │ │ │ │ -00067fb0: 2d69 6669 6361 7469 6f6e 206f 6620 4d20 -ification of M │ │ │ │ -00067fc0: 6973 205c 7375 6d5f 7b64 0a5c 696e 205a is \sum_{d.\in Z │ │ │ │ -00067fd0: 5a7d 2048 5e30 2828 7368 6561 6620 4d29 Z} H^0((sheaf M) │ │ │ │ -00067fe0: 2864 2929 2c20 7768 6963 6820 6d61 7920 (d)), which may │ │ │ │ -00067ff0: 6265 2063 6f6d 7075 7465 6420 6173 206c be computed as l │ │ │ │ -00068000: 696d 5f7b 642d 3e5c 696e 6674 797d 2048 im_{d->\infty} H │ │ │ │ -00068010: 6f6d 2849 5f64 2c4d 292c 0a77 6865 7265 om(I_d,M),.where │ │ │ │ -00068020: 2049 5f64 2069 7320 616e 7920 7365 7175 I_d is any sequ │ │ │ │ -00068030: 656e 6365 206f 6620 6964 6561 6c73 2063 ence of ideals c │ │ │ │ -00068040: 6f6e 7461 696e 6564 2069 6e20 6869 6768 ontained in high │ │ │ │ -00068050: 6572 2061 6e64 2068 6967 6865 7220 706f er and higher po │ │ │ │ -00068060: 7765 7273 206f 660a 535f 2b2e 2054 6865 wers of.S_+. The │ │ │ │ -00068070: 7265 2069 7320 6120 6e61 7475 7261 6c20 re is a natural │ │ │ │ -00068080: 7265 7374 7269 6374 696f 6e20 6d61 7020 restriction map │ │ │ │ -00068090: 663a 204d 203d 2048 6f6d 2853 2c4d 2920 f: M = Hom(S,M) │ │ │ │ -000680a0: 5c74 6f20 486f 6d28 495f 642c 4d29 2e20 \to Hom(I_d,M). │ │ │ │ -000680b0: 5765 0a63 6f6d 7075 7465 2061 6c6c 2074 We.compute all t │ │ │ │ -000680c0: 6869 7320 7573 696e 6720 7468 6520 6964 his using the id │ │ │ │ -000680d0: 6561 6c73 2049 5f64 2067 656e 6572 6174 eals I_d generat │ │ │ │ -000680e0: 6564 2062 7920 7468 6520 642d 7468 2070 ed by the d-th p │ │ │ │ -000680f0: 6f77 6572 7320 6f66 2074 6865 0a76 6172 owers of the.var │ │ │ │ -00068100: 6961 626c 6573 2069 6e20 532e 0a0a 5369 iables in S...Si │ │ │ │ -00068110: 6e63 6520 7468 6520 7265 7375 6c74 206d nce the result m │ │ │ │ -00068120: 6179 206e 6f74 2062 6520 6669 6e69 7465 ay not be finite │ │ │ │ -00068130: 6c79 2067 656e 6572 6174 6564 2028 7468 ly generated (th │ │ │ │ -00068140: 6973 2068 6170 7065 6e73 2069 6620 616e is happens if an │ │ │ │ -00068150: 6420 6f6e 6c79 2069 6620 4d0a 6861 7320 d only if M.has │ │ │ │ -00068160: 616e 2061 7373 6f63 6961 7465 6420 7072 an associated pr │ │ │ │ -00068170: 696d 6520 6f66 2064 696d 656e 7369 6f6e ime of dimension │ │ │ │ -00068180: 2031 292c 2077 6520 636f 6d70 7574 6520 1), we compute │ │ │ │ -00068190: 6f6e 6c79 2075 7020 746f 2061 2073 7065 only up to a spe │ │ │ │ -000681a0: 6369 6669 6564 0a64 6567 7265 6520 626f cified.degree bo │ │ │ │ -000681b0: 756e 6420 622e 2046 6f72 2074 6865 2072 und b. For the r │ │ │ │ -000681c0: 6573 756c 7420 746f 2062 6520 636f 7272 esult to be corr │ │ │ │ -000681d0: 6563 7420 646f 776e 2074 6f20 6465 6772 ect down to degr │ │ │ │ -000681e0: 6565 2062 2c20 6974 2069 7320 7375 6666 ee b, it is suff │ │ │ │ -000681f0: 6963 6965 6e74 0a74 6f20 636f 6d70 7574 icient.to comput │ │ │ │ -00068200: 6520 486f 6d28 492c 4d29 2077 6865 7265 e Hom(I,M) where │ │ │ │ -00068210: 2049 205c 7375 6273 6574 2028 535f 2b29 I \subset (S_+) │ │ │ │ -00068220: 5e7b 722d 627d 2e0a 0a2b 2d2d 2d2d 2d2d ^{r-b}...+------ │ │ │ │ +00067df0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +00067e00: 7361 6765 3a20 0a20 2020 2020 2020 2066 sage: . f │ │ │ │ +00067e10: 203d 2053 3228 622c 4d29 0a20 202a 2049 = S2(b,M). * I │ │ │ │ +00067e20: 6e70 7574 733a 0a20 2020 2020 202a 2062 nputs:. * b │ │ │ │ +00067e30: 2c20 616e 202a 6e6f 7465 2069 6e74 6567 , an *note integ │ │ │ │ +00067e40: 6572 3a20 284d 6163 6175 6c61 7932 446f er: (Macaulay2Do │ │ │ │ +00067e50: 6329 5a5a 2c2c 2064 6567 7265 6520 626f c)ZZ,, degree bo │ │ │ │ +00067e60: 756e 6420 746f 2077 6869 6368 2074 6f20 und to which to │ │ │ │ +00067e70: 6361 7272 790a 2020 2020 2020 2020 7468 carry. th │ │ │ │ +00067e80: 6520 636f 6d70 7574 6174 696f 6e0a 2020 e computation. │ │ │ │ +00067e90: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +00067ea0: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +00067eb0: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +00067ec0: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00067ed0: 2020 2020 2a20 662c 2061 202a 6e6f 7465 * f, a *note │ │ │ │ +00067ee0: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ +00067ef0: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ +00067f00: 6465 6669 6e69 6e67 2061 206d 6170 204d defining a map M │ │ │ │ +00067f10: 2d2d 3e4d 2720 7468 6174 0a20 2020 2020 -->M' that. │ │ │ │ +00067f20: 2020 2061 6772 6565 7320 7769 7468 2074 agrees with t │ │ │ │ +00067f30: 6865 2053 322d 6966 6963 6174 696f 6e20 he S2-ification │ │ │ │ +00067f40: 6f66 204d 2069 6e20 6465 6772 6565 7320 of M in degrees │ │ │ │ +00067f50: 245c 6765 7120 6224 0a0a 4465 7363 7269 $\geq b$..Descri │ │ │ │ +00067f60: 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d ption.========== │ │ │ │ +00067f70: 3d0a 0a49 6620 4d20 6973 2061 2067 7261 =..If M is a gra │ │ │ │ +00067f80: 6465 6420 6d6f 6475 6c65 206f 7665 7220 ded module over │ │ │ │ +00067f90: 6120 7269 6e67 2053 2c20 7468 656e 2074 a ring S, then t │ │ │ │ +00067fa0: 6865 2053 322d 6966 6963 6174 696f 6e20 he S2-ification │ │ │ │ +00067fb0: 6f66 204d 2069 7320 5c73 756d 5f7b 640a of M is \sum_{d. │ │ │ │ +00067fc0: 5c69 6e20 5a5a 7d20 485e 3028 2873 6865 \in ZZ} H^0((she │ │ │ │ +00067fd0: 6166 204d 2928 6429 292c 2077 6869 6368 af M)(d)), which │ │ │ │ +00067fe0: 206d 6179 2062 6520 636f 6d70 7574 6564 may be computed │ │ │ │ +00067ff0: 2061 7320 6c69 6d5f 7b64 2d3e 5c69 6e66 as lim_{d->\inf │ │ │ │ +00068000: 7479 7d20 486f 6d28 495f 642c 4d29 2c0a ty} Hom(I_d,M),. │ │ │ │ +00068010: 7768 6572 6520 495f 6420 6973 2061 6e79 where I_d is any │ │ │ │ +00068020: 2073 6571 7565 6e63 6520 6f66 2069 6465 sequence of ide │ │ │ │ +00068030: 616c 7320 636f 6e74 6169 6e65 6420 696e als contained in │ │ │ │ +00068040: 2068 6967 6865 7220 616e 6420 6869 6768 higher and high │ │ │ │ +00068050: 6572 2070 6f77 6572 7320 6f66 0a53 5f2b er powers of.S_+ │ │ │ │ +00068060: 2e20 5468 6572 6520 6973 2061 206e 6174 . There is a nat │ │ │ │ +00068070: 7572 616c 2072 6573 7472 6963 7469 6f6e ural restriction │ │ │ │ +00068080: 206d 6170 2066 3a20 4d20 3d20 486f 6d28 map f: M = Hom( │ │ │ │ +00068090: 532c 4d29 205c 746f 2048 6f6d 2849 5f64 S,M) \to Hom(I_d │ │ │ │ +000680a0: 2c4d 292e 2057 650a 636f 6d70 7574 6520 ,M). We.compute │ │ │ │ +000680b0: 616c 6c20 7468 6973 2075 7369 6e67 2074 all this using t │ │ │ │ +000680c0: 6865 2069 6465 616c 7320 495f 6420 6765 he ideals I_d ge │ │ │ │ +000680d0: 6e65 7261 7465 6420 6279 2074 6865 2064 nerated by the d │ │ │ │ +000680e0: 2d74 6820 706f 7765 7273 206f 6620 7468 -th powers of th │ │ │ │ +000680f0: 650a 7661 7269 6162 6c65 7320 696e 2053 e.variables in S │ │ │ │ +00068100: 2e0a 0a53 696e 6365 2074 6865 2072 6573 ...Since the res │ │ │ │ +00068110: 756c 7420 6d61 7920 6e6f 7420 6265 2066 ult may not be f │ │ │ │ +00068120: 696e 6974 656c 7920 6765 6e65 7261 7465 initely generate │ │ │ │ +00068130: 6420 2874 6869 7320 6861 7070 656e 7320 d (this happens │ │ │ │ +00068140: 6966 2061 6e64 206f 6e6c 7920 6966 204d if and only if M │ │ │ │ +00068150: 0a68 6173 2061 6e20 6173 736f 6369 6174 .has an associat │ │ │ │ +00068160: 6564 2070 7269 6d65 206f 6620 6469 6d65 ed prime of dime │ │ │ │ +00068170: 6e73 696f 6e20 3129 2c20 7765 2063 6f6d nsion 1), we com │ │ │ │ +00068180: 7075 7465 206f 6e6c 7920 7570 2074 6f20 pute only up to │ │ │ │ +00068190: 6120 7370 6563 6966 6965 640a 6465 6772 a specified.degr │ │ │ │ +000681a0: 6565 2062 6f75 6e64 2062 2e20 466f 7220 ee bound b. For │ │ │ │ +000681b0: 7468 6520 7265 7375 6c74 2074 6f20 6265 the result to be │ │ │ │ +000681c0: 2063 6f72 7265 6374 2064 6f77 6e20 746f correct down to │ │ │ │ +000681d0: 2064 6567 7265 6520 622c 2069 7420 6973 degree b, it is │ │ │ │ +000681e0: 2073 7566 6669 6369 656e 740a 746f 2063 sufficient.to c │ │ │ │ +000681f0: 6f6d 7075 7465 2048 6f6d 2849 2c4d 2920 ompute Hom(I,M) │ │ │ │ +00068200: 7768 6572 6520 4920 5c73 7562 7365 7420 where I \subset │ │ │ │ +00068210: 2853 5f2b 295e 7b72 2d62 7d2e 0a0a 2b2d (S_+)^{r-b}...+- │ │ │ │ +00068220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068250: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068270: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 206b -------+.|i1 : k │ │ │ │ -00068280: 6b3d 5a5a 2f31 3031 2020 2020 2020 2020 k=ZZ/101 │ │ │ │ +00068260: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068270: 3120 3a20 6b6b 3d5a 5a2f 3130 3120 2020 1 : kk=ZZ/101 │ │ │ │ +00068280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000682b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000682c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000682b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000682c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000682f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068310: 2020 2020 2020 207c 0a7c 6f31 203d 206b |.|o1 = k │ │ │ │ -00068320: 6b20 2020 2020 2020 2020 2020 2020 2020 k │ │ │ │ +00068300: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068310: 3120 3d20 6b6b 2020 2020 2020 2020 2020 1 = kk │ │ │ │ +00068320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068360: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068350: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683b0: 2020 2020 2020 207c 0a7c 6f31 203a 2051 |.|o1 : Q │ │ │ │ -000683c0: 756f 7469 656e 7452 696e 6720 2020 2020 uotientRing │ │ │ │ +000683a0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000683b0: 3120 3a20 5175 6f74 6965 6e74 5269 6e67 1 : QuotientRing │ │ │ │ +000683c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000683d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000683e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000683f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068400: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000683f0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068400: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068410: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068450: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2053 -------+.|i2 : S │ │ │ │ -00068460: 203d 206b 6b5b 612c 622c 632c 645d 2020 = kk[a,b,c,d] │ │ │ │ +00068440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068450: 3220 3a20 5320 3d20 6b6b 5b61 2c62 2c63 2 : S = kk[a,b,c │ │ │ │ +00068460: 2c64 5d20 2020 2020 2020 2020 2020 2020 ,d] │ │ │ │ 00068470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068490: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000684a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000684d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000684f0: 2020 2020 2020 207c 0a7c 6f32 203d 2053 |.|o2 = S │ │ │ │ +000684e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000684f0: 3220 3d20 5320 2020 2020 2020 2020 2020 2 = S │ │ │ │ 00068500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068540: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068530: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068590: 2020 2020 2020 207c 0a7c 6f32 203a 2050 |.|o2 : P │ │ │ │ -000685a0: 6f6c 796e 6f6d 6961 6c52 696e 6720 2020 olynomialRing │ │ │ │ +00068580: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068590: 3220 3a20 506f 6c79 6e6f 6d69 616c 5269 2 : PolynomialRi │ │ │ │ +000685a0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ 000685b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000685c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000685d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000685e0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000685d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000685e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000685f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068630: 2d2d 2d2d 2d2d 2d2b 0a7c 6933 203a 204d -------+.|i3 : M │ │ │ │ -00068640: 203d 2074 7275 6e63 6174 6528 332c 535e = truncate(3,S^ │ │ │ │ -00068650: 3129 2020 2020 2020 2020 2020 2020 2020 1) │ │ │ │ +00068620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068630: 3320 3a20 4d20 3d20 7472 756e 6361 7465 3 : M = truncate │ │ │ │ +00068640: 2833 2c53 5e31 2920 2020 2020 2020 2020 (3,S^1) │ │ │ │ +00068650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068680: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068670: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000686a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000686b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000686c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000686d0: 2020 2020 2020 207c 0a7c 6f33 203d 2069 |.|o3 = i │ │ │ │ -000686e0: 6d61 6765 207c 2064 3320 6364 3220 6264 mage | d3 cd2 bd │ │ │ │ -000686f0: 3220 6164 3220 6332 6420 6263 6420 6163 2 ad2 c2d bcd ac │ │ │ │ -00068700: 6420 6232 6420 6162 6420 6132 6420 6333 d b2d abd a2d c3 │ │ │ │ -00068710: 2062 6332 2061 6332 2062 3263 2061 6263 bc2 ac2 b2c abc │ │ │ │ -00068720: 2061 3263 2062 337c 0a7c 2020 2020 2020 a2c b3|.| │ │ │ │ +000686c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000686d0: 3320 3d20 696d 6167 6520 7c20 6433 2063 3 = image | d3 c │ │ │ │ +000686e0: 6432 2062 6432 2061 6432 2063 3264 2062 d2 bd2 ad2 c2d b │ │ │ │ +000686f0: 6364 2061 6364 2062 3264 2061 6264 2061 cd acd b2d abd a │ │ │ │ +00068700: 3264 2063 3320 6263 3220 6163 3220 6232 2d c3 bc2 ac2 b2 │ │ │ │ +00068710: 6320 6162 6320 6132 6320 6233 7c0a 7c20 c abc a2c b3|.| │ │ │ │ +00068720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068770: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068790: 2020 2020 2020 2031 2020 2020 2020 2020 1 │ │ │ │ +00068760: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00068780: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ +00068790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000687a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000687b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000687c0: 2020 2020 2020 207c 0a7c 6f33 203a 2053 |.|o3 : S │ │ │ │ -000687d0: 2d6d 6f64 756c 652c 2073 7562 6d6f 6475 -module, submodu │ │ │ │ -000687e0: 6c65 206f 6620 5320 2020 2020 2020 2020 le of S │ │ │ │ +000687b0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000687c0: 3320 3a20 532d 6d6f 6475 6c65 2c20 7375 3 : S-module, su │ │ │ │ +000687d0: 626d 6f64 756c 6520 6f66 2053 2020 2020 bmodule of S │ │ │ │ +000687e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000687f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068810: 2020 2020 2020 207c 0a7c 2d2d 2d2d 2d2d |.|------ │ │ │ │ +00068800: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +00068810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068860: 2d2d 2d2d 2d2d 2d7c 0a7c 6162 3220 6132 -------|.|ab2 a2 │ │ │ │ -00068870: 6220 6133 207c 2020 2020 2020 2020 2020 b a3 | │ │ │ │ +00068850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c61 ------------|.|a │ │ │ │ +00068860: 6232 2061 3262 2061 3320 7c20 2020 2020 b2 a2b a3 | │ │ │ │ +00068870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000688a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000688b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000688a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000688b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000688e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068900: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2062 -------+.|i4 : b │ │ │ │ -00068910: 6574 7469 206d 6174 7269 7820 5332 2830 etti matrix S2(0 │ │ │ │ -00068920: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +000688f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068900: 3420 3a20 6265 7474 6920 6d61 7472 6978 4 : betti matrix │ │ │ │ +00068910: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +00068920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068950: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068940: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000689b0: 2020 2020 2020 3020 2031 2020 2020 2020 0 1 │ │ │ │ +00068990: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000689a0: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ +000689b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000689c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000689d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000689f0: 2020 2020 2020 207c 0a7c 6f34 203d 2074 |.|o4 = t │ │ │ │ -00068a00: 6f74 616c 3a20 3120 3230 2020 2020 2020 otal: 1 20 │ │ │ │ +000689e0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000689f0: 3420 3d20 746f 7461 6c3a 2031 2032 3020 4 = total: 1 20 │ │ │ │ +00068a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a40: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068a50: 2020 2030 3a20 3120 202e 2020 2020 2020 0: 1 . │ │ │ │ +00068a30: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068a40: 2020 2020 2020 2020 303a 2031 2020 2e20 0: 1 . │ │ │ │ +00068a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068a90: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068aa0: 2020 2031 3a20 2e20 202e 2020 2020 2020 1: . . │ │ │ │ +00068a80: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068a90: 2020 2020 2020 2020 313a 202e 2020 2e20 1: . . │ │ │ │ +00068aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068ad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068ae0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068af0: 2020 2032 3a20 2e20 3230 2020 2020 2020 2: . 20 │ │ │ │ +00068ad0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068ae0: 2020 2020 2020 2020 323a 202e 2032 3020 2: . 20 │ │ │ │ +00068af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068b20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068b30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068b80: 2020 2020 2020 207c 0a7c 6f34 203a 2042 |.|o4 : B │ │ │ │ -00068b90: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +00068b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068b80: 3420 3a20 4265 7474 6954 616c 6c79 2020 4 : BettiTally │ │ │ │ +00068b90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068bd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00068bc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068c20: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2062 -------+.|i5 : b │ │ │ │ -00068c30: 6574 7469 206d 6174 7269 7820 5332 2831 etti matrix S2(1 │ │ │ │ -00068c40: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00068c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068c20: 3520 3a20 6265 7474 6920 6d61 7472 6978 5 : betti matrix │ │ │ │ +00068c30: 2053 3228 312c 4d29 2020 2020 2020 2020 S2(1,M) │ │ │ │ +00068c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068c70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00068c60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068cc0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068cd0: 2020 2020 2020 3020 2031 2020 2020 2020 0 1 │ │ │ │ +00068cb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068cc0: 2020 2020 2020 2020 2020 2030 2020 3120 0 1 │ │ │ │ +00068cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068cf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d10: 2020 2020 2020 207c 0a7c 6f35 203d 2074 |.|o5 = t │ │ │ │ -00068d20: 6f74 616c 3a20 3120 3230 2020 2020 2020 otal: 1 20 │ │ │ │ +00068d00: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068d10: 3520 3d20 746f 7461 6c3a 2031 2032 3020 5 = total: 1 20 │ │ │ │ +00068d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068d60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068d70: 2020 2030 3a20 3120 202e 2020 2020 2020 0: 1 . │ │ │ │ +00068d50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068d60: 2020 2020 2020 2020 303a 2031 2020 2e20 0: 1 . │ │ │ │ +00068d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068db0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068dc0: 2020 2031 3a20 2e20 202e 2020 2020 2020 1: . . │ │ │ │ +00068da0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068db0: 2020 2020 2020 2020 313a 202e 2020 2e20 1: . . │ │ │ │ +00068dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068e00: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00068e10: 2020 2032 3a20 2e20 3230 2020 2020 2020 2: . 20 │ │ │ │ +00068df0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00068e00: 2020 2020 2020 2020 323a 202e 2032 3020 2: . 20 │ │ │ │ +00068e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068e40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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+00068ee0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00068ef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00068f20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00068f40: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 204d -------+.|i6 : M │ │ │ │ -00068f50: 203d 2053 5e31 2f69 6e74 6572 7365 6374 = S^1/intersect │ │ │ │ -00068f60: 2869 6465 616c 2261 2c62 2c63 222c 2069 (ideal"a,b,c", i │ │ │ │ -00068f70: 6465 616c 2262 2c63 2c64 222c 6964 6561 deal"b,c,d",idea │ │ │ │ -00068f80: 6c22 632c 642c 6122 2c69 6465 616c 2264 l"c,d,a",ideal"d │ │ │ │ -00068f90: 2c61 2c62 2229 207c 0a7c 2020 2020 2020 ,a,b") |.| │ │ │ │ +00068f30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00068f40: 3620 3a20 4d20 3d20 535e 312f 696e 7465 6 : M = S^1/inte │ │ │ │ +00068f50: 7273 6563 7428 6964 6561 6c22 612c 622c rsect(ideal"a,b, │ │ │ │ +00068f60: 6322 2c20 6964 6561 6c22 622c 632c 6422 c", ideal"b,c,d" │ │ │ │ +00068f70: 2c69 6465 616c 2263 2c64 2c61 222c 6964 ,ideal"c,d,a",id │ │ │ │ +00068f80: 6561 6c22 642c 612c 6222 2920 7c0a 7c20 eal"d,a,b") |.| │ │ │ │ +00068f90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00068fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00068fe0: 2020 2020 2020 207c 0a7c 6f36 203d 2063 |.|o6 = c │ │ │ │ -00068ff0: 6f6b 6572 6e65 6c20 7c20 6364 2062 6420 okernel | cd bd │ │ │ │ -00069000: 6164 2062 6320 6163 2061 6220 7c20 2020 ad bc ac ab | │ │ │ │ +00068fd0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00068fe0: 3620 3d20 636f 6b65 726e 656c 207c 2063 6 = cokernel | c │ │ │ │ +00068ff0: 6420 6264 2061 6420 6263 2061 6320 6162 d bd ad bc ac ab │ │ │ │ +00069000: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 00069010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069030: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069020: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069080: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690a0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +00069070: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069090: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +000690a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000690b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000690d0: 2020 2020 2020 207c 0a7c 6f36 203a 2053 |.|o6 : S │ │ │ │ -000690e0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -000690f0: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +000690c0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +000690d0: 3620 3a20 532d 6d6f 6475 6c65 2c20 7175 6 : S-module, qu │ │ │ │ +000690e0: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +000690f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069120: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069110: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069150: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069170: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2070 -------+.|i7 : p │ │ │ │ -00069180: 7275 6e65 2073 6f75 7263 6520 5332 2830 rune source S2(0 │ │ │ │ -00069190: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00069160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +00069170: 3720 3a20 7072 756e 6520 736f 7572 6365 7 : prune source │ │ │ │ +00069180: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +00069190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000691a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000691b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000691c0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 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0a7c 2020 2020 2020 |.| │ │ │ │ +00069250: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692b0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000692c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692d0: 2020 2020 2020 3120 2020 2020 2020 2020 1 │ │ │ │ +000692a0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000692b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000692c0: 2020 2020 2020 2020 2020 2031 2020 2020 1 │ │ │ │ +000692d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000692e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000692f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069300: 2020 2020 2020 207c 0a7c 6f37 203a 2053 |.|o7 : S │ │ │ │ -00069310: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00069320: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +000692f0: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069300: 3720 3a20 532d 6d6f 6475 6c65 2c20 7175 7 : S-module, qu │ │ │ │ +00069310: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +00069320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069350: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069340: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069380: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000693a0: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2070 -------+.|i8 : p │ │ │ │ -000693b0: 7275 6e65 2074 6172 6765 7420 5332 2830 rune target S2(0 │ │ │ │ -000693c0: 2c4d 2920 2020 2020 2020 2020 2020 2020 ,M) │ │ │ │ +00069390: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000693a0: 3820 3a20 7072 756e 6520 7461 7267 6574 8 : prune target │ │ │ │ +000693b0: 2053 3228 302c 4d29 2020 2020 2020 2020 S2(0,M) │ │ │ │ +000693c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000693d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000693e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000693f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000693e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000693f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069440: 2020 2020 2020 207c 0a7c 6f38 203d 2063 |.|o8 = c │ │ │ │ -00069450: 6f6b 6572 6e65 6c20 7b2d 317d 207c 2064 okernel {-1} | d │ │ │ │ -00069460: 2063 2062 2030 2030 2030 2030 2030 2030 c b 0 0 0 0 0 0 │ │ │ │ -00069470: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00069480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069490: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000694a0: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -000694b0: 2030 2030 2064 2063 2061 2030 2030 2030 0 0 d c a 0 0 0 │ │ │ │ -000694c0: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -000694d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000694e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000694f0: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -00069500: 2030 2030 2030 2030 2030 2064 2062 2061 0 0 0 0 0 d b a │ │ │ │ -00069510: 2030 2030 2030 207c 2020 2020 2020 2020 0 0 0 | │ │ │ │ -00069520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069530: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069540: 2020 2020 2020 2020 7b2d 317d 207c 2030 {-1} | 0 │ │ │ │ -00069550: 2030 2030 2030 2030 2030 2030 2030 2030 0 0 0 0 0 0 0 0 │ │ │ │ -00069560: 2063 2062 2061 207c 2020 2020 2020 2020 c b a | │ │ │ │ -00069570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069580: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069430: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069440: 3820 3d20 636f 6b65 726e 656c 207b 2d31 8 = cokernel {-1 │ │ │ │ +00069450: 7d20 7c20 6420 6320 6220 3020 3020 3020 } | d c b 0 0 0 │ │ │ │ +00069460: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +00069470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069480: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069490: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +000694a0: 7d20 7c20 3020 3020 3020 6420 6320 6120 } | 0 0 0 d c a │ │ │ │ +000694b0: 3020 3020 3020 3020 3020 3020 7c20 2020 0 0 0 0 0 0 | │ │ │ │ +000694c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000694d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000694e0: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +000694f0: 7d20 7c20 3020 3020 3020 3020 3020 3020 } | 0 0 0 0 0 0 │ │ │ │ +00069500: 6420 6220 6120 3020 3020 3020 7c20 2020 d b a 0 0 0 | │ │ │ │ +00069510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069520: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069530: 2020 2020 2020 2020 2020 2020 207b 2d31 {-1 │ │ │ │ +00069540: 7d20 7c20 3020 3020 3020 3020 3020 3020 } | 0 0 0 0 0 0 │ │ │ │ +00069550: 3020 3020 3020 6320 6220 6120 7c20 2020 0 0 0 c b a | │ │ │ │ +00069560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069570: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00069580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000695a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000695b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -000695e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000695f0: 2020 2020 2020 3420 2020 2020 2020 2020 4 │ │ │ │ +000695c0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000695d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000695e0: 2020 2020 2020 2020 2020 2034 2020 2020 4 │ │ │ │ +000695f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069620: 2020 2020 2020 207c 0a7c 6f38 203a 2053 |.|o8 : S │ │ │ │ -00069630: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ -00069640: 7420 6f66 2053 2020 2020 2020 2020 2020 t of S │ │ │ │ +00069610: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +00069620: 3820 3a20 532d 6d6f 6475 6c65 2c20 7175 8 : S-module, qu │ │ │ │ +00069630: 6f74 6965 6e74 206f 6620 5320 2020 2020 otient of S │ │ │ │ +00069640: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069670: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069660: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000696a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000696c0: 2d2d 2d2d 2d2d 2d2b 0a0a 4174 206f 6e65 -------+..At one │ │ │ │ -000696d0: 2074 696d 6520 4445 2068 6f70 6564 2074 time DE hoped t │ │ │ │ -000696e0: 6861 742c 2069 6620 4d20 7765 7265 2061 hat, if M were a │ │ │ │ -000696f0: 206d 6f64 756c 6520 6f76 6572 2074 6865 module over the │ │ │ │ -00069700: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -00069710: 6563 7469 6f6e 2052 0a77 6974 6820 7265 ection R.with re │ │ │ │ -00069720: 7369 6475 6520 6669 656c 6420 6b2c 2074 sidue field k, t │ │ │ │ -00069730: 6865 6e20 7468 6520 6e61 7475 7261 6c20 hen the natural │ │ │ │ -00069740: 6d61 7020 6672 6f6d 2022 636f 6d70 6c65 map from "comple │ │ │ │ -00069750: 7465 2220 4578 7420 6d6f 6475 6c65 2022 te" Ext module " │ │ │ │ -00069760: 2877 6964 6568 6174 0a45 7874 295f 5228 (widehat.Ext)_R( │ │ │ │ -00069770: 4d2c 6b29 2220 746f 2074 6865 2053 322d M,k)" to the S2- │ │ │ │ -00069780: 6966 6963 6174 696f 6e20 6f66 2045 7874 ification of Ext │ │ │ │ -00069790: 5f52 284d 2c6b 2920 776f 756c 6420 6265 _R(M,k) would be │ │ │ │ -000697a0: 2073 7572 6a65 6374 6976 653b 0a65 7175 surjective;.equ │ │ │ │ -000697b0: 6976 616c 656e 746c 792c 2069 6620 4e20 ivalently, if N │ │ │ │ -000697c0: 7765 7265 2061 2073 7566 6669 6369 656e were a sufficien │ │ │ │ -000697d0: 746c 7920 6e65 6761 7469 7665 2073 797a tly negative syz │ │ │ │ -000697e0: 7967 7920 6f66 204d 2c20 7468 656e 2074 ygy of M, then t │ │ │ │ -000697f0: 6865 2066 6972 7374 0a6c 6f63 616c 2063 he first.local c │ │ │ │ -00069800: 6f68 6f6d 6f6c 6f67 7920 6d6f 6475 6c65 ohomology module │ │ │ │ -00069810: 206f 6620 4578 745f 5228 4d2c 6b29 2077 of Ext_R(M,k) w │ │ │ │ -00069820: 6f75 6c64 2062 6520 7a65 726f 2e20 5468 ould be zero. Th │ │ │ │ -00069830: 6973 2069 7320 6661 6c73 652c 2061 7320 is is false, as │ │ │ │ -00069840: 7368 6f77 6e20 6279 0a74 6865 2066 6f6c shown by.the fol │ │ │ │ -00069850: 6c6f 7769 6e67 2065 7861 6d70 6c65 3a0a lowing example:. │ │ │ │ -00069860: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000696b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a41 ------------+..A │ │ │ │ +000696c0: 7420 6f6e 6520 7469 6d65 2044 4520 686f t one time DE ho │ │ │ │ +000696d0: 7065 6420 7468 6174 2c20 6966 204d 2077 ped that, if M w │ │ │ │ +000696e0: 6572 6520 6120 6d6f 6475 6c65 206f 7665 ere a module ove │ │ │ │ +000696f0: 7220 7468 6520 636f 6d70 6c65 7465 2069 r the complete i │ │ │ │ +00069700: 6e74 6572 7365 6374 696f 6e20 520a 7769 ntersection R.wi │ │ │ │ +00069710: 7468 2072 6573 6964 7565 2066 6965 6c64 th residue field │ │ │ │ +00069720: 206b 2c20 7468 656e 2074 6865 206e 6174 k, then the nat │ │ │ │ +00069730: 7572 616c 206d 6170 2066 726f 6d20 2263 ural map from "c │ │ │ │ +00069740: 6f6d 706c 6574 6522 2045 7874 206d 6f64 omplete" Ext mod │ │ │ │ +00069750: 756c 6520 2228 7769 6465 6861 740a 4578 ule "(widehat.Ex │ │ │ │ +00069760: 7429 5f52 284d 2c6b 2922 2074 6f20 7468 t)_R(M,k)" to th │ │ │ │ +00069770: 6520 5332 2d69 6669 6361 7469 6f6e 206f e S2-ification o │ │ │ │ +00069780: 6620 4578 745f 5228 4d2c 6b29 2077 6f75 f Ext_R(M,k) wou │ │ │ │ +00069790: 6c64 2062 6520 7375 726a 6563 7469 7665 ld be surjective │ │ │ │ +000697a0: 3b0a 6571 7569 7661 6c65 6e74 6c79 2c20 ;.equivalently, │ │ │ │ +000697b0: 6966 204e 2077 6572 6520 6120 7375 6666 if N were a suff │ │ │ │ +000697c0: 6963 6965 6e74 6c79 206e 6567 6174 6976 iciently negativ │ │ │ │ +000697d0: 6520 7379 7a79 6779 206f 6620 4d2c 2074 e syzygy of M, t │ │ │ │ +000697e0: 6865 6e20 7468 6520 6669 7273 740a 6c6f hen the first.lo │ │ │ │ +000697f0: 6361 6c20 636f 686f 6d6f 6c6f 6779 206d cal cohomology m │ │ │ │ +00069800: 6f64 756c 6520 6f66 2045 7874 5f52 284d odule of Ext_R(M │ │ │ │ +00069810: 2c6b 2920 776f 756c 6420 6265 207a 6572 ,k) would be zer │ │ │ │ +00069820: 6f2e 2054 6869 7320 6973 2066 616c 7365 o. This is false │ │ │ │ +00069830: 2c20 6173 2073 686f 776e 2062 790a 7468 , as shown by.th │ │ │ │ +00069840: 6520 666f 6c6c 6f77 696e 6720 6578 616d e following exam │ │ │ │ +00069850: 706c 653a 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d ple:..+--------- │ │ │ │ +00069860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069870: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069880: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069890: 2d2d 2d2b 0a7c 6939 203a 2053 203d 205a ---+.|i9 : S = Z │ │ │ │ -000698a0: 5a2f 3130 315b 785f 302e 2e78 5f32 5d3b Z/101[x_0..x_2]; │ │ │ │ -000698b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000698c0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069880: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3920 3a20 --------+.|i9 : │ │ │ │ +00069890: 5320 3d20 5a5a 2f31 3031 5b78 5f30 2e2e S = ZZ/101[x_0.. │ │ │ │ +000698a0: 785f 325d 3b20 2020 2020 2020 2020 2020 x_2]; │ │ │ │ +000698b0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +000698c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000698d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000698e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000698f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069900: 3020 3a20 6666 203d 2061 7070 6c79 2833 0 : ff = apply(3 │ │ │ │ -00069910: 2c20 692d 3e78 5f69 5e32 293b 2020 2020 , i->x_i^2); │ │ │ │ -00069920: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069930: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000698f0: 2b0a 7c69 3130 203a 2066 6620 3d20 6170 +.|i10 : ff = ap │ │ │ │ +00069900: 706c 7928 332c 2069 2d3e 785f 695e 3229 ply(3, i->x_i^2) │ │ │ │ +00069910: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00069920: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069930: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069940: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069950: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069960: 2d2d 2d2b 0a7c 6931 3120 3a20 5220 3d20 ---+.|i11 : R = │ │ │ │ -00069970: 532f 6964 6561 6c20 6666 3b20 2020 2020 S/ideal ff; │ │ │ │ -00069980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069990: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069950: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3131 203a --------+.|i11 : │ │ │ │ +00069960: 2052 203d 2053 2f69 6465 616c 2066 663b R = S/ideal ff; │ │ │ │ +00069970: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069980: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069990: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000699a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000699b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000699c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -000699d0: 3220 3a20 4d20 3d20 636f 6b65 726e 656c 2 : M = cokernel │ │ │ │ -000699e0: 206d 6174 7269 7820 7b7b 785f 302c 2078 matrix {{x_0, x │ │ │ │ -000699f0: 5f31 2a78 5f32 7d7d 3b20 2020 2020 207c _1*x_2}}; | │ │ │ │ -00069a00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000699c0: 2b0a 7c69 3132 203a 204d 203d 2063 6f6b +.|i12 : M = cok │ │ │ │ +000699d0: 6572 6e65 6c20 6d61 7472 6978 207b 7b78 ernel matrix {{x │ │ │ │ +000699e0: 5f30 2c20 785f 312a 785f 327d 7d3b 2020 _0, x_1*x_2}}; │ │ │ │ +000699f0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a30: 2d2d 2d2b 0a7c 6931 3320 3a20 6220 3d20 ---+.|i13 : b = │ │ │ │ -00069a40: 353b 2020 2020 2020 2020 2020 2020 2020 5; │ │ │ │ -00069a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069a60: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069a20: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3133 203a --------+.|i13 : │ │ │ │ +00069a30: 2062 203d 2035 3b20 2020 2020 2020 2020 b = 5; │ │ │ │ +00069a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069a50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069a60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069aa0: 3420 3a20 4d62 203d 2070 7275 6e65 2073 4 : Mb = prune s │ │ │ │ -00069ab0: 797a 7967 794d 6f64 756c 6528 2d62 2c4d yzygyModule(-b,M │ │ │ │ -00069ac0: 293b 2020 2020 2020 2020 2020 2020 207c ); | │ │ │ │ -00069ad0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069a90: 2b0a 7c69 3134 203a 204d 6220 3d20 7072 +.|i14 : Mb = pr │ │ │ │ +00069aa0: 756e 6520 7379 7a79 6779 4d6f 6475 6c65 une syzygyModule │ │ │ │ +00069ab0: 282d 622c 4d29 3b20 2020 2020 2020 2020 (-b,M); │ │ │ │ +00069ac0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069ad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069ae0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069af0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069b00: 2d2d 2d2b 0a7c 6931 3520 3a20 4520 3d20 ---+.|i15 : E = │ │ │ │ -00069b10: 7072 756e 6520 6576 656e 4578 744d 6f64 prune evenExtMod │ │ │ │ -00069b20: 756c 6520 4d62 3b20 2020 2020 2020 2020 ule Mb; │ │ │ │ -00069b30: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069af0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3135 203a --------+.|i15 : │ │ │ │ +00069b00: 2045 203d 2070 7275 6e65 2065 7665 6e45 E = prune evenE │ │ │ │ +00069b10: 7874 4d6f 6475 6c65 204d 623b 2020 2020 xtModule Mb; │ │ │ │ +00069b20: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069b30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069b40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069b70: 3620 3a20 5332 6d61 7020 3d20 5332 2830 6 : S2map = S2(0 │ │ │ │ -00069b80: 2c45 293b 2020 2020 2020 2020 2020 2020 ,E); │ │ │ │ -00069b90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069ba0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069b60: 2b0a 7c69 3136 203a 2053 326d 6170 203d +.|i16 : S2map = │ │ │ │ +00069b70: 2053 3228 302c 4529 3b20 2020 2020 2020 S2(0,E); │ │ │ │ +00069b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069b90: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069bb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069bd0: 2020 207c 0a7c 6f31 3620 3a20 4d61 7472 |.|o16 : Matr │ │ │ │ -00069be0: 6978 2020 2020 2020 2020 2020 2020 2020 ix │ │ │ │ -00069bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069c00: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069bc0: 2020 2020 2020 2020 7c0a 7c6f 3136 203a |.|o16 : │ │ │ │ +00069bd0: 204d 6174 7269 7820 2020 2020 2020 2020 Matrix │ │ │ │ +00069be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069bf0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069c30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069c40: 3720 3a20 5345 203d 2070 7275 6e65 2074 7 : SE = prune t │ │ │ │ -00069c50: 6172 6765 7420 5332 6d61 703b 2020 2020 arget S2map; │ │ │ │ -00069c60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069c70: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069c30: 2b0a 7c69 3137 203a 2053 4520 3d20 7072 +.|i17 : SE = pr │ │ │ │ +00069c40: 756e 6520 7461 7267 6574 2053 326d 6170 une target S2map │ │ │ │ +00069c50: 3b20 2020 2020 2020 2020 2020 2020 2020 ; │ │ │ │ +00069c60: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069c80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069c90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069ca0: 2d2d 2d2b 0a7c 6931 3820 3a20 6578 7472 ---+.|i18 : extr │ │ │ │ -00069cb0: 6120 3d20 7072 756e 6520 636f 6b65 7220 a = prune coker │ │ │ │ -00069cc0: 5332 6d61 703b 2020 2020 2020 2020 2020 S2map; │ │ │ │ -00069cd0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069c90: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3138 203a --------+.|i18 : │ │ │ │ +00069ca0: 2065 7874 7261 203d 2070 7275 6e65 2063 extra = prune c │ │ │ │ +00069cb0: 6f6b 6572 2053 326d 6170 3b20 2020 2020 oker S2map; │ │ │ │ +00069cc0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069ce0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ -00069d10: 3920 3a20 4b45 203d 2070 7275 6e65 206b 9 : KE = prune k │ │ │ │ -00069d20: 6572 2053 326d 6170 3b20 2020 2020 2020 er S2map; │ │ │ │ -00069d30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00069d40: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00069d00: 2b0a 7c69 3139 203a 204b 4520 3d20 7072 +.|i19 : KE = pr │ │ │ │ +00069d10: 756e 6520 6b65 7220 5332 6d61 703b 2020 une ker S2map; │ │ │ │ +00069d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069d30: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00069d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069d70: 2d2d 2d2b 0a7c 6932 3020 3a20 6265 7474 ---+.|i20 : bett │ │ │ │ -00069d80: 6920 6672 6565 5265 736f 6c75 7469 6f6e i freeResolution │ │ │ │ -00069d90: 284d 622c 204c 656e 6774 684c 696d 6974 (Mb, LengthLimit │ │ │ │ -00069da0: 203d 3e20 3130 297c 0a7c 2020 2020 2020 => 10)|.| │ │ │ │ +00069d60: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 203a --------+.|i20 : │ │ │ │ +00069d70: 2062 6574 7469 2066 7265 6552 6573 6f6c betti freeResol │ │ │ │ +00069d80: 7574 696f 6e28 4d62 2c20 4c65 6e67 7468 ution(Mb, Length │ │ │ │ +00069d90: 4c69 6d69 7420 3d3e 2031 3029 7c0a 7c20 Limit => 10)|.| │ │ │ │ +00069da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069db0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069dd0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00069de0: 2020 2020 2020 2020 2020 2020 3020 2031 0 1 │ │ │ │ -00069df0: 2032 2033 2034 2035 2036 2037 2038 2020 2 3 4 5 6 7 8 │ │ │ │ -00069e00: 3920 3130 2020 2020 2020 2020 2020 207c 9 10 | │ │ │ │ -00069e10: 0a7c 6f32 3020 3d20 746f 7461 6c3a 2032 .|o20 = total: 2 │ │ │ │ -00069e20: 3020 3134 2039 2035 2032 2031 2032 2034 0 14 9 5 2 1 2 4 │ │ │ │ -00069e30: 2037 2031 3120 3136 2020 2020 2020 2020 7 11 16 │ │ │ │ -00069e40: 2020 207c 0a7c 2020 2020 2020 2020 202d |.| - │ │ │ │ -00069e50: 363a 2032 3020 3134 2039 2035 2032 202e 6: 20 14 9 5 2 . │ │ │ │ -00069e60: 202e 202e 202e 2020 2e20 202e 2020 2020 . . . . . │ │ │ │ -00069e70: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00069e80: 2020 202d 353a 2020 2e20 202e 202e 202e -5: . . . . │ │ │ │ -00069e90: 202e 2031 2031 2031 2031 2020 3120 2031 . 1 1 1 1 1 1 │ │ │ │ -00069ea0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00069eb0: 2020 2020 2020 202d 343a 2020 2e20 202e -4: . . │ │ │ │ -00069ec0: 202e 202e 202e 202e 2031 2033 2036 2031 . . . . 1 3 6 1 │ │ │ │ -00069ed0: 3020 3135 2020 2020 2020 2020 2020 207c 0 15 | │ │ │ │ -00069ee0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069dd0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00069de0: 2030 2020 3120 3220 3320 3420 3520 3620 0 1 2 3 4 5 6 │ │ │ │ +00069df0: 3720 3820 2039 2031 3020 2020 2020 2020 7 8 9 10 │ │ │ │ +00069e00: 2020 2020 7c0a 7c6f 3230 203d 2074 6f74 |.|o20 = tot │ │ │ │ +00069e10: 616c 3a20 3230 2031 3420 3920 3520 3220 al: 20 14 9 5 2 │ │ │ │ +00069e20: 3120 3220 3420 3720 3131 2031 3620 2020 1 2 4 7 11 16 │ │ │ │ +00069e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00069e40: 2020 2020 2d36 3a20 3230 2031 3420 3920 -6: 20 14 9 │ │ │ │ +00069e50: 3520 3220 2e20 2e20 2e20 2e20 202e 2020 5 2 . . . . . │ │ │ │ +00069e60: 2e20 2020 2020 2020 2020 2020 7c0a 7c20 . |.| │ │ │ │ +00069e70: 2020 2020 2020 2020 2d35 3a20 202e 2020 -5: . │ │ │ │ +00069e80: 2e20 2e20 2e20 2e20 3120 3120 3120 3120 . . . . 1 1 1 1 │ │ │ │ +00069e90: 2031 2020 3120 2020 2020 2020 2020 2020 1 1 │ │ │ │ +00069ea0: 7c0a 7c20 2020 2020 2020 2020 2d34 3a20 |.| -4: │ │ │ │ +00069eb0: 202e 2020 2e20 2e20 2e20 2e20 2e20 3120 . . . . . . 1 │ │ │ │ +00069ec0: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +00069ed0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f10: 2020 207c 0a7c 6f32 3020 3a20 4265 7474 |.|o20 : Bett │ │ │ │ -00069f20: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ -00069f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069f40: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +00069f00: 2020 2020 2020 2020 7c0a 7c6f 3230 203a |.|o20 : │ │ │ │ +00069f10: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +00069f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00069f30: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +00069f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00069f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00069f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -00069f80: 3120 3a20 6170 706c 7920 2835 2c20 692d 1 : apply (5, i- │ │ │ │ -00069f90: 3e20 6869 6c62 6572 7446 756e 6374 696f > hilbertFunctio │ │ │ │ -00069fa0: 6e28 692c 204b 4529 2920 2020 2020 207c n(i, KE)) | │ │ │ │ -00069fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00069f70: 2b0a 7c69 3231 203a 2061 7070 6c79 2028 +.|i21 : apply ( │ │ │ │ +00069f80: 352c 2069 2d3e 2068 696c 6265 7274 4675 5, i-> hilbertFu │ │ │ │ +00069f90: 6e63 7469 6f6e 2869 2c20 4b45 2929 2020 nction(i, KE)) │ │ │ │ +00069fa0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +00069fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00069fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00069fe0: 2020 207c 0a7c 6f32 3120 3d20 7b32 302c |.|o21 = {20, │ │ │ │ -00069ff0: 2039 2c20 322c 2030 2c20 307d 2020 2020 9, 2, 0, 0} │ │ │ │ -0006a000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a010: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00069fd0: 2020 2020 2020 2020 7c0a 7c6f 3231 203d |.|o21 = │ │ │ │ +00069fe0: 207b 3230 2c20 392c 2032 2c20 302c 2030 {20, 9, 2, 0, 0 │ │ │ │ +00069ff0: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +0006a000: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a040: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a050: 3120 3a20 4c69 7374 2020 2020 2020 2020 1 : List │ │ │ │ +0006a040: 7c0a 7c6f 3231 203a 204c 6973 7420 2020 |.|o21 : List │ │ │ │ +0006a050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a080: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0006a070: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006a080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a0b0: 2d2d 2d2b 0a7c 6932 3220 3a20 6170 706c ---+.|i22 : appl │ │ │ │ -0006a0c0: 7920 2835 2c20 692d 3e20 6869 6c62 6572 y (5, i-> hilber │ │ │ │ -0006a0d0: 7446 756e 6374 696f 6e28 692c 2045 2929 tFunction(i, E)) │ │ │ │ -0006a0e0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006a0a0: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3232 203a --------+.|i22 : │ │ │ │ +0006a0b0: 2061 7070 6c79 2028 352c 2069 2d3e 2068 apply (5, i-> h │ │ │ │ +0006a0c0: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ +0006a0d0: 2c20 4529 2920 2020 2020 2020 7c0a 7c20 , E)) |.| │ │ │ │ +0006a0e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a110: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a120: 3220 3d20 7b32 302c 2039 2c20 322c 2032 2 = {20, 9, 2, 2 │ │ │ │ -0006a130: 2c20 377d 2020 2020 2020 2020 2020 2020 , 7} │ │ │ │ -0006a140: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a150: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a110: 7c0a 7c6f 3232 203d 207b 3230 2c20 392c |.|o22 = {20, 9, │ │ │ │ +0006a120: 2032 2c20 322c 2037 7d20 2020 2020 2020 2, 2, 7} │ │ │ │ +0006a130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a140: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a180: 2020 207c 0a7c 6f32 3220 3a20 4c69 7374 |.|o22 : List │ │ │ │ +0006a170: 2020 2020 2020 2020 7c0a 7c6f 3232 203a |.|o22 : │ │ │ │ +0006a180: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0006a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a1b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006a1a0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006a1b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a1c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a1e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ -0006a1f0: 3320 3a20 6170 706c 7920 2835 2c20 692d 3 : apply (5, i- │ │ │ │ -0006a200: 3e20 6869 6c62 6572 7446 756e 6374 696f > hilbertFunctio │ │ │ │ -0006a210: 6e28 692c 2053 4529 2920 2020 2020 207c n(i, SE)) | │ │ │ │ -0006a220: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a1e0: 2b0a 7c69 3233 203a 2061 7070 6c79 2028 +.|i23 : apply ( │ │ │ │ +0006a1f0: 352c 2069 2d3e 2068 696c 6265 7274 4675 5, i-> hilbertFu │ │ │ │ +0006a200: 6e63 7469 6f6e 2869 2c20 5345 2929 2020 nction(i, SE)) │ │ │ │ +0006a210: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a250: 2020 207c 0a7c 6f32 3320 3d20 7b31 2c20 |.|o23 = {1, │ │ │ │ -0006a260: 312c 2031 2c20 322c 2037 7d20 2020 2020 1, 1, 2, 7} │ │ │ │ -0006a270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a280: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006a240: 2020 2020 2020 2020 7c0a 7c6f 3233 203d |.|o23 = │ │ │ │ +0006a250: 207b 312c 2031 2c20 312c 2032 2c20 377d {1, 1, 1, 2, 7} │ │ │ │ +0006a260: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a270: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006a280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a2b0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a2c0: 3320 3a20 4c69 7374 2020 2020 2020 2020 3 : List │ │ │ │ +0006a2b0: 7c0a 7c6f 3233 203a 204c 6973 7420 2020 |.|o23 : List │ │ │ │ +0006a2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a2d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a2e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a2f0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0006a2e0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +0006a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006a320: 2d2d 2d2b 0a7c 6932 3420 3a20 6170 706c ---+.|i24 : appl │ │ │ │ -0006a330: 7920 2835 2c20 692d 3e20 6869 6c62 6572 y (5, i-> hilber │ │ │ │ -0006a340: 7446 756e 6374 696f 6e28 692c 2065 7874 tFunction(i, ext │ │ │ │ -0006a350: 7261 2929 2020 207c 0a7c 2020 2020 2020 ra)) |.| │ │ │ │ +0006a310: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3234 203a --------+.|i24 : │ │ │ │ +0006a320: 2061 7070 6c79 2028 352c 2069 2d3e 2068 apply (5, i-> h │ │ │ │ +0006a330: 696c 6265 7274 4675 6e63 7469 6f6e 2869 ilbertFunction(i │ │ │ │ +0006a340: 2c20 6578 7472 6129 2920 2020 7c0a 7c20 , extra)) |.| │ │ │ │ +0006a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a380: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006a390: 3420 3d20 7b31 2c20 312c 2031 2c20 302c 4 = {1, 1, 1, 0, │ │ │ │ -0006a3a0: 2030 7d20 2020 2020 2020 2020 2020 2020 0} │ │ │ │ -0006a3b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006a3c0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006a380: 7c0a 7c6f 3234 203d 207b 312c 2031 2c20 |.|o24 = {1, 1, │ │ │ │ +0006a390: 312c 2030 2c20 307d 2020 2020 2020 2020 1, 0, 0} │ │ │ │ +0006a3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006a3b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006a3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006a3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006a3f0: 2020 207c 0a7c 6f32 3420 3a20 4c69 7374 |.|o24 : List │ │ │ │ +0006a3e0: 2020 2020 2020 2020 7c0a 7c6f 3234 203a |.|o24 : │ │ │ │ +0006a3f0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ 0006a400: 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e object *note S │ │ │ │ -0006a740: 323a 2053 322c 2069 7320 6120 2a6e 6f74 2: S2, is a *not │ │ │ │ -0006a750: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ -0006a760: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ -0006a770: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ -0006a780: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ +0006a450: 2b0a 0a43 6176 6561 740a 3d3d 3d3d 3d3d +..Caveat.====== │ │ │ │ +0006a460: 0a0a 5465 7874 2053 322d 6966 6963 6174 ..Text S2-ificat │ │ │ │ +0006a470: 696f 6e20 6973 2072 656c 6174 6564 2074 ion is related t │ │ │ │ +0006a480: 6f20 636f 6d70 7574 696e 6720 636f 686f o computing coho │ │ │ │ +0006a490: 6d6f 6c6f 6779 2061 6e64 2074 6f20 636f mology and to co │ │ │ │ +0006a4a0: 6d70 7574 696e 6720 696e 7465 6772 616c mputing integral │ │ │ │ +0006a4b0: 0a63 6c6f 7375 7265 3b20 7468 6572 6520 .closure; there │ │ │ │ +0006a4c0: 6172 6520 7363 7269 7074 7320 696e 2074 are scripts 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acaulay2-1.26.06 │ │ │ │ -0006a820: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ -0006a830: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ -0006a840: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006a850: 6573 6f6c 7574 696f 6e73 2e6d 323a 3338 esolutions.m2:38 │ │ │ │ -0006a860: 3833 3a30 2e0a 1f0a 4669 6c65 3a20 436f 83:0....File: Co │ │ │ │ -0006a870: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006a880: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ -0006a890: 666f 2c20 4e6f 6465 3a20 5368 616d 6173 fo, Node: Shamas │ │ │ │ -0006a8a0: 682c 204e 6578 743a 2073 706c 6974 7469 h, Next: splitti │ │ │ │ -0006a8b0: 6e67 732c 2050 7265 763a 2053 322c 2055 ngs, Prev: S2, U │ │ │ │ -0006a8c0: 703a 2054 6f70 0a0a 5368 616d 6173 6820 p: Top..Shamash │ │ │ │ -0006a8d0: 2d2d 2043 6f6d 7075 7465 7320 7468 6520 -- Computes the │ │ │ │ -0006a8e0: 5368 616d 6173 6820 436f 6d70 6c65 780a Shamash Complex. │ │ │ │ +0006a7c0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +0006a7d0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +0006a7e0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +0006a7f0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +0006a800: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +0006a810: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +0006a820: 756c 6179 322f 7061 636b 6167 6573 2f0a ulay2/packages/. │ │ │ │ +0006a830: 436f 6d70 6c65 7465 496e 7465 7273 6563 CompleteIntersec │ │ │ │ +0006a840: 7469 6f6e 5265 736f 6c75 7469 6f6e 732e tionResolutions. │ │ │ │ +0006a850: 6d32 3a33 3838 333a 302e 0a1f 0a46 696c m2:3883:0....Fil │ │ │ │ +0006a860: 653a 2043 6f6d 706c 6574 6549 6e74 6572 e: CompleteInter │ │ │ │ +0006a870: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006a880: 6e73 2e69 6e66 6f2c 204e 6f64 653a 2053 ns.info, Node: S │ │ │ │ +0006a890: 6861 6d61 7368 2c20 4e65 7874 3a20 7370 hamash, Next: sp │ │ │ │ +0006a8a0: 6c69 7474 696e 6773 2c20 5072 6576 3a20 littings, Prev: │ │ │ │ +0006a8b0: 5332 2c20 5570 3a20 546f 700a 0a53 6861 S2, Up: Top..Sha │ │ │ │ +0006a8c0: 6d61 7368 202d 2d20 436f 6d70 7574 6573 mash -- Computes │ │ │ │ +0006a8d0: 2074 6865 2053 6861 6d61 7368 2043 6f6d the Shamash Com │ │ │ │ +0006a8e0: 706c 6578 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a plex.*********** │ │ │ │ 0006a8f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006a900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006a910: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ -0006a920: 6765 3a20 0a20 2020 2020 2020 2046 4620 ge: . FF │ │ │ │ -0006a930: 3d20 5368 616d 6173 6828 6666 2c46 2c6c = Shamash(ff,F,l │ │ │ │ -0006a940: 656e 290a 2020 2020 2020 2020 4646 203d en). FF = │ │ │ │ -0006a950: 2053 6861 6d61 7368 2852 6261 722c 462c Shamash(Rbar,F, │ │ │ │ -0006a960: 6c65 6e29 0a20 202a 2049 6e70 7574 733a len). * Inputs: │ │ │ │ -0006a970: 0a20 2020 2020 202a 2066 662c 2061 202a . * ff, a * │ │ │ │ -0006a980: 6e6f 7465 206d 6174 7269 783a 2028 4d61 note matrix: (Ma │ │ │ │ -0006a990: 6361 756c 6179 3244 6f63 294d 6174 7269 caulay2Doc)Matri │ │ │ │ -0006a9a0: 782c 2c20 3120 7820 3120 4d61 7472 6978 x,, 1 x 1 Matrix │ │ │ │ -0006a9b0: 206f 7665 7220 7269 6e67 2046 2e0a 2020 over ring F.. │ │ │ │ -0006a9c0: 2020 2020 2a20 5262 6172 2c20 6120 2a6e * Rbar, a *n │ │ │ │ -0006a9d0: 6f74 6520 7269 6e67 3a20 284d 6163 6175 ote ring: (Macau │ │ │ │ -0006a9e0: 6c61 7932 446f 6329 5269 6e67 2c2c 2072 lay2Doc)Ring,, r │ │ │ │ -0006a9f0: 696e 6720 4620 6d6f 6420 6964 6561 6c20 ing F mod ideal │ │ │ │ -0006aa00: 6666 0a20 2020 2020 202a 2046 2c20 6120 ff. * F, a │ │ │ │ -0006aa10: 2a6e 6f74 6520 636f 6d70 6c65 783a 2028 *note complex: ( │ │ │ │ -0006aa20: 436f 6d70 6c65 7865 7329 436f 6d70 6c65 Complexes)Comple │ │ │ │ -0006aa30: 782c 2c20 6465 6669 6e65 6420 6f76 6572 x,, defined over │ │ │ │ -0006aa40: 2072 696e 6720 6666 0a20 2020 2020 202a ring ff. * │ │ │ │ -0006aa50: 206c 656e 2c20 616e 202a 6e6f 7465 2069 len, an *note i │ │ │ │ -0006aa60: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ -0006aa70: 7932 446f 6329 5a5a 2c2c 200a 2020 2a20 y2Doc)ZZ,, . * │ │ │ │ -0006aa80: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ -0006aa90: 2046 462c 2061 202a 6e6f 7465 2063 6f6d FF, a *note com │ │ │ │ -0006aaa0: 706c 6578 3a20 2843 6f6d 706c 6578 6573 plex: (Complexes │ │ │ │ -0006aab0: 2943 6f6d 706c 6578 2c2c 2063 6861 696e )Complex,, chain │ │ │ │ -0006aac0: 2063 6f6d 706c 6578 206f 7665 7220 2872 complex over (r │ │ │ │ -0006aad0: 696e 670a 2020 2020 2020 2020 4629 2f28 ing. F)/( │ │ │ │ -0006aae0: 6964 6561 6c20 6666 290a 0a44 6573 6372 ideal ff)..Descr │ │ │ │ -0006aaf0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -0006ab00: 3d3d 0a0a 4c65 7420 5220 3d20 7269 6e67 ==..Let R = ring │ │ │ │ -0006ab10: 2046 203d 2072 696e 6720 6666 2c20 616e F = ring ff, an │ │ │ │ -0006ab20: 6420 5262 6172 203d 2052 2f28 6964 6561 d Rbar = R/(idea │ │ │ │ -0006ab30: 6c20 6629 2c20 7768 6572 6520 6666 203d l f), where ff = │ │ │ │ -0006ab40: 206d 6174 7269 787b 7b66 7d7d 2069 7320 matrix{{f}} is │ │ │ │ -0006ab50: 610a 3178 3120 6d61 7472 6978 2077 686f a.1x1 matrix who │ │ │ │ -0006ab60: 7365 2065 6e74 7279 2069 7320 6120 6e6f se entry is a no │ │ │ │ -0006ab70: 6e7a 6572 6f64 6976 6973 6f72 2069 6e20 nzerodivisor in │ │ │ │ -0006ab80: 522e 2054 6865 2063 6f6d 706c 6578 2046 R. The complex F │ │ │ │ -0006ab90: 2073 686f 756c 6420 6164 6d69 7420 610a should admit a. │ │ │ │ -0006aba0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ -0006abb0: 2068 6f6d 6f74 6f70 6965 7320 666f 7220 homotopies for │ │ │ │ -0006abc0: 7468 6520 656e 7472 7920 6f66 2066 662c the entry of ff, │ │ │ │ -0006abd0: 2072 6574 7572 6e65 6420 6279 2074 6865 returned by the │ │ │ │ -0006abe0: 2063 616c 6c0a 6d61 6b65 486f 6d6f 746f call.makeHomoto │ │ │ │ -0006abf0: 7069 6573 2866 662c 4629 2e0a 0a54 6865 pies(ff,F)...The │ │ │ │ -0006ac00: 2063 6f6d 706c 6578 2046 4620 6861 7320 complex FF has │ │ │ │ -0006ac10: 7465 726d 730a 0a46 465f 7b32 2a69 7d20 terms..FF_{2*i} │ │ │ │ -0006ac20: 3d20 5262 6172 2a2a 2846 5f30 202b 2b20 = Rbar**(F_0 ++ │ │ │ │ -0006ac30: 465f 3220 2b2b 202e 2e20 2b2b 2046 5f69 F_2 ++ .. ++ F_i │ │ │ │ -0006ac40: 290a 0a46 465f 7b32 2a69 2b31 7d20 3d20 )..FF_{2*i+1} = │ │ │ │ -0006ac50: 5262 6172 2a2a 2846 5f31 202b 2b20 465f Rbar**(F_1 ++ F_ │ │ │ │ -0006ac60: 3320 2b2b 2e2e 2b2b 465f 7b32 2a69 2b31 3 ++..++F_{2*i+1 │ │ │ │ -0006ac70: 7d29 0a0a 616e 6420 6d61 7073 206d 6164 })..and maps mad │ │ │ │ -0006ac80: 6520 6672 6f6d 2074 6865 2068 6967 6865 e from the highe │ │ │ │ -0006ac90: 7220 686f 6d6f 746f 7069 6573 2e0a 0a46 r homotopies...F │ │ │ │ -0006aca0: 6f72 2074 6865 2063 6173 6520 6f66 2061 or the case of a │ │ │ │ -0006acb0: 2063 6f6d 706c 6574 6520 696e 7465 7273 complete inters │ │ │ │ -0006acc0: 6563 7469 6f6e 206f 6620 6869 6768 6572 ection of higher │ │ │ │ -0006acd0: 2063 6f64 696d 656e 7369 6f6e 2c20 6f72 codimension, or │ │ │ │ -0006ace0: 2074 6f20 7365 6520 7468 650a 636f 6d70 to see the.comp │ │ │ │ -0006acf0: 6f6e 656e 7473 206f 6620 7468 6520 7265 onents of the re │ │ │ │ -0006ad00: 736f 6c75 7469 6f6e 2061 7320 7375 6d6d solution as summ │ │ │ │ -0006ad10: 616e 6473 206f 6620 4646 5f6a 2c20 7573 ands of FF_j, us │ │ │ │ -0006ad20: 6520 7468 6520 726f 7574 696e 650a 4569 e the routine.Ei │ │ │ │ -0006ad30: 7365 6e62 7564 5368 616d 6173 6820 696e senbudShamash in │ │ │ │ -0006ad40: 7374 6561 642e 0a0a 2b2d 2d2d 2d2d 2d2d stead...+------- │ │ │ │ +0006a900: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +0006a910: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ +0006a920: 2020 4646 203d 2053 6861 6d61 7368 2866 FF = Shamash(f │ │ │ │ +0006a930: 662c 462c 6c65 6e29 0a20 2020 2020 2020 f,F,len). │ │ │ │ +0006a940: 2046 4620 3d20 5368 616d 6173 6828 5262 FF = Shamash(Rb │ │ │ │ +0006a950: 6172 2c46 2c6c 656e 290a 2020 2a20 496e ar,F,len). * In │ │ │ │ +0006a960: 7075 7473 3a0a 2020 2020 2020 2a20 6666 puts:. * ff │ │ │ │ +0006a970: 2c20 6120 2a6e 6f74 6520 6d61 7472 6978 , a *note matrix │ │ │ │ +0006a980: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006a990: 4d61 7472 6978 2c2c 2031 2078 2031 204d Matrix,, 1 x 1 M │ │ │ │ +0006a9a0: 6174 7269 7820 6f76 6572 2072 696e 6720 atrix over ring │ │ │ │ +0006a9b0: 462e 0a20 2020 2020 202a 2052 6261 722c F.. * Rbar, │ │ │ │ +0006a9c0: 2061 202a 6e6f 7465 2072 696e 673a 2028 a *note ring: ( │ │ │ │ +0006a9d0: 4d61 6361 756c 6179 3244 6f63 2952 696e Macaulay2Doc)Rin │ │ │ │ +0006a9e0: 672c 2c20 7269 6e67 2046 206d 6f64 2069 g,, ring F mod i │ │ │ │ +0006a9f0: 6465 616c 2066 660a 2020 2020 2020 2a20 deal ff. * │ │ │ │ +0006aa00: 462c 2061 202a 6e6f 7465 2063 6f6d 706c F, a *note compl │ │ │ │ +0006aa10: 6578 3a20 2843 6f6d 706c 6578 6573 2943 ex: (Complexes)C │ │ │ │ +0006aa20: 6f6d 706c 6578 2c2c 2064 6566 696e 6564 omplex,, defined │ │ │ │ +0006aa30: 206f 7665 7220 7269 6e67 2066 660a 2020 over ring ff. │ │ │ │ +0006aa40: 2020 2020 2a20 6c65 6e2c 2061 6e20 2a6e * len, an *n │ │ │ │ +0006aa50: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +0006aa60: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +0006aa70: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +0006aa80: 2020 2020 2a20 4646 2c20 6120 2a6e 6f74 * FF, a *not │ │ │ │ +0006aa90: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +0006aaa0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +0006aab0: 6368 6169 6e20 636f 6d70 6c65 7820 6f76 chain complex ov │ │ │ │ +0006aac0: 6572 2028 7269 6e67 0a20 2020 2020 2020 er (ring. │ │ │ │ +0006aad0: 2046 292f 2869 6465 616c 2066 6629 0a0a F)/(ideal ff).. │ │ │ │ +0006aae0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ +0006aaf0: 3d3d 3d3d 3d3d 3d0a 0a4c 6574 2052 203d =======..Let R = │ │ │ │ +0006ab00: 2072 696e 6720 4620 3d20 7269 6e67 2066 ring F = ring f │ │ │ │ +0006ab10: 662c 2061 6e64 2052 6261 7220 3d20 522f f, and Rbar = R/ │ │ │ │ +0006ab20: 2869 6465 616c 2066 292c 2077 6865 7265 (ideal f), where │ │ │ │ +0006ab30: 2066 6620 3d20 6d61 7472 6978 7b7b 667d ff = matrix{{f} │ │ │ │ +0006ab40: 7d20 6973 2061 0a31 7831 206d 6174 7269 } is a.1x1 matri │ │ │ │ +0006ab50: 7820 7768 6f73 6520 656e 7472 7920 6973 x whose entry is │ │ │ │ +0006ab60: 2061 206e 6f6e 7a65 726f 6469 7669 736f a nonzerodiviso │ │ │ │ +0006ab70: 7220 696e 2052 2e20 5468 6520 636f 6d70 r in R. The comp │ │ │ │ +0006ab80: 6c65 7820 4620 7368 6f75 6c64 2061 646d lex F should adm │ │ │ │ +0006ab90: 6974 2061 0a73 7973 7465 6d20 6f66 2068 it a.system of h │ │ │ │ +0006aba0: 6967 6865 7220 686f 6d6f 746f 7069 6573 igher homotopies │ │ │ │ +0006abb0: 2066 6f72 2074 6865 2065 6e74 7279 206f for the entry o │ │ │ │ +0006abc0: 6620 6666 2c20 7265 7475 726e 6564 2062 f ff, returned b │ │ │ │ +0006abd0: 7920 7468 6520 6361 6c6c 0a6d 616b 6548 y the call.makeH │ │ │ │ +0006abe0: 6f6d 6f74 6f70 6965 7328 6666 2c46 292e omotopies(ff,F). │ │ │ │ +0006abf0: 0a0a 5468 6520 636f 6d70 6c65 7820 4646 ..The complex FF │ │ │ │ +0006ac00: 2068 6173 2074 6572 6d73 0a0a 4646 5f7b has terms..FF_{ │ │ │ │ +0006ac10: 322a 697d 203d 2052 6261 722a 2a28 465f 2*i} = Rbar**(F_ │ │ │ │ +0006ac20: 3020 2b2b 2046 5f32 202b 2b20 2e2e 202b 0 ++ F_2 ++ .. + │ │ │ │ +0006ac30: 2b20 465f 6929 0a0a 4646 5f7b 322a 692b + F_i)..FF_{2*i+ │ │ │ │ +0006ac40: 317d 203d 2052 6261 722a 2a28 465f 3120 1} = Rbar**(F_1 │ │ │ │ +0006ac50: 2b2b 2046 5f33 202b 2b2e 2e2b 2b46 5f7b ++ F_3 ++..++F_{ │ │ │ │ +0006ac60: 322a 692b 317d 290a 0a61 6e64 206d 6170 2*i+1})..and map │ │ │ │ +0006ac70: 7320 6d61 6465 2066 726f 6d20 7468 6520 s made from the │ │ │ │ +0006ac80: 6869 6768 6572 2068 6f6d 6f74 6f70 6965 higher homotopie │ │ │ │ +0006ac90: 732e 0a0a 466f 7220 7468 6520 6361 7365 s...For the case │ │ │ │ +0006aca0: 206f 6620 6120 636f 6d70 6c65 7465 2069 of a complete i │ │ │ │ +0006acb0: 6e74 6572 7365 6374 696f 6e20 6f66 2068 ntersection of h │ │ │ │ +0006acc0: 6967 6865 7220 636f 6469 6d65 6e73 696f igher codimensio │ │ │ │ +0006acd0: 6e2c 206f 7220 746f 2073 6565 2074 6865 n, or to see the │ │ │ │ +0006ace0: 0a63 6f6d 706f 6e65 6e74 7320 6f66 2074 .components of t │ │ │ │ +0006acf0: 6865 2072 6573 6f6c 7574 696f 6e20 6173 he resolution as │ │ │ │ +0006ad00: 2073 756d 6d61 6e64 7320 6f66 2046 465f summands of FF_ │ │ │ │ +0006ad10: 6a2c 2075 7365 2074 6865 2072 6f75 7469 j, use the routi │ │ │ │ +0006ad20: 6e65 0a45 6973 656e 6275 6453 6861 6d61 ne.EisenbudShama │ │ │ │ +0006ad30: 7368 2069 6e73 7465 6164 2e0a 0a2b 2d2d sh instead...+-- │ │ │ │ +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 2d2b 0a7c -------------+.| │ │ │ │ -0006ad80: 6931 203a 2053 203d 205a 5a2f 3130 315b i1 : S = ZZ/101[ │ │ │ │ -0006ad90: 782c 792c 7a5d 2020 2020 2020 2020 2020 x,y,z] │ │ │ │ -0006ada0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006adb0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006ad70: 2d2d 2b0a 7c69 3120 3a20 5320 3d20 5a5a --+.|i1 : S = ZZ │ │ │ │ +0006ad80: 2f31 3031 5b78 2c79 2c7a 5d20 2020 2020 /101[x,y,z] │ │ │ │ +0006ad90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ada0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006adb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006adc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006add0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ade0: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ -0006adf0: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0006ade0: 7c0a 7c6f 3120 3d20 5320 2020 2020 2020 |.|o1 = S │ │ │ │ +0006adf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006ae10: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006ae20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ae30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae50: 2020 2020 2020 2020 207c 0a7c 6f31 203a |.|o1 : │ │ │ │ -0006ae60: 2050 6f6c 796e 6f6d 6961 6c52 696e 6720 PolynomialRing │ │ │ │ +0006ae40: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006ae50: 7c6f 3120 3a20 506f 6c79 6e6f 6d69 616c |o1 : Polynomial │ │ │ │ +0006ae60: 5269 6e67 2020 2020 2020 2020 2020 2020 Ring │ │ │ │ 0006ae70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ae90: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +0006ae80: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006ae90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006aea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006aec0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2052 -------+.|i2 : R │ │ │ │ -0006aed0: 203d 2053 2f69 6465 616c 2278 332c 7933 = S/ideal"x3,y3 │ │ │ │ -0006aee0: 2220 2020 2020 2020 2020 2020 2020 2020 " │ │ │ │ -0006aef0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006af00: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006aeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0006aec0: 3220 3a20 5220 3d20 532f 6964 6561 6c22 2 : R = S/ideal" │ │ │ │ +0006aed0: 7833 2c79 3322 2020 2020 2020 2020 2020 x3,y3" │ │ │ │ +0006aee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006aef0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006af00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af30: 2020 2020 207c 0a7c 6f32 203d 2052 2020 |.|o2 = R │ │ │ │ +0006af20: 2020 2020 2020 2020 2020 7c0a 7c6f 3220 |.|o2 │ │ │ │ +0006af30: 3d20 5220 2020 2020 2020 2020 2020 2020 = R │ │ │ │ 0006af40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006af60: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006af70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006af80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006af90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006afa0: 2020 207c 0a7c 6f32 203a 2051 756f 7469 |.|o2 : Quoti │ │ │ │ -0006afb0: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ -0006afc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006afd0: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006af90: 2020 2020 2020 2020 7c0a 7c6f 3220 3a20 |.|o2 : │ │ │ │ +0006afa0: 5175 6f74 6965 6e74 5269 6e67 2020 2020 QuotientRing │ │ │ │ +0006afb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006afc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006afd0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0006afe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006aff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b010: 2d2b 0a7c 6933 203a 204d 203d 2052 5e31 -+.|i3 : M = R^1 │ │ │ │ -0006b020: 2f69 6465 616c 2878 2c79 2c7a 2920 2020 /ideal(x,y,z) │ │ │ │ -0006b030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b040: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b000: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 4d20 ------+.|i3 : M │ │ │ │ +0006b010: 3d20 525e 312f 6964 6561 6c28 782c 792c = R^1/ideal(x,y, │ │ │ │ +0006b020: 7a29 2020 2020 2020 2020 2020 2020 2020 z) │ │ │ │ +0006b030: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b070: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b080: 0a7c 6f33 203d 2063 6f6b 6572 6e65 6c20 .|o3 = cokernel │ │ │ │ -0006b090: 7c20 7820 7920 7a20 7c20 2020 2020 2020 | x y z | │ │ │ │ -0006b0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b0b0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006b070: 2020 2020 7c0a 7c6f 3320 3d20 636f 6b65 |.|o3 = coke │ │ │ │ +0006b080: 726e 656c 207c 2078 2079 207a 207c 2020 rnel | x y z | │ │ │ │ +0006b090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b0a0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b0e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b0e0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b100: 2020 2020 2020 2020 2020 2020 3120 2020 1 │ │ │ │ -0006b110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b120: 2020 2020 7c0a 7c6f 3320 3a20 522d 6d6f |.|o3 : R-mo │ │ │ │ -0006b130: 6475 6c65 2c20 7175 6f74 6965 6e74 206f dule, quotient o │ │ │ │ -0006b140: 6620 5220 2020 2020 2020 2020 2020 2020 f R │ │ │ │ -0006b150: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b100: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ +0006b110: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ +0006b120: 2052 2d6d 6f64 756c 652c 2071 756f 7469 R-module, quoti │ │ │ │ +0006b130: 656e 7420 6f66 2052 2020 2020 2020 2020 ent of R │ │ │ │ +0006b140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b150: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b160: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b190: 2d2d 2b0a 7c69 3420 3a20 4620 3d20 6672 --+.|i4 : F = fr │ │ │ │ -0006b1a0: 6565 5265 736f 6c75 7469 6f6e 284d 2c20 eeResolution(M, │ │ │ │ -0006b1b0: 4c65 6e67 7468 4c69 6d69 7420 3d3e 2034 LengthLimit => 4 │ │ │ │ -0006b1c0: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +0006b180: 2d2d 2d2d 2d2d 2d2b 0a7c 6934 203a 2046 -------+.|i4 : F │ │ │ │ +0006b190: 203d 2066 7265 6552 6573 6f6c 7574 696f = freeResolutio │ │ │ │ +0006b1a0: 6e28 4d2c 204c 656e 6774 684c 696d 6974 n(M, LengthLimit │ │ │ │ +0006b1b0: 203d 3e20 3429 2020 2020 2020 2020 7c0a => 4) |. │ │ │ │ +0006b1c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b1d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b1e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b200: 7c0a 7c20 2020 2020 2031 2020 2020 2020 |.| 1 │ │ │ │ -0006b210: 3320 2020 2020 2035 2020 2020 2020 3720 3 5 7 │ │ │ │ -0006b220: 2020 2020 2039 2020 2020 2020 2020 2020 9 │ │ │ │ -0006b230: 2020 2020 2020 207c 0a7c 6f34 203d 2052 |.|o4 = R │ │ │ │ -0006b240: 2020 3c2d 2d20 5220 203c 2d2d 2052 2020 <-- R <-- R │ │ │ │ -0006b250: 3c2d 2d20 5220 203c 2d2d 2052 2020 2020 <-- R <-- R │ │ │ │ -0006b260: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006b270: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006b1f0: 2020 2020 207c 0a7c 2020 2020 2020 3120 |.| 1 │ │ │ │ +0006b200: 2020 2020 2033 2020 2020 2020 3520 2020 3 5 │ │ │ │ +0006b210: 2020 2037 2020 2020 2020 3920 2020 2020 7 9 │ │ │ │ +0006b220: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0006b230: 3420 3d20 5220 203c 2d2d 2052 2020 3c2d 4 = R <-- R <- │ │ │ │ +0006b240: 2d20 5220 203c 2d2d 2052 2020 3c2d 2d20 - R <-- R <-- │ │ │ │ +0006b250: 5220 2020 2020 2020 2020 2020 2020 2020 R │ │ │ │ +0006b260: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b2a0: 2020 2020 207c 0a7c 2020 2020 2030 2020 |.| 0 │ │ │ │ -0006b2b0: 2020 2020 3120 2020 2020 2032 2020 2020 1 2 │ │ │ │ -0006b2c0: 2020 3320 2020 2020 2034 2020 2020 2020 3 4 │ │ │ │ -0006b2d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006b290: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006b2a0: 2020 3020 2020 2020 2031 2020 2020 2020 0 1 │ │ │ │ +0006b2b0: 3220 2020 2020 2033 2020 2020 2020 3420 2 3 4 │ │ │ │ +0006b2c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b2d0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b2e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b310: 2020 207c 0a7c 6f34 203a 2043 6f6d 706c |.|o4 : Compl │ │ │ │ -0006b320: 6578 2020 2020 2020 2020 2020 2020 2020 ex │ │ │ │ -0006b330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b340: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006b300: 2020 2020 2020 2020 7c0a 7c6f 3420 3a20 |.|o4 : │ │ │ │ +0006b310: 436f 6d70 6c65 7820 2020 2020 2020 2020 Complex │ │ │ │ +0006b320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b330: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b340: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 0006b350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b380: 2d2b 0a7c 6935 203a 2066 6620 3d20 6d61 -+.|i5 : ff = ma │ │ │ │ -0006b390: 7472 6978 7b7b 7a5e 337d 7d20 2020 2020 trix{{z^3}} │ │ │ │ -0006b3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b3b0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b370: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 6666 ------+.|i5 : ff │ │ │ │ +0006b380: 203d 206d 6174 7269 787b 7b7a 5e33 7d7d = matrix{{z^3}} │ │ │ │ +0006b390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b3a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b3e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b3f0: 0a7c 6f35 203d 207c 207a 3320 7c20 2020 .|o5 = | z3 | │ │ │ │ +0006b3e0: 2020 2020 7c0a 7c6f 3520 3d20 7c20 7a33 |.|o5 = | z3 │ │ │ │ +0006b3f0: 207c 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b420: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006b410: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b450: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006b460: 2020 2020 2020 2020 2020 2020 2031 2020 1 │ │ │ │ -0006b470: 2020 2020 3120 2020 2020 2020 2020 2020 1 │ │ │ │ -0006b480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b490: 2020 2020 7c0a 7c6f 3520 3a20 4d61 7472 |.|o5 : Matr │ │ │ │ -0006b4a0: 6978 2052 2020 3c2d 2d20 5220 2020 2020 ix R <-- R │ │ │ │ +0006b450: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b460: 2020 3120 2020 2020 2031 2020 2020 2020 1 1 │ │ │ │ +0006b470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b480: 2020 2020 2020 2020 207c 0a7c 6f35 203a |.|o5 : │ │ │ │ +0006b490: 204d 6174 7269 7820 5220 203c 2d2d 2052 Matrix R <-- R │ │ │ │ +0006b4a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b4b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b4c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b4c0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b500: 2d2d 2b0a 7c69 3620 3a20 5231 203d 2052 --+.|i6 : R1 = R │ │ │ │ -0006b510: 2f69 6465 616c 2066 6620 2020 2020 2020 /ideal ff │ │ │ │ -0006b520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b530: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006b4f0: 2d2d 2d2d 2d2d 2d2b 0a7c 6936 203a 2052 -------+.|i6 : R │ │ │ │ +0006b500: 3120 3d20 522f 6964 6561 6c20 6666 2020 1 = R/ideal ff │ │ │ │ +0006b510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b520: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006b530: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b570: 7c0a 7c6f 3620 3d20 5231 2020 2020 2020 |.|o6 = R1 │ │ │ │ +0006b560: 2020 2020 207c 0a7c 6f36 203d 2052 3120 |.|o6 = R1 │ │ │ │ +0006b570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5a0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006b590: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006b5a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b5d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006b5e0: 7c6f 3620 3a20 5175 6f74 6965 6e74 5269 |o6 : QuotientRi │ │ │ │ -0006b5f0: 6e67 2020 2020 2020 2020 2020 2020 2020 ng │ │ │ │ -0006b600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b610: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006b5d0: 2020 207c 0a7c 6f36 203a 2051 756f 7469 |.|o6 : Quoti │ │ │ │ +0006b5e0: 656e 7452 696e 6720 2020 2020 2020 2020 entRing │ │ │ │ +0006b5f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b600: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ +0006b610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ -0006b650: 3720 3a20 6265 7474 6920 4620 2020 2020 7 : betti F │ │ │ │ +0006b640: 2d2b 0a7c 6937 203a 2062 6574 7469 2046 -+.|i7 : betti F │ │ │ │ +0006b650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b670: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b680: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006b670: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6b0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0006b6c0: 2020 2020 2020 2020 2030 2031 2032 2033 0 1 2 3 │ │ │ │ -0006b6d0: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ -0006b6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b6f0: 207c 0a7c 6f37 203d 2074 6f74 616c 3a20 |.|o7 = total: │ │ │ │ -0006b700: 3120 3320 3520 3720 3920 2020 2020 2020 1 3 5 7 9 │ │ │ │ -0006b710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b720: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006b730: 2020 2020 303a 2031 2033 2033 2031 202e 0: 1 3 3 1 . │ │ │ │ +0006b6a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006b6b0: 0a7c 2020 2020 2020 2020 2020 2020 3020 .| 0 │ │ │ │ +0006b6c0: 3120 3220 3320 3420 2020 2020 2020 2020 1 2 3 4 │ │ │ │ +0006b6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b6e0: 2020 2020 2020 7c0a 7c6f 3720 3d20 746f |.|o7 = to │ │ │ │ +0006b6f0: 7461 6c3a 2031 2033 2035 2037 2039 2020 tal: 1 3 5 7 9 │ │ │ │ +0006b700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b710: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b720: 2020 2020 2020 2020 2030 3a20 3120 3320 0: 1 3 │ │ │ │ +0006b730: 3320 3120 2e20 2020 2020 2020 2020 2020 3 1 . │ │ │ │ 0006b740: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006b760: 0a7c 2020 2020 2020 2020 2031 3a20 2e20 .| 1: . │ │ │ │ -0006b770: 2e20 3220 3620 3620 2020 2020 2020 2020 . 2 6 6 │ │ │ │ -0006b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b790: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -0006b7a0: 2020 323a 202e 202e 202e 202e 2033 2020 2: . . . . 3 │ │ │ │ +0006b750: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006b760: 313a 202e 202e 2032 2036 2036 2020 2020 1: . . 2 6 6 │ │ │ │ +0006b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006b780: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006b790: 2020 2020 2020 2032 3a20 2e20 2e20 2e20 2: . . . │ │ │ │ +0006b7a0: 2e20 3320 2020 2020 2020 2020 2020 2020 . 3 │ │ │ │ 0006b7b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b7c0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006b7c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b7d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b800: 2020 2020 7c0a 7c6f 3720 3a20 4265 7474 |.|o7 : Bett │ │ │ │ -0006b810: 6954 616c 6c79 2020 2020 2020 2020 2020 iTally │ │ │ │ +0006b7f0: 2020 2020 2020 2020 207c 0a7c 6f37 203a |.|o7 : │ │ │ │ +0006b800: 2042 6574 7469 5461 6c6c 7920 2020 2020 BettiTally │ │ │ │ +0006b810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b830: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ +0006b830: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 0006b840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006b850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006b870: 2d2d 2b0a 7c69 3820 3a20 4646 203d 2053 --+.|i8 : FF = S │ │ │ │ -0006b880: 6861 6d61 7368 2866 662c 462c 3429 2020 hamash(ff,F,4) │ │ │ │ -0006b890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b8a0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006b860: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2046 -------+.|i8 : F │ │ │ │ +0006b870: 4620 3d20 5368 616d 6173 6828 6666 2c46 F = Shamash(ff,F │ │ │ │ +0006b880: 2c34 2920 2020 2020 2020 2020 2020 2020 ,4) │ │ │ │ +0006b890: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006b8a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006b8b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b8c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b8e0: 7c0a 7c20 2020 2020 2f20 525c 3120 2020 |.| / R\1 │ │ │ │ -0006b8f0: 2020 2f20 525c 3320 2020 2020 2f20 525c / R\3 / R\ │ │ │ │ -0006b900: 3620 2020 2020 2f20 525c 3130 2020 2020 6 / R\10 │ │ │ │ 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2020 2f20 525c 3135 7c0a 7c6f 0 / R\15|.|o │ │ │ │ +0006b910: 3820 3d20 7c2d 2d7c 2020 3c2d 2d20 7c2d 8 = |--| <-- |- │ │ │ │ +0006b920: 2d7c 2020 3c2d 2d20 7c2d 2d7c 2020 3c2d -| <-- |--| <- │ │ │ │ +0006b930: 2d20 7c2d 2d7c 2020 203c 2d2d 207c 2d2d - |--| <-- |-- │ │ │ │ +0006b940: 7c20 207c 0a7c 2020 2020 207c 2033 7c20 | |.| | 3| │ │ │ │ +0006b950: 2020 2020 207c 2033 7c20 2020 2020 207c | 3| | │ │ │ │ +0006b960: 2033 7c20 2020 2020 207c 2033 7c20 2020 3| | 3| │ │ │ │ +0006b970: 2020 2020 7c20 337c 2020 7c0a 7c20 2020 | 3| |.| │ │ │ │ +0006b980: 2020 5c7a 202f 2020 2020 2020 5c7a 202f \z / \z / │ │ │ │ +0006b990: 2020 2020 2020 5c7a 202f 2020 2020 2020 \z / │ │ │ │ +0006b9a0: 5c7a 202f 2020 2020 2020 205c 7a20 2f20 \z / \z / │ │ │ │ +0006b9b0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006b9d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b9e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006b9f0: 2020 207c 0a7c 2020 2020 2030 2020 2020 |.| 0 │ │ │ │ -0006ba00: 2020 2020 2031 2020 2020 2020 2020 2032 1 2 │ │ │ │ -0006ba10: 2020 2020 2020 2020 2033 2020 2020 2020 3 │ │ │ │ -0006ba20: 2020 2020 3420 2020 2020 7c0a 7c20 2020 4 |.| │ │ │ │ +0006b9e0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006b9f0: 3020 2020 2020 2020 2020 3120 2020 2020 0 1 │ │ │ │ +0006ba00: 2020 2020 3220 2020 2020 2020 2020 3320 2 3 │ │ │ │ +0006ba10: 2020 2020 2020 2020 2034 2020 2020 207c 4 | │ │ │ │ +0006ba20: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0006ba30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ba40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba60: 207c 0a7c 6f38 203a 2043 6f6d 706c 6578 |.|o8 : Complex │ │ │ │ +0006ba50: 2020 2020 2020 7c0a 7c6f 3820 3a20 436f |.|o8 : Co │ │ │ │ +0006ba60: 6d70 6c65 7820 2020 2020 2020 2020 2020 mplex │ │ │ │ 0006ba70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ba90: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +0006ba80: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0006ba90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006baa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0006bad0: 0a7c 6939 203a 2047 4720 3d20 5368 616d .|i9 : GG = Sham │ │ │ │ -0006bae0: 6173 6828 5231 2c46 2c34 2920 2020 2020 ash(R1,F,4) │ │ │ │ -0006baf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bb00: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006bac0: 2d2d 2d2d 2b0a 7c69 3920 3a20 4747 203d ----+.|i9 : GG = │ │ │ │ +0006bad0: 2053 6861 6d61 7368 2852 312c 462c 3429 Shamash(R1,F,4) │ │ │ │ +0006bae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006baf0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006bb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bb10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bb30: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006bb40: 2020 2020 2020 2031 2020 2020 2020 2033 1 3 │ │ │ │ -0006bb50: 2020 2020 2020 2036 2020 2020 2020 2031 6 1 │ │ │ │ -0006bb60: 3020 2020 2020 2020 3135 2020 2020 2020 0 15 │ │ │ │ -0006bb70: 2020 2020 7c0a 7c6f 3920 3d20 5231 2020 |.|o9 = R1 │ │ │ │ -0006bb80: 3c2d 2d20 5231 2020 3c2d 2d20 5231 2020 <-- R1 <-- R1 │ │ │ │ -0006bb90: 3c2d 2d20 5231 2020 203c 2d2d 2052 3120 <-- R1 <-- R1 │ │ │ │ -0006bba0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +0006bb30: 2020 7c0a 7c20 2020 2020 2020 3120 2020 |.| 1 │ │ │ │ +0006bb40: 2020 2020 3320 2020 2020 2020 3620 2020 3 6 │ │ │ │ +0006bb50: 2020 2020 3130 2020 2020 2020 2031 3520 10 15 │ │ │ │ +0006bb60: 2020 2020 2020 2020 207c 0a7c 6f39 203d |.|o9 = │ │ │ │ +0006bb70: 2052 3120 203c 2d2d 2052 3120 203c 2d2d R1 <-- R1 <-- │ │ │ │ +0006bb80: 2052 3120 203c 2d2d 2052 3120 2020 3c2d R1 <-- R1 <- │ │ │ │ +0006bb90: 2d20 5231 2020 2020 2020 2020 2020 2020 - R1 │ │ │ │ +0006bba0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006bbb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bbe0: 2020 7c0a 7c20 2020 2020 3020 2020 2020 |.| 0 │ │ │ │ -0006bbf0: 2020 3120 2020 2020 2020 3220 2020 2020 1 2 │ │ │ │ -0006bc00: 2020 3320 2020 2020 2020 2034 2020 2020 3 4 │ │ │ │ -0006bc10: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006bbd0: 2020 2020 2020 207c 0a7c 2020 2020 2030 |.| 0 │ │ │ │ +0006bbe0: 2020 2020 2020 2031 2020 2020 2020 2032 1 2 │ │ │ │ +0006bbf0: 2020 2020 2020 2033 2020 2020 2020 2020 3 │ │ │ │ +0006bc00: 3420 2020 2020 2020 2020 2020 2020 7c0a 4 |. │ │ │ │ +0006bc10: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006bc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bc30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc50: 7c0a 7c6f 3920 3a20 436f 6d70 6c65 7820 |.|o9 : Complex │ │ │ │ +0006bc40: 2020 2020 207c 0a7c 6f39 203a 2043 6f6d |.|o9 : Com │ │ │ │ +0006bc50: 706c 6578 2020 2020 2020 2020 2020 2020 plex │ │ │ │ 0006bc60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bc80: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006bc70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ +0006bc80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bc90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bca0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -0006bcc0: 7c69 3130 203a 2062 6574 7469 2046 4620 |i10 : betti FF │ │ │ │ +0006bcb0: 2d2d 2d2b 0a7c 6931 3020 3a20 6265 7474 ---+.|i10 : bett │ │ │ │ +0006bcc0: 6920 4646 2020 2020 2020 2020 2020 2020 i FF │ │ │ │ 0006bcd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bcf0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bce0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006bcf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006bd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd20: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006bd30: 2020 2020 2020 2020 2020 2020 3020 3120 0 1 │ │ │ │ -0006bd40: 3220 2033 2020 3420 2020 2020 2020 2020 2 3 4 │ │ │ │ -0006bd50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd60: 2020 207c 0a7c 6f31 3020 3d20 746f 7461 |.|o10 = tota │ │ │ │ -0006bd70: 6c3a 2031 2033 2036 2031 3020 3135 2020 l: 1 3 6 10 15 │ │ │ │ -0006bd80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bd90: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -0006bda0: 2020 2020 2020 2030 3a20 3120 3320 3320 0: 1 3 3 │ │ │ │ -0006bdb0: 2031 2020 2e20 2020 2020 2020 2020 2020 1 . │ │ │ │ -0006bdc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bdd0: 207c 0a7c 2020 2020 2020 2020 2020 313a |.| 1: │ │ │ │ -0006bde0: 202e 202e 2033 2020 3920 2039 2020 2020 . . 3 9 9 │ │ │ │ -0006bdf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be00: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -0006be10: 2020 2020 2032 3a20 2e20 2e20 2e20 202e 2: . . . . │ │ │ │ -0006be20: 2020 3620 2020 2020 2020 2020 2020 2020 6 │ │ │ │ -0006be30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006be40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006bd20: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006bd30: 2030 2031 2032 2020 3320 2034 2020 2020 0 1 2 3 4 │ │ │ │ +0006bd40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bd50: 2020 2020 2020 2020 7c0a 7c6f 3130 203d |.|o10 = │ │ │ │ +0006bd60: 2074 6f74 616c 3a20 3120 3320 3620 3130 total: 1 3 6 10 │ │ │ │ +0006bd70: 2031 3520 2020 2020 2020 2020 2020 2020 15 │ │ │ │ +0006bd80: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006bd90: 0a7c 2020 2020 2020 2020 2020 303a 2031 .| 0: 1 │ │ │ │ +0006bda0: 2033 2033 2020 3120 202e 2020 2020 2020 3 3 1 . │ │ │ │ +0006bdb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bdc0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0006bdd0: 2020 2031 3a20 2e20 2e20 3320 2039 2020 1: . . 3 9 │ │ │ │ +0006bde0: 3920 2020 2020 2020 2020 2020 2020 2020 9 │ │ │ │ +0006bdf0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006be00: 2020 2020 2020 2020 2020 323a 202e 202e 2: . . │ │ │ │ +0006be10: 202e 2020 2e20 2036 2020 2020 2020 2020 . . 6 │ │ │ │ +0006be20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006be30: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006be40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006be50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006be70: 2020 2020 2020 7c0a 7c6f 3130 203a 2042 |.|o10 : B │ │ │ │ -0006be80: 6574 7469 5461 6c6c 7920 2020 2020 2020 ettiTally │ │ │ │ +0006be60: 2020 2020 2020 2020 2020 207c 0a7c 6f31 |.|o1 │ │ │ │ +0006be70: 3020 3a20 4265 7474 6954 616c 6c79 2020 0 : BettiTally │ │ │ │ +0006be80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006be90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006bea0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ +0006bea0: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 0006beb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006bec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006bee0: 2d2d 2d2d 2b0a 7c69 3131 203a 2062 6574 ----+.|i11 : bet │ │ │ │ -0006bef0: 7469 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2020 2020 7c0a |. │ │ │ │ +0006bf80: 7c6f 3131 203d 2074 6f74 616c 3a20 3120 |o11 = total: 1 │ │ │ │ +0006bf90: 3320 3620 3130 2031 3520 2020 2020 2020 3 6 10 15 │ │ │ │ +0006bfa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bfb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006bfc0: 2020 303a 2031 2033 2033 2020 3120 202e 0: 1 3 3 1 . │ │ │ │ +0006bfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006bfe0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006bff0: 2020 2020 2020 2020 2031 3a20 2e20 2e20 1: . . │ │ │ │ +0006c000: 3320 2039 2020 3920 2020 2020 2020 2020 3 9 9 │ │ │ │ +0006c010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006c030: 323a 202e 202e 202e 2020 2e20 2036 2020 2: . . . . 6 │ │ │ │ +0006c040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006c050: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006c060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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The │ │ │ │ -0006c3f0: 6d61 7472 6978 2066 6620 6d75 7374 0a62 matrix ff must.b │ │ │ │ -0006c400: 6520 3178 312e 0a0a 5365 6520 616c 736f e 1x1...See also │ │ │ │ -0006c410: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ -0006c420: 6e6f 7465 2045 6973 656e 6275 6453 6861 note EisenbudSha │ │ │ │ -0006c430: 6d61 7368 3a20 4569 7365 6e62 7564 5368 mash: EisenbudSh │ │ │ │ -0006c440: 616d 6173 682c 202d 2d20 436f 6d70 7574 amash, -- Comput │ │ │ │ -0006c450: 6573 2074 6865 2045 6973 656e 6275 642d es the Eisenbud- │ │ │ │ -0006c460: 5368 616d 6173 680a 2020 2020 436f 6d70 Shamash. Comp │ │ │ │ -0006c470: 6c65 780a 2020 2a20 2a6e 6f74 6520 6d61 lex. * *note ma │ │ │ │ -0006c480: 6b65 486f 6d6f 746f 7069 6573 3a20 6d61 keHomotopies: ma │ │ │ │ -0006c490: 6b65 486f 6d6f 746f 7069 6573 2c20 2d2d keHomotopies, -- │ │ │ │ -0006c4a0: 2072 6574 7572 6e73 2061 2073 7973 7465 returns a syste │ │ │ │ -0006c4b0: 6d20 6f66 2068 6967 6865 720a 2020 2020 m of higher. │ │ │ │ -0006c4c0: 686f 6d6f 746f 7069 6573 0a0a 5761 7973 homotopies..Ways │ │ │ │ -0006c4d0: 2074 6f20 7573 6520 5368 616d 6173 683a to use Shamash: │ │ │ │ -0006c4e0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -0006c4f0: 3d3d 3d3d 3d0a 0a20 202a 2022 5368 616d =====.. * "Sham │ │ │ │ -0006c500: 6173 6828 4d61 7472 6978 2c43 6f6d 706c ash(Matrix,Compl │ │ │ │ -0006c510: 6578 2c5a 5a29 220a 2020 2a20 2253 6861 ex,ZZ)". * "Sha │ │ │ │ -0006c520: 6d61 7368 2852 696e 672c 436f 6d70 6c65 mash(Ring,Comple │ │ │ │ -0006c530: 782c 5a5a 2922 0a0a 466f 7220 7468 6520 x,ZZ)"..For the │ │ │ │ -0006c540: 7072 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d programmer.===== │ │ │ │ -0006c550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 =============..T │ │ │ │ -0006c560: 6865 206f 626a 6563 7420 2a6e 6f74 6520 he object *note │ │ │ │ -0006c570: 5368 616d 6173 683a 2053 6861 6d61 7368 Shamash: Shamash │ │ │ │ -0006c580: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ -0006c590: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ -0006c5a0: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -0006c5b0: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ +0006c390: 2d2d 2d2b 0a0a 4361 7665 6174 0a3d 3d3d ---+..Caveat.=== │ │ │ │ +0006c3a0: 3d3d 3d0a 0a46 2069 7320 6173 7375 6d65 ===..F is assume │ │ │ │ +0006c3b0: 6420 746f 2062 6520 6120 686f 6d6f 6c6f d to be a homolo │ │ │ │ +0006c3c0: 6769 6361 6c20 636f 6d70 6c65 7820 7374 gical complex st │ │ │ │ +0006c3d0: 6172 7469 6e67 2066 726f 6d20 465f 302e arting from F_0. │ │ │ │ +0006c3e0: 2054 6865 206d 6174 7269 7820 6666 206d The matrix ff m │ │ │ │ +0006c3f0: 7573 740a 6265 2031 7831 2e0a 0a53 6565 ust.be 1x1...See │ │ │ │ +0006c400: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ +0006c410: 2020 2a20 2a6e 6f74 6520 4569 7365 6e62 * *note Eisenb │ │ │ │ +0006c420: 7564 5368 616d 6173 683a 2045 6973 656e udShamash: Eisen │ │ │ │ +0006c430: 6275 6453 6861 6d61 7368 2c20 2d2d 2043 budShamash, -- C │ │ │ │ +0006c440: 6f6d 7075 7465 7320 7468 6520 4569 7365 omputes the Eise │ │ │ │ +0006c450: 6e62 7564 2d53 6861 6d61 7368 0a20 2020 nbud-Shamash. │ │ │ │ +0006c460: 2043 6f6d 706c 6578 0a20 202a 202a 6e6f Complex. * *no │ │ │ │ +0006c470: 7465 206d 616b 6548 6f6d 6f74 6f70 6965 te makeHomotopie │ │ │ │ +0006c480: 733a 206d 616b 6548 6f6d 6f74 6f70 6965 s: makeHomotopie │ │ │ │ +0006c490: 732c 202d 2d20 7265 7475 726e 7320 6120 s, -- returns a │ │ │ │ +0006c4a0: 7379 7374 656d 206f 6620 6869 6768 6572 system of higher │ │ │ │ +0006c4b0: 0a20 2020 2068 6f6d 6f74 6f70 6965 730a . homotopies. │ │ │ │ +0006c4c0: 0a57 6179 7320 746f 2075 7365 2053 6861 .Ways to use Sha │ │ │ │ +0006c4d0: 6d61 7368 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d mash:.========== │ │ │ │ +0006c4e0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +0006c4f0: 2253 6861 6d61 7368 284d 6174 7269 782c "Shamash(Matrix, │ │ │ │ +0006c500: 436f 6d70 6c65 782c 5a5a 2922 0a20 202a Complex,ZZ)". * │ │ │ │ +0006c510: 2022 5368 616d 6173 6828 5269 6e67 2c43 "Shamash(Ring,C │ │ │ │ +0006c520: 6f6d 706c 6578 2c5a 5a29 220a 0a46 6f72 omplex,ZZ)"..For │ │ │ │ +0006c530: 2074 6865 2070 726f 6772 616d 6d65 720a the programmer. │ │ │ │ +0006c540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006c550: 3d3d 0a0a 5468 6520 6f62 6a65 6374 202a ==..The object * │ │ │ │ +0006c560: 6e6f 7465 2053 6861 6d61 7368 3a20 5368 note Shamash: Sh │ │ │ │ +0006c570: 616d 6173 682c 2069 7320 6120 2a6e 6f74 amash, is a *not │ │ │ │ +0006c580: 6520 6d65 7468 6f64 2066 756e 6374 696f e method functio │ │ │ │ +0006c590: 6e3a 0a28 4d61 6361 756c 6179 3244 6f63 n:.(Macaulay2Doc │ │ │ │ +0006c5a0: 294d 6574 686f 6446 756e 6374 696f 6e2c )MethodFunction, │ │ │ │ +0006c5b0: 2e0a 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...------------- │ │ │ │ 0006c5c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c5d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c5e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006c5f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006c600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ -0006c610: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ -0006c620: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ -0006c630: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ -0006c640: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ -0006c650: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ -0006c660: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ -0006c670: 6167 6573 2f0a 436f 6d70 6c65 7465 496e ages/.CompleteIn │ │ │ │ -0006c680: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ -0006c690: 7469 6f6e 732e 6d32 3a34 3736 303a 302e tions.m2:4760:0. │ │ │ │ -0006c6a0: 0a1f 0a46 696c 653a 2043 6f6d 706c 6574 ...File: Complet │ │ │ │ -0006c6b0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006c6c0: 6f6c 7574 696f 6e73 2e69 6e66 6f2c 204e olutions.info, N │ │ │ │ -0006c6d0: 6f64 653a 2073 706c 6974 7469 6e67 732c ode: splittings, │ │ │ │ -0006c6e0: 204e 6578 743a 2073 7461 626c 6548 6f6d Next: stableHom │ │ │ │ -0006c6f0: 2c20 5072 6576 3a20 5368 616d 6173 682c , Prev: Shamash, │ │ │ │ -0006c700: 2055 703a 2054 6f70 0a0a 7370 6c69 7474 Up: Top..splitt │ │ │ │ -0006c710: 696e 6773 202d 2d20 636f 6d70 7574 6520 ings -- compute │ │ │ │ -0006c720: 7468 6520 7370 6c69 7474 696e 6773 206f the splittings o │ │ │ │ -0006c730: 6620 6120 7370 6c69 7420 7269 6768 7420 f a split right │ │ │ │ -0006c740: 6578 6163 7420 7365 7175 656e 6365 0a2a exact sequence.* │ │ │ │ +0006c600: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +0006c610: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +0006c620: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +0006c630: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +0006c640: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +0006c650: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +0006c660: 2f70 6163 6b61 6765 732f 0a43 6f6d 706c /packages/.Compl │ │ │ │ +0006c670: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ +0006c680: 6573 6f6c 7574 696f 6e73 2e6d 323a 3437 esolutions.m2:47 │ │ │ │ +0006c690: 3630 3a30 2e0a 1f0a 4669 6c65 3a20 436f 60:0....File: Co │ │ │ │ +0006c6a0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006c6b0: 6f6e 5265 736f 6c75 7469 6f6e 732e 696e onResolutions.in │ │ │ │ +0006c6c0: 666f 2c20 4e6f 6465 3a20 7370 6c69 7474 fo, Node: splitt │ │ │ │ +0006c6d0: 696e 6773 2c20 4e65 7874 3a20 7374 6162 ings, Next: stab │ │ │ │ +0006c6e0: 6c65 486f 6d2c 2050 7265 763a 2053 6861 leHom, Prev: Sha │ │ │ │ +0006c6f0: 6d61 7368 2c20 5570 3a20 546f 700a 0a73 mash, Up: Top..s │ │ │ │ +0006c700: 706c 6974 7469 6e67 7320 2d2d 2063 6f6d plittings -- com │ │ │ │ +0006c710: 7075 7465 2074 6865 2073 706c 6974 7469 pute the splitti │ │ │ │ +0006c720: 6e67 7320 6f66 2061 2073 706c 6974 2072 ngs of a split r │ │ │ │ +0006c730: 6967 6874 2065 7861 6374 2073 6571 7565 ight exact seque │ │ │ │ +0006c740: 6e63 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a nce.************ │ │ │ │ 0006c750: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006c760: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006c770: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006c780: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006c790: 2a2a 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -0006c7a0: 0a20 2020 2020 2020 2078 203d 2073 706c . x = spl │ │ │ │ -0006c7b0: 6974 7469 6e67 7328 612c 6229 0a20 202a ittings(a,b). * │ │ │ │ -0006c7c0: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0006c7d0: 2061 2c20 6120 2a6e 6f74 6520 6d61 7472 a, a *note matr │ │ │ │ -0006c7e0: 6978 3a20 284d 6163 6175 6c61 7932 446f ix: (Macaulay2Do │ │ │ │ -0006c7f0: 6329 4d61 7472 6978 2c2c 206d 6170 7320 c)Matrix,, maps │ │ │ │ -0006c800: 696e 746f 2074 6865 206b 6572 6e65 6c20 into the kernel │ │ │ │ -0006c810: 6f66 2062 0a20 2020 2020 202a 2062 2c20 of b. * b, │ │ │ │ -0006c820: 6120 2a6e 6f74 6520 6d61 7472 6978 3a20 a *note matrix: │ │ │ │ -0006c830: 284d 6163 6175 6c61 7932 446f 6329 4d61 (Macaulay2Doc)Ma │ │ │ │ -0006c840: 7472 6978 2c2c 2072 6570 7265 7365 6e74 trix,, represent │ │ │ │ -0006c850: 696e 6720 6120 7375 726a 6563 7469 6f6e ing a surjection │ │ │ │ -0006c860: 0a20 2020 2020 2020 2066 726f 6d20 7461 . from ta │ │ │ │ -0006c870: 7267 6574 2061 2074 6f20 6120 6672 6565 rget a to a free │ │ │ │ -0006c880: 206d 6f64 756c 650a 2020 2a20 4f75 7470 module. * Outp │ │ │ │ -0006c890: 7574 733a 0a20 2020 2020 202a 204c 2c20 uts:. * L, │ │ │ │ -0006c8a0: 6120 2a6e 6f74 6520 6c69 7374 3a20 284d a *note list: (M │ │ │ │ -0006c8b0: 6163 6175 6c61 7932 446f 6329 4c69 7374 acaulay2Doc)List │ │ │ │ -0006c8c0: 2c2c 204c 203d 205c 7b73 6967 6d61 2c74 ,, L = \{sigma,t │ │ │ │ -0006c8d0: 6175 5c7d 2c20 7370 6c69 7474 696e 6773 au\}, splittings │ │ │ │ -0006c8e0: 206f 660a 2020 2020 2020 2020 612c 6220 of. a,b │ │ │ │ -0006c8f0: 7265 7370 6563 7469 7665 6c79 0a0a 4465 respectively..De │ │ │ │ -0006c900: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -0006c910: 3d3d 3d3d 3d0a 0a41 7373 756d 696e 6720 =====..Assuming │ │ │ │ -0006c920: 7468 6174 2028 612c 6229 2061 7265 2074 that (a,b) are t │ │ │ │ -0006c930: 6865 206d 6170 7320 6f66 2061 2072 6967 he maps of a rig │ │ │ │ -0006c940: 6874 2065 7861 6374 2073 6571 7565 6e63 ht exact sequenc │ │ │ │ -0006c950: 650a 0a24 305c 746f 2041 5c74 6f20 425c e..$0\to A\to B\ │ │ │ │ -0006c960: 746f 2043 205c 746f 2030 240a 0a77 6974 to C \to 0$..wit │ │ │ │ -0006c970: 6820 422c 2043 2066 7265 652c 2074 6865 h B, C free, the │ │ │ │ -0006c980: 2073 6372 6970 7420 7072 6f64 7563 6573 script produces │ │ │ │ -0006c990: 2061 2070 6169 7220 6f66 206d 6170 7320 a pair of maps │ │ │ │ -0006c9a0: 7369 676d 612c 2074 6175 2077 6974 6820 sigma, tau with │ │ │ │ -0006c9b0: 2474 6175 3a20 4320 5c74 6f0a 4224 2061 $tau: C \to.B$ a │ │ │ │ -0006c9c0: 2073 706c 6974 7469 6e67 206f 6620 6220 splitting of b │ │ │ │ -0006c9d0: 616e 6420 2473 6967 6d61 3a20 4220 5c74 and $sigma: B \t │ │ │ │ -0006c9e0: 6f20 4124 2061 2073 706c 6974 7469 6e67 o A$ a splitting │ │ │ │ -0006c9f0: 206f 6620 613b 2074 6861 7420 6973 2c0a of a; that is,. │ │ │ │ -0006ca00: 0a24 612a 7369 676d 612b 7461 752a 6220 .$a*sigma+tau*b │ │ │ │ -0006ca10: 3d20 315f 4224 0a0a 2473 6967 6d61 2a61 = 1_B$..$sigma*a │ │ │ │ -0006ca20: 203d 2031 5f41 240a 0a24 622a 7461 7520 = 1_A$..$b*tau │ │ │ │ -0006ca30: 3d20 315f 4324 0a0a 2b2d 2d2d 2d2d 2d2d = 1_C$..+------- │ │ │ │ +0006c780: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +0006c790: 6167 653a 200a 2020 2020 2020 2020 7820 age: . x │ │ │ │ +0006c7a0: 3d20 7370 6c69 7474 696e 6773 2861 2c62 = splittings(a,b │ │ │ │ +0006c7b0: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006c7c0: 2020 2020 2a20 612c 2061 202a 6e6f 7465 * a, a *note │ │ │ │ +0006c7d0: 206d 6174 7269 783a 2028 4d61 6361 756c matrix: (Macaul │ │ │ │ +0006c7e0: 6179 3244 6f63 294d 6174 7269 782c 2c20 ay2Doc)Matrix,, │ │ │ │ +0006c7f0: 6d61 7073 2069 6e74 6f20 7468 6520 6b65 maps into the ke │ │ │ │ +0006c800: 726e 656c 206f 6620 620a 2020 2020 2020 rnel of b. │ │ │ │ +0006c810: 2a20 622c 2061 202a 6e6f 7465 206d 6174 * b, a *note mat │ │ │ │ +0006c820: 7269 783a 2028 4d61 6361 756c 6179 3244 rix: (Macaulay2D │ │ │ │ +0006c830: 6f63 294d 6174 7269 782c 2c20 7265 7072 oc)Matrix,, repr │ │ │ │ +0006c840: 6573 656e 7469 6e67 2061 2073 7572 6a65 esenting a surje │ │ │ │ +0006c850: 6374 696f 6e0a 2020 2020 2020 2020 6672 ction. fr │ │ │ │ +0006c860: 6f6d 2074 6172 6765 7420 6120 746f 2061 om target a to a │ │ │ │ +0006c870: 2066 7265 6520 6d6f 6475 6c65 0a20 202a free module. * │ │ │ │ +0006c880: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ +0006c890: 2a20 4c2c 2061 202a 6e6f 7465 206c 6973 * L, a *note lis │ │ │ │ +0006c8a0: 743a 2028 4d61 6361 756c 6179 3244 6f63 t: (Macaulay2Doc │ │ │ │ +0006c8b0: 294c 6973 742c 2c20 4c20 3d20 5c7b 7369 )List,, L = \{si │ │ │ │ +0006c8c0: 676d 612c 7461 755c 7d2c 2073 706c 6974 gma,tau\}, split │ │ │ │ +0006c8d0: 7469 6e67 7320 6f66 0a20 2020 2020 2020 tings of. │ │ │ │ +0006c8e0: 2061 2c62 2072 6573 7065 6374 6976 656c a,b respectivel │ │ │ │ +0006c8f0: 790a 0a44 6573 6372 6970 7469 6f6e 0a3d y..Description.= │ │ │ │ +0006c900: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 4173 7375 ==========..Assu │ │ │ │ +0006c910: 6d69 6e67 2074 6861 7420 2861 2c62 2920 ming that (a,b) │ │ │ │ +0006c920: 6172 6520 7468 6520 6d61 7073 206f 6620 are the maps of │ │ │ │ +0006c930: 6120 7269 6768 7420 6578 6163 7420 7365 a right exact se │ │ │ │ +0006c940: 7175 656e 6365 0a0a 2430 5c74 6f20 415c quence..$0\to A\ │ │ │ │ +0006c950: 746f 2042 5c74 6f20 4320 5c74 6f20 3024 to B\to C \to 0$ │ │ │ │ +0006c960: 0a0a 7769 7468 2042 2c20 4320 6672 6565 ..with B, C free │ │ │ │ +0006c970: 2c20 7468 6520 7363 7269 7074 2070 726f , the script pro │ │ │ │ +0006c980: 6475 6365 7320 6120 7061 6972 206f 6620 duces a pair of │ │ │ │ +0006c990: 6d61 7073 2073 6967 6d61 2c20 7461 7520 maps sigma, tau │ │ │ │ +0006c9a0: 7769 7468 2024 7461 753a 2043 205c 746f with $tau: C \to │ │ │ │ +0006c9b0: 0a42 2420 6120 7370 6c69 7474 696e 6720 .B$ a splitting │ │ │ │ +0006c9c0: 6f66 2062 2061 6e64 2024 7369 676d 613a of b and $sigma: │ │ │ │ +0006c9d0: 2042 205c 746f 2041 2420 6120 7370 6c69 B \to A$ a spli │ │ │ │ +0006c9e0: 7474 696e 6720 6f66 2061 3b20 7468 6174 tting of a; that │ │ │ │ +0006c9f0: 2069 732c 0a0a 2461 2a73 6967 6d61 2b74 is,..$a*sigma+t │ │ │ │ +0006ca00: 6175 2a62 203d 2031 5f42 240a 0a24 7369 au*b = 1_B$..$si │ │ │ │ +0006ca10: 676d 612a 6120 3d20 315f 4124 0a0a 2462 gma*a = 1_A$..$b │ │ │ │ +0006ca20: 2a74 6175 203d 2031 5f43 240a 0a2b 2d2d *tau = 1_C$..+-- │ │ │ │ +0006ca30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ca60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ca70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ca80: 2d2d 2d2b 0a7c 6931 203a 206b 6b3d 205a ---+.|i1 : kk= Z │ │ │ │ -0006ca90: 5a2f 3130 3120 2020 2020 2020 2020 2020 Z/101 │ │ │ │ +0006ca70: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 --------+.|i1 : │ │ │ │ +0006ca80: 6b6b 3d20 5a5a 2f31 3031 2020 2020 2020 kk= ZZ/101 │ │ │ │ +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 2020 2020 2020 2020 │ │ │ │ -0006cad0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006cac0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006cad0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006caf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cb00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cb10: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -0006cb20: 6f31 203d 206b 6b20 2020 2020 2020 2020 o1 = kk │ │ │ │ +0006cb10: 2020 7c0a 7c6f 3120 3d20 6b6b 2020 2020 |.|o1 = kk │ │ │ │ +0006cb20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2020 2020 2020 2020 │ │ │ │ 0006cc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cc90: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006cca0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ +0006cc90: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006cca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ccd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cce0: 2020 2020 2020 2020 2020 207c 0a7c 6f32 |.|o2 │ │ │ │ -0006ccf0: 203d 2053 2020 2020 2020 2020 2020 2020 = S │ │ │ │ +0006cce0: 7c0a 7c6f 3220 3d20 5320 2020 2020 2020 |.|o2 = S │ │ │ │ +0006ccf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006cd10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cd30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 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|.+------ │ │ │ │ +0006cdd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cde0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006cdf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ce00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ce10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ -0006ce20: 0a7c 6933 203a 2073 6574 5261 6e64 6f6d .|i3 : setRandom │ │ │ │ -0006ce30: 5365 6564 2030 2020 2020 2020 2020 2020 Seed 0 │ │ │ │ +0006ce10: 2d2d 2d2d 2b0a 7c69 3320 3a20 7365 7452 ----+.|i3 : setR │ │ │ │ +0006ce20: 616e 646f 6d53 6565 6420 3020 2020 2020 andomSeed 0 │ │ │ │ +0006ce30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ce40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ce50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ce60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006ce70: 2d2d 2073 6574 7469 6e67 2072 616e 646f -- setting rando │ │ │ │ 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+0006cf90: 2d2d 2d2d 2d2b 0a7c 6934 203a 2074 203d -----+.|i4 : t = │ │ │ │ +0006cfa0: 2072 616e 646f 6d28 535e 7b32 3a2d 312c random(S^{2:-1, │ │ │ │ +0006cfb0: 323a 2d32 7d2c 2053 5e7b 333a 2d31 2c34 2:-2}, S^{3:-1,4 │ │ │ │ +0006cfc0: 3a2d 327d 2920 2020 2020 2020 2020 2020 :-2}) │ │ │ │ 0006cfd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006cfe0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +0006cfe0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 0006cff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d030: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ -0006d040: 3d20 7b31 7d20 7c20 3234 2020 2d33 3620 = {1} | 24 -36 │ │ │ │ -0006d050: 2d33 3020 3339 782d 3433 792b 3435 7a20 -30 39x-43y+45z │ │ │ │ -0006d060: 2032 3178 2d31 3579 2d33 347a 2033 3478 21x-15y-34z 34x │ │ │ │ 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│ │ │ │ -0006d2f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +0006d2e0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006d2f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d330: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -0006d340: 3520 3d20 7b7b 317d 207c 2030 2030 2031 5 = {{1} | 0 0 1 │ │ │ │ -0006d350: 2030 2030 2030 2020 2030 2020 7c2c 207b 0 0 0 0 |, { │ │ │ │ -0006d360: 317d 207c 202d 3237 2032 2020 3133 782d 1} | -27 2 13x- │ │ │ │ -0006d370: 3130 792b 3433 7a20 3530 782d 3334 792d 10y+43z 50x-34y- │ │ │ │ -0006d380: 3530 7a20 7c7d 2020 207c 0a7c 2020 2020 50z |} |.| │ │ │ │ -0006d390: 2020 7b32 7d20 7c20 3020 3020 3020 3020 {2} | 0 0 0 0 │ │ │ │ -0006d3a0: 3020 2d33 3120 2d36 207c 2020 7b31 7d20 0 -31 -6 | {1} │ │ │ │ -0006d3b0: 7c20 2d34 2020 3335 2032 3278 2b33 3279 | 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2020 2020 2020 2020 │ │ │ │ -0006d770: 2020 7c0a 7c20 2020 2020 2020 2020 2030 |.| 0 │ │ │ │ -0006d780: 3a20 2e20 3320 2020 2020 2030 3a20 2e20 : . 3 0: . │ │ │ │ -0006d790: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0006d760: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006d770: 2020 2020 303a 202e 2033 2020 2020 2020 0: . 3 │ │ │ │ +0006d780: 303a 202e 2032 2020 2020 2020 2020 2020 0: . 2 │ │ │ │ +0006d790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d7a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d7b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0006d7c0: 0a7c 2020 2020 2020 2020 2020 313a 2031 .| 1: 1 │ │ │ │ -0006d7d0: 2034 2020 2020 2020 313a 2033 2032 2020 4 1: 3 2 │ │ │ │ +0006d7b0: 2020 2020 7c0a 7c20 2020 2020 2020 2020 |.| │ │ │ │ +0006d7c0: 2031 3a20 3120 3420 2020 2020 2031 3a20 1: 1 4 1: │ │ │ │ +0006d7d0: 3320 3220 2020 2020 2020 2020 2020 2020 3 2 │ │ │ │ 0006d7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d800: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0006d810: 2020 2020 2020 2020 2032 3a20 3220 2e20 2: 2 . │ │ │ │ -0006d820: 2020 2020 2032 3a20 3420 2e20 2020 2020 2: 4 . │ │ │ │ +0006d800: 207c 0a7c 2020 2020 2020 2020 2020 323a |.| 2: │ │ │ │ +0006d810: 2032 202e 2020 2020 2020 323a 2034 202e 2 . 2: 4 . │ │ │ │ +0006d820: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d830: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d850: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ +0006d840: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006d850: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 0006d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006d880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006d8a0: 2020 2020 2020 7c0a 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0a57 6179 7320 746f 2075 7365 2073 +..Ways to use s │ │ │ │ -0006d950: 706c 6974 7469 6e67 733a 0a3d 3d3d 3d3d plittings:.===== │ │ │ │ -0006d960: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006d970: 3d3d 0a0a 2020 2a20 2273 706c 6974 7469 ==.. * "splitti │ │ │ │ -0006d980: 6e67 7328 4d61 7472 6978 2c4d 6174 7269 ngs(Matrix,Matri │ │ │ │ -0006d990: 7829 220a 0a46 6f72 2074 6865 2070 726f x)"..For the pro │ │ │ │ -0006d9a0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -0006d9b0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -0006d9c0: 6f62 6a65 6374 202a 6e6f 7465 2073 706c object *note spl │ │ │ │ -0006d9d0: 6974 7469 6e67 733a 2073 706c 6974 7469 ittings: splitti │ │ │ │ -0006d9e0: 6e67 732c 2069 7320 6120 2a6e 6f74 6520 ngs, is a *note │ │ │ │ -0006d9f0: 6d65 7468 6f64 2066 756e 6374 696f 6e3a method function: │ │ │ │ -0006da00: 0a28 4d61 6361 756c 6179 3244 6f63 294d .(Macaulay2Doc)M │ │ │ │ -0006da10: 6574 686f 6446 756e 6374 696f 6e2c 2e0a ethodFunction,.. │ │ │ │ -0006da20: 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .--------------- │ │ │ │ +0006d930: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ +0006d940: 7573 6520 7370 6c69 7474 696e 6773 3a0a use splittings:. │ │ │ │ +0006d950: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006d960: 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 7370 =======.. * "sp │ │ │ │ +0006d970: 6c69 7474 696e 6773 284d 6174 7269 782c littings(Matrix, │ │ │ │ +0006d980: 4d61 7472 6978 2922 0a0a 466f 7220 7468 Matrix)"..For th │ │ │ │ +0006d990: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ +0006d9a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006d9b0: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ +0006d9c0: 6520 7370 6c69 7474 696e 6773 3a20 7370 e splittings: sp │ │ │ │ +0006d9d0: 6c69 7474 696e 6773 2c20 6973 2061 202a littings, is a * │ │ │ │ +0006d9e0: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ +0006d9f0: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ +0006da00: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0006da10: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006da20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006da50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006da60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006da70: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -0006da80: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -0006da90: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -0006daa0: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -0006dab0: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ -0006dac0: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ -0006dad0: 6163 6b61 6765 732f 0a43 6f6d 706c 6574 ackages/.Complet │ │ │ │ -0006dae0: 6549 6e74 6572 7365 6374 696f 6e52 6573 eIntersectionRes │ │ │ │ -0006daf0: 6f6c 7574 696f 6e73 2e6d 323a 3339 3235 olutions.m2:3925 │ │ │ │ -0006db00: 3a30 2e0a 1f0a 4669 6c65 3a20 436f 6d70 :0....File: Comp │ │ │ │ -0006db10: 6c65 7465 496e 7465 7273 6563 7469 6f6e leteIntersection │ │ │ │ -0006db20: 5265 736f 6c75 7469 6f6e 732e 696e 666f Resolutions.info │ │ │ │ -0006db30: 2c20 4e6f 6465 3a20 7374 6162 6c65 486f , Node: stableHo │ │ │ │ -0006db40: 6d2c 204e 6578 743a 2073 756d 5477 6f4d m, Next: sumTwoM │ │ │ │ -0006db50: 6f6e 6f6d 6961 6c73 2c20 5072 6576 3a20 onomials, Prev: │ │ │ │ -0006db60: 7370 6c69 7474 696e 6773 2c20 5570 3a20 splittings, Up: │ │ │ │ -0006db70: 546f 700a 0a73 7461 626c 6548 6f6d 202d Top..stableHom - │ │ │ │ -0006db80: 2d20 6d61 7020 6672 6f6d 2048 6f6d 284d - map from Hom(M │ │ │ │ -0006db90: 2c4e 2920 746f 2074 6865 2073 7461 626c ,N) to the stabl │ │ │ │ -0006dba0: 6520 486f 6d20 6d6f 6475 6c65 0a2a 2a2a e Hom module.*** │ │ │ │ +0006da60: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0006da70: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0006da80: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0006da90: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0006daa0: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0006dab0: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0006dac0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ +0006dad0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ +0006dae0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ +0006daf0: 3a33 3932 353a 302e 0a1f 0a46 696c 653a :3925:0....File: │ │ │ │ +0006db00: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ +0006db10: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ +0006db20: 2e69 6e66 6f2c 204e 6f64 653a 2073 7461 .info, Node: sta │ │ │ │ +0006db30: 626c 6548 6f6d 2c20 4e65 7874 3a20 7375 bleHom, Next: su │ │ │ │ +0006db40: 6d54 776f 4d6f 6e6f 6d69 616c 732c 2050 mTwoMonomials, P │ │ │ │ +0006db50: 7265 763a 2073 706c 6974 7469 6e67 732c rev: splittings, │ │ │ │ +0006db60: 2055 703a 2054 6f70 0a0a 7374 6162 6c65 Up: Top..stable │ │ │ │ +0006db70: 486f 6d20 2d2d 206d 6170 2066 726f 6d20 Hom -- map from │ │ │ │ +0006db80: 486f 6d28 4d2c 4e29 2074 6f20 7468 6520 Hom(M,N) to the │ │ │ │ +0006db90: 7374 6162 6c65 2048 6f6d 206d 6f64 756c stable Hom modul │ │ │ │ +0006dba0: 650a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a e.************** │ │ │ │ 0006dbb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006dbc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006dbd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006dbe0: 2a2a 2a2a 0a0a 2020 2a20 5573 6167 653a ****.. * Usage: │ │ │ │ -0006dbf0: 200a 2020 2020 2020 2020 7020 3d20 7374 . p = st │ │ │ │ -0006dc00: 6162 6c65 486f 6d28 4d2c 4e29 0a20 202a ableHom(M,N). * │ │ │ │ -0006dc10: 2049 6e70 7574 733a 0a20 2020 2020 202a Inputs:. * │ │ │ │ -0006dc20: 204d 2c20 6120 2a6e 6f74 6520 6d6f 6475 M, a *note modu │ │ │ │ -0006dc30: 6c65 3a20 284d 6163 6175 6c61 7932 446f le: (Macaulay2Do │ │ │ │ -0006dc40: 6329 4d6f 6475 6c65 2c2c 200a 2020 2020 c)Module,, . │ │ │ │ -0006dc50: 2020 2a20 4e2c 2061 202a 6e6f 7465 206d * N, a *note m │ │ │ │ -0006dc60: 6f64 756c 653a 2028 4d61 6361 756c 6179 odule: (Macaulay │ │ │ │ -0006dc70: 3244 6f63 294d 6f64 756c 652c 2c20 0a20 2Doc)Module,, . │ │ │ │ -0006dc80: 202a 204f 7574 7075 7473 3a0a 2020 2020 * Outputs:. │ │ │ │ -0006dc90: 2020 2a20 702c 2061 202a 6e6f 7465 206d * p, a *note m │ │ │ │ -0006dca0: 6174 7269 783a 2028 4d61 6361 756c 6179 atrix: (Macaulay │ │ │ │ -0006dcb0: 3244 6f63 294d 6174 7269 782c 2c20 7072 2Doc)Matrix,, pr │ │ │ │ -0006dcc0: 6f6a 6563 7469 6f6e 2066 726f 6d20 486f ojection from Ho │ │ │ │ -0006dcd0: 6d28 4d2c 4e29 2074 6f0a 2020 2020 2020 m(M,N) to. │ │ │ │ -0006dce0: 2020 7468 6520 7374 6162 6c65 2048 6f6d the stable Hom │ │ │ │ -0006dcf0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ -0006dd00: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 2073 =========..The s │ │ │ │ -0006dd10: 7461 626c 6520 486f 6d20 6973 2048 6f6d table Hom is Hom │ │ │ │ -0006dd20: 284d 2c4e 292f 5420 7768 6572 6520 5420 (M,N)/T where T │ │ │ │ -0006dd30: 6973 2074 6865 2073 7562 6d6f 6475 6c65 is the submodule │ │ │ │ -0006dd40: 206f 6620 686f 6d6f 6d6f 7270 6869 736d of homomorphism │ │ │ │ -0006dd50: 7320 7468 6174 0a66 6163 746f 7220 7468 s that.factor th │ │ │ │ -0006dd60: 726f 7567 6820 6120 6672 6565 2063 6f76 rough a free cov │ │ │ │ -0006dd70: 6572 206f 6620 4e20 286f 722c 2065 7175 er of N (or, equ │ │ │ │ -0006dd80: 6976 616c 656e 746c 792c 2074 6872 6f75 ivalently, throu │ │ │ │ -0006dd90: 6768 2061 6e79 2070 726f 6a65 6374 6976 gh any projectiv │ │ │ │ -0006dda0: 6529 0a0a 5365 6520 616c 736f 0a3d 3d3d e)..See also.=== │ │ │ │ -0006ddb0: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ -0006ddc0: 2069 7353 7461 626c 7954 7269 7669 616c isStablyTrivial │ │ │ │ -0006ddd0: 3a20 6973 5374 6162 6c79 5472 6976 6961 : isStablyTrivia │ │ │ │ -0006dde0: 6c2c 202d 2d20 7265 7475 726e 7320 7472 l, -- returns tr │ │ │ │ -0006ddf0: 7565 2069 6620 7468 6520 6d61 7020 676f ue if the map go │ │ │ │ -0006de00: 6573 2074 6f0a 2020 2020 3020 756e 6465 es to. 0 unde │ │ │ │ -0006de10: 7220 7374 6162 6c65 486f 6d0a 0a57 6179 r stableHom..Way │ │ │ │ -0006de20: 7320 746f 2075 7365 2073 7461 626c 6548 s to use stableH │ │ │ │ -0006de30: 6f6d 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d om:.============ │ │ │ │ -0006de40: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0006de50: 2273 7461 626c 6548 6f6d 284d 6f64 756c "stableHom(Modul │ │ │ │ -0006de60: 652c 4d6f 6475 6c65 2922 0a0a 466f 7220 e,Module)"..For │ │ │ │ -0006de70: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0006de80: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006de90: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0006dea0: 6f74 6520 7374 6162 6c65 486f 6d3a 2073 ote stableHom: s │ │ │ │ -0006deb0: 7461 626c 6548 6f6d 2c20 6973 2061 202a tableHom, is a * │ │ │ │ -0006dec0: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ -0006ded0: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ -0006dee0: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ -0006def0: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ +0006dbd0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ +0006dbe0: 7361 6765 3a20 0a20 2020 2020 2020 2070 sage: . p │ │ │ │ +0006dbf0: 203d 2073 7461 626c 6548 6f6d 284d 2c4e = stableHom(M,N │ │ │ │ +0006dc00: 290a 2020 2a20 496e 7075 7473 3a0a 2020 ). * Inputs:. │ │ │ │ +0006dc10: 2020 2020 2a20 4d2c 2061 202a 6e6f 7465 * M, a *note │ │ │ │ +0006dc20: 206d 6f64 756c 653a 2028 4d61 6361 756c module: (Macaul │ │ │ │ +0006dc30: 6179 3244 6f63 294d 6f64 756c 652c 2c20 ay2Doc)Module,, │ │ │ │ +0006dc40: 0a20 2020 2020 202a 204e 2c20 6120 2a6e . * N, a *n │ │ │ │ +0006dc50: 6f74 6520 6d6f 6475 6c65 3a20 284d 6163 ote module: (Mac │ │ │ │ +0006dc60: 6175 6c61 7932 446f 6329 4d6f 6475 6c65 aulay2Doc)Module │ │ │ │ +0006dc70: 2c2c 200a 2020 2a20 4f75 7470 7574 733a ,, . * Outputs: │ │ │ │ +0006dc80: 0a20 2020 2020 202a 2070 2c20 6120 2a6e . * p, a *n │ │ │ │ +0006dc90: 6f74 6520 6d61 7472 6978 3a20 284d 6163 ote matrix: (Mac │ │ │ │ +0006dca0: 6175 6c61 7932 446f 6329 4d61 7472 6978 aulay2Doc)Matrix │ │ │ │ +0006dcb0: 2c2c 2070 726f 6a65 6374 696f 6e20 6672 ,, projection fr │ │ │ │ +0006dcc0: 6f6d 2048 6f6d 284d 2c4e 2920 746f 0a20 om Hom(M,N) to. │ │ │ │ +0006dcd0: 2020 2020 2020 2074 6865 2073 7461 626c the stabl │ │ │ │ +0006dce0: 6520 486f 6d0a 0a44 6573 6372 6970 7469 e Hom..Descripti │ │ │ │ +0006dcf0: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ +0006dd00: 5468 6520 7374 6162 6c65 2048 6f6d 2069 The stable Hom i │ │ │ │ +0006dd10: 7320 486f 6d28 4d2c 4e29 2f54 2077 6865 s Hom(M,N)/T whe │ │ │ │ +0006dd20: 7265 2054 2069 7320 7468 6520 7375 626d re T is the subm │ │ │ │ +0006dd30: 6f64 756c 6520 6f66 2068 6f6d 6f6d 6f72 odule of homomor │ │ │ │ +0006dd40: 7068 6973 6d73 2074 6861 740a 6661 6374 phisms that.fact │ │ │ │ +0006dd50: 6f72 2074 6872 6f75 6768 2061 2066 7265 or through a fre │ │ │ │ +0006dd60: 6520 636f 7665 7220 6f66 204e 2028 6f72 e cover of N (or │ │ │ │ +0006dd70: 2c20 6571 7569 7661 6c65 6e74 6c79 2c20 , equivalently, │ │ │ │ +0006dd80: 7468 726f 7567 6820 616e 7920 7072 6f6a through any proj │ │ │ │ +0006dd90: 6563 7469 7665 290a 0a53 6565 2061 6c73 ective)..See als │ │ │ │ +0006dda0: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0006ddb0: 2a6e 6f74 6520 6973 5374 6162 6c79 5472 *note isStablyTr │ │ │ │ +0006ddc0: 6976 6961 6c3a 2069 7353 7461 626c 7954 ivial: isStablyT │ │ │ │ +0006ddd0: 7269 7669 616c 2c20 2d2d 2072 6574 7572 rivial, -- retur │ │ │ │ +0006dde0: 6e73 2074 7275 6520 6966 2074 6865 206d ns true if the m │ │ │ │ +0006ddf0: 6170 2067 6f65 7320 746f 0a20 2020 2030 ap goes to. 0 │ │ │ │ +0006de00: 2075 6e64 6572 2073 7461 626c 6548 6f6d under stableHom │ │ │ │ +0006de10: 0a0a 5761 7973 2074 6f20 7573 6520 7374 ..Ways to use st │ │ │ │ +0006de20: 6162 6c65 486f 6d3a 0a3d 3d3d 3d3d 3d3d ableHom:.======= │ │ │ │ +0006de30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ +0006de40: 0a20 202a 2022 7374 6162 6c65 486f 6d28 . * "stableHom( │ │ │ │ +0006de50: 4d6f 6475 6c65 2c4d 6f64 756c 6529 220a Module,Module)". │ │ │ │ +0006de60: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +0006de70: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +0006de80: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +0006de90: 6374 202a 6e6f 7465 2073 7461 626c 6548 ct *note stableH │ │ │ │ +0006dea0: 6f6d 3a20 7374 6162 6c65 486f 6d2c 2069 om: stableHom, i │ │ │ │ +0006deb0: 7320 6120 2a6e 6f74 6520 6d65 7468 6f64 s a *note method │ │ │ │ +0006dec0: 2066 756e 6374 696f 6e3a 0a28 4d61 6361 function:.(Maca │ │ │ │ +0006ded0: 756c 6179 3244 6f63 294d 6574 686f 6446 ulay2Doc)MethodF │ │ │ │ +0006dee0: 756e 6374 696f 6e2c 2e0a 0a2d 2d2d 2d2d unction,...----- │ │ │ │ +0006def0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006df20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006df30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006df40: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -0006df50: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -0006df60: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -0006df70: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -0006df80: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -0006df90: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -0006dfa0: 6179 322f 7061 636b 6167 6573 2f0a 436f ay2/packages/.Co │ │ │ │ -0006dfb0: 6d70 6c65 7465 496e 7465 7273 6563 7469 mpleteIntersecti │ │ │ │ -0006dfc0: 6f6e 5265 736f 6c75 7469 6f6e 732e 6d32 onResolutions.m2 │ │ │ │ -0006dfd0: 3a34 3634 393a 302e 0a1f 0a46 696c 653a :4649:0....File: │ │ │ │ -0006dfe0: 2043 6f6d 706c 6574 6549 6e74 6572 7365 CompleteInterse │ │ │ │ -0006dff0: 6374 696f 6e52 6573 6f6c 7574 696f 6e73 ctionResolutions │ │ │ │ -0006e000: 2e69 6e66 6f2c 204e 6f64 653a 2073 756d .info, Node: sum │ │ │ │ -0006e010: 5477 6f4d 6f6e 6f6d 6961 6c73 2c20 4e65 TwoMonomials, Ne │ │ │ │ -0006e020: 7874 3a20 5461 7465 5265 736f 6c75 7469 xt: TateResoluti │ │ │ │ -0006e030: 6f6e 2c20 5072 6576 3a20 7374 6162 6c65 on, Prev: stable │ │ │ │ -0006e040: 486f 6d2c 2055 703a 2054 6f70 0a0a 7375 Hom, Up: Top..su │ │ │ │ -0006e050: 6d54 776f 4d6f 6e6f 6d69 616c 7320 2d2d mTwoMonomials -- │ │ │ │ -0006e060: 2074 616c 6c79 2074 6865 2073 6571 7565 tally the seque │ │ │ │ -0006e070: 6e63 6573 206f 6620 4252 616e 6b73 2066 nces of BRanks f │ │ │ │ -0006e080: 6f72 2063 6572 7461 696e 2065 7861 6d70 or certain examp │ │ │ │ -0006e090: 6c65 730a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a les.************ │ │ │ │ +0006df30: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0006df40: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0006df50: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0006df60: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0006df70: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0006df80: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +0006df90: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0006dfa0: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +0006dfb0: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006dfc0: 6e73 2e6d 323a 3436 3439 3a30 2e0a 1f0a ns.m2:4649:0.... │ │ │ │ +0006dfd0: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +0006dfe0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ +0006dff0: 7469 6f6e 732e 696e 666f 2c20 4e6f 6465 tions.info, Node │ │ │ │ +0006e000: 3a20 7375 6d54 776f 4d6f 6e6f 6d69 616c : sumTwoMonomial │ │ │ │ +0006e010: 732c 204e 6578 743a 2054 6174 6552 6573 s, Next: TateRes │ │ │ │ +0006e020: 6f6c 7574 696f 6e2c 2050 7265 763a 2073 olution, Prev: s │ │ │ │ +0006e030: 7461 626c 6548 6f6d 2c20 5570 3a20 546f tableHom, Up: To │ │ │ │ +0006e040: 700a 0a73 756d 5477 6f4d 6f6e 6f6d 6961 p..sumTwoMonomia │ │ │ │ +0006e050: 6c73 202d 2d20 7461 6c6c 7920 7468 6520 ls -- tally the │ │ │ │ +0006e060: 7365 7175 656e 6365 7320 6f66 2042 5261 sequences of BRa │ │ │ │ +0006e070: 6e6b 7320 666f 7220 6365 7274 6169 6e20 nks for certain │ │ │ │ +0006e080: 6578 616d 706c 6573 0a2a 2a2a 2a2a 2a2a examples.******* │ │ │ │ +0006e090: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006e0a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006e0b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006e0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006e0d0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 *********.. * U │ │ │ │ -0006e0e0: 7361 6765 3a20 0a20 2020 2020 2020 2073 sage: . s │ │ │ │ -0006e0f0: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2863 umTwoMonomials(c │ │ │ │ -0006e100: 2c64 290a 2020 2a20 496e 7075 7473 3a0a ,d). * Inputs:. │ │ │ │ -0006e110: 2020 2020 2020 2a20 632c 2061 6e20 2a6e * c, an *n │ │ │ │ -0006e120: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ -0006e130: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ -0006e140: 636f 6469 6d65 6e73 696f 6e20 696e 2077 codimension in w │ │ │ │ -0006e150: 6869 6368 2074 6f20 776f 726b 0a20 2020 hich to work. │ │ │ │ -0006e160: 2020 202a 2064 2c20 616e 202a 6e6f 7465 * d, an *note │ │ │ │ -0006e170: 2069 6e74 6567 6572 3a20 284d 6163 6175 integer: (Macau │ │ │ │ -0006e180: 6c61 7932 446f 6329 5a5a 2c2c 2064 6567 lay2Doc)ZZ,, deg │ │ │ │ -0006e190: 7265 6520 6f66 2074 6865 206d 6f6e 6f6d ree of the monom │ │ │ │ -0006e1a0: 6961 6c73 2074 6f20 7461 6b65 0a20 202a ials to take. * │ │ │ │ -0006e1b0: 204f 7574 7075 7473 3a0a 2020 2020 2020 Outputs:. │ │ │ │ -0006e1c0: 2a20 542c 2061 202a 6e6f 7465 2074 616c * T, a *note tal │ │ │ │ -0006e1d0: 6c79 3a20 284d 6163 6175 6c61 7932 446f ly: (Macaulay2Do │ │ │ │ -0006e1e0: 6329 5461 6c6c 792c 2c20 0a0a 4465 7363 c)Tally,, ..Desc │ │ │ │ -0006e1f0: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0006e200: 3d3d 3d0a 0a74 616c 6c69 6573 2074 6865 ===..tallies the │ │ │ │ -0006e210: 2073 6571 7565 6e63 6573 206f 6620 422d sequences of B- │ │ │ │ -0006e220: 7261 6e6b 7320 7468 6174 206f 6363 7572 ranks that occur │ │ │ │ -0006e230: 2066 6f72 2073 756d 7320 6f66 2070 6169 for sums of pai │ │ │ │ -0006e240: 7273 206f 6620 6d6f 6e6f 6d69 616c 7320 rs of monomials │ │ │ │ -0006e250: 696e 2052 0a3d 2053 2f28 642d 7468 2070 in R.= S/(d-th p │ │ │ │ -0006e260: 6f77 6572 7320 6f66 2074 6865 2076 6172 owers of the var │ │ │ │ -0006e270: 6961 626c 6573 292c 2077 6974 6820 6675 iables), with fu │ │ │ │ -0006e280: 6c6c 2063 6f6d 706c 6578 6974 7920 283d ll complexity (= │ │ │ │ -0006e290: 6329 3b20 7468 6174 2069 732c 2066 6f72 c); that is, for │ │ │ │ -0006e2a0: 2061 6e0a 6170 7072 6f70 7269 6174 6520 an.appropriate │ │ │ │ -0006e2b0: 7379 7a79 6779 204d 206f 6620 4d30 203d syzygy M of M0 = │ │ │ │ -0006e2c0: 2052 2f28 6d31 2b6d 3229 2077 6865 7265 R/(m1+m2) where │ │ │ │ -0006e2d0: 206d 3120 616e 6420 6d32 2061 7265 206d m1 and m2 are m │ │ │ │ -0006e2e0: 6f6e 6f6d 6961 6c73 206f 6620 7468 650a onomials of the. │ │ │ │ -0006e2f0: 7361 6d65 2064 6567 7265 652e 0a0a 2b2d same degree...+- │ │ │ │ +0006e0c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +0006e0d0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +0006e0e0: 2020 2020 7375 6d54 776f 4d6f 6e6f 6d69 sumTwoMonomi │ │ │ │ +0006e0f0: 616c 7328 632c 6429 0a20 202a 2049 6e70 als(c,d). * Inp │ │ │ │ +0006e100: 7574 733a 0a20 2020 2020 202a 2063 2c20 uts:. * c, │ │ │ │ +0006e110: 616e 202a 6e6f 7465 2069 6e74 6567 6572 an *note integer │ │ │ │ +0006e120: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +0006e130: 5a5a 2c2c 2063 6f64 696d 656e 7369 6f6e ZZ,, codimension │ │ │ │ +0006e140: 2069 6e20 7768 6963 6820 746f 2077 6f72 in which to wor │ │ │ │ +0006e150: 6b0a 2020 2020 2020 2a20 642c 2061 6e20 k. * d, an │ │ │ │ +0006e160: 2a6e 6f74 6520 696e 7465 6765 723a 2028 *note integer: ( │ │ │ │ +0006e170: 4d61 6361 756c 6179 3244 6f63 295a 5a2c Macaulay2Doc)ZZ, │ │ │ │ +0006e180: 2c20 6465 6772 6565 206f 6620 7468 6520 , degree of the │ │ │ │ +0006e190: 6d6f 6e6f 6d69 616c 7320 746f 2074 616b monomials to tak │ │ │ │ +0006e1a0: 650a 2020 2a20 4f75 7470 7574 733a 0a20 e. * Outputs:. │ │ │ │ +0006e1b0: 2020 2020 202a 2054 2c20 6120 2a6e 6f74 * T, a *not │ │ │ │ +0006e1c0: 6520 7461 6c6c 793a 2028 4d61 6361 756c e tally: (Macaul │ │ │ │ +0006e1d0: 6179 3244 6f63 2954 616c 6c79 2c2c 200a ay2Doc)Tally,, . │ │ │ │ +0006e1e0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +0006e1f0: 3d3d 3d3d 3d3d 3d3d 0a0a 7461 6c6c 6965 ========..tallie │ │ │ │ +0006e200: 7320 7468 6520 7365 7175 656e 6365 7320 s the sequences │ │ │ │ +0006e210: 6f66 2042 2d72 616e 6b73 2074 6861 7420 of B-ranks that │ │ │ │ +0006e220: 6f63 6375 7220 666f 7220 7375 6d73 206f occur for sums o │ │ │ │ +0006e230: 6620 7061 6972 7320 6f66 206d 6f6e 6f6d f pairs of monom │ │ │ │ +0006e240: 6961 6c73 2069 6e20 520a 3d20 532f 2864 ials in R.= S/(d │ │ │ │ +0006e250: 2d74 6820 706f 7765 7273 206f 6620 7468 -th powers of th │ │ │ │ +0006e260: 6520 7661 7269 6162 6c65 7329 2c20 7769 e variables), wi │ │ │ │ +0006e270: 7468 2066 756c 6c20 636f 6d70 6c65 7869 th full complexi │ │ │ │ +0006e280: 7479 2028 3d63 293b 2074 6861 7420 6973 ty (=c); that is │ │ │ │ +0006e290: 2c20 666f 7220 616e 0a61 7070 726f 7072 , for an.appropr │ │ │ │ +0006e2a0: 6961 7465 2073 797a 7967 7920 4d20 6f66 iate syzygy M of │ │ │ │ +0006e2b0: 204d 3020 3d20 522f 286d 312b 6d32 2920 M0 = R/(m1+m2) │ │ │ │ +0006e2c0: 7768 6572 6520 6d31 2061 6e64 206d 3220 where m1 and m2 │ │ │ │ +0006e2d0: 6172 6520 6d6f 6e6f 6d69 616c 7320 6f66 are monomials of │ │ │ │ +0006e2e0: 2074 6865 0a73 616d 6520 6465 6772 6565 the.same degree │ │ │ │ +0006e2f0: 2e0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ...+------------ │ │ │ │ 0006e300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e330: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7365 ------+.|i1 : se │ │ │ │ -0006e340: 7452 616e 646f 6d53 6565 6420 3020 2020 tRandomSeed 0 │ │ │ │ +0006e320: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +0006e330: 3a20 7365 7452 616e 646f 6d53 6565 6420 : setRandomSeed │ │ │ │ +0006e340: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ 0006e350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e370: 7c0a 7c20 2d2d 2073 6574 7469 6e67 2072 |.| -- setting r │ │ │ │ -0006e380: 616e 646f 6d20 7365 6564 2074 6f20 3020 andom seed to 0 │ │ │ │ -0006e390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +0006e360: 2020 207c 0a7c 202d 2d20 7365 7474 696e |.| -- settin │ │ │ │ +0006e370: 6720 7261 6e64 6f6d 2073 6565 6420 746f g random seed to │ │ │ │ +0006e380: 2030 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +0006e390: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0006e3a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e3b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e3c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e3e0: 2020 2020 7c0a 7c6f 3120 3d20 3020 2020 |.|o1 = 0 │ │ │ │ +0006e3d0: 2020 2020 207c 0a7c 6f31 203d 2030 2020 |.|o1 = 0 │ │ │ │ +0006e3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e410: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006e420: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0006e400: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +0006e410: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +0006e420: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -0006e460: 7375 6d54 776f 4d6f 6e6f 6d69 616c 7328 sumTwoMonomials( │ │ │ │ -0006e470: 322c 3329 2020 2020 2020 2020 2020 2020 2,3) │ │ │ │ -0006e480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e490: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -0006e4a0: 3534 3735 3131 7320 2863 7075 293b 2030 547511s (cpu); 0 │ │ │ │ -0006e4b0: 2e34 3130 3232 7320 2874 6872 6561 6429 .41022s (thread) │ │ │ │ -0006e4c0: 3b20 3073 2028 6763 2920 2020 7c0a 7c32 ; 0s (gc) |.|2 │ │ │ │ +0006e440: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 2073 -------+.|i2 : s │ │ │ │ +0006e450: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 2832 umTwoMonomials(2 │ │ │ │ +0006e460: 2c33 2920 2020 2020 2020 2020 2020 2020 ,3) │ │ │ │ +0006e470: 2020 2020 2020 2020 2020 2020 2020 2020 │ 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0a7c 2020 |.| │ │ │ │ 0006e530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e540: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0006e540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e570: 2020 2020 2020 2020 2020 7c0a 7c20 2d2d |.| -- │ │ │ │ -0006e580: 2075 7365 6420 302e 3130 3638 3434 7320 used 0.106844s │ │ │ │ -0006e590: 2863 7075 293b 2030 2e31 3036 3431 7320 (cpu); 0.10641s │ │ │ │ -0006e5a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -0006e5b0: 2920 2020 7c0a 7c33 2020 2020 2020 2020 ) |.|3 │ │ │ │ +0006e560: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +0006e570: 302e 3135 3638 3973 2028 6370 7529 3b20 0.15689s (cpu); │ │ │ │ +0006e580: 302e 3131 3034 3132 7320 2874 6872 6561 0.110412s (threa │ │ │ │ +0006e590: 6429 3b20 3073 2028 6763 2920 207c 0a7c d); 0s (gc) |.| │ │ │ │ +0006e5a0: 3320 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ +0006e5b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e5c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e5e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -0006e5f0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20 |Tally{{{2, 2}, │ │ │ │ -0006e600: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020 {1, 2}} => 1} │ │ │ │ -0006e610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e620: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +0006e5d0: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b |.|Tally{{ │ │ │ │ +0006e5e0: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ +0006e5f0: 3d3e 2031 7d20 2020 2020 2020 2020 2020 => 1} │ │ │ │ +0006e600: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +0006e610: 0a7c 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 2020 2020 2020 2020 │ │ │ │ -0006e650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e660: 2020 7c0a 7c20 2d2d 2075 7365 6420 342e |.| -- used 4. │ │ │ │ -0006e670: 3035 3865 2d30 3673 2028 6370 7529 3b20 058e-06s (cpu); │ │ │ │ -0006e680: 332e 3438 3665 2d30 3673 2028 7468 7265 3.486e-06s (thre │ │ │ │ -0006e690: 6164 293b 2030 7320 2867 6329 7c0a 7c34 ad); 0s (gc)|.|4 │ │ │ │ +0006e640: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ +0006e650: 7365 6420 332e 3032 3365 2d30 3673 2028 sed 3.023e-06s ( │ │ │ │ +0006e660: 6370 7529 3b20 322e 3632 652d 3036 7320 cpu); 2.62e-06s │ │ │ │ +0006e670: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +0006e680: 297c 0a7c 3420 2020 2020 2020 2020 2020 )|.|4 │ │ │ │ +0006e690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6d0: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7d |.|Tally{} │ │ │ │ +0006e6b0: 2020 2020 2020 2020 2020 7c0a 7c54 616c |.|Tal │ │ │ │ +0006e6c0: 6c79 7b7d 2020 2020 2020 2020 2020 2020 ly{} │ │ │ │ +0006e6d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006e6e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006e710: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ -0006e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e740: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 6565 ----------+..See │ │ │ │ -0006e750: 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a also.========.. │ │ │ │ -0006e760: 2020 2a20 2a6e 6f74 6520 7477 6f4d 6f6e * *note twoMon │ │ │ │ -0006e770: 6f6d 6961 6c73 3a20 7477 6f4d 6f6e 6f6d omials: twoMonom │ │ │ │ -0006e780: 6961 6c73 2c20 2d2d 2074 616c 6c79 2074 ials, -- tally t │ │ │ │ -0006e790: 6865 2073 6571 7565 6e63 6573 206f 6620 he sequences of │ │ │ │ -0006e7a0: 4252 616e 6b73 2066 6f72 0a20 2020 2063 BRanks for. c │ │ │ │ -0006e7b0: 6572 7461 696e 2065 7861 6d70 6c65 730a ertain examples. │ │ │ │ -0006e7c0: 0a57 6179 7320 746f 2075 7365 2073 756d .Ways to use sum │ │ │ │ -0006e7d0: 5477 6f4d 6f6e 6f6d 6961 6c73 3a0a 3d3d TwoMonomials:.== │ │ │ │ -0006e7e0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006e7f0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -0006e800: 2273 756d 5477 6f4d 6f6e 6f6d 6961 6c73 "sumTwoMonomials │ │ │ │ -0006e810: 285a 5a2c 5a5a 2922 0a0a 466f 7220 7468 (ZZ,ZZ)"..For th │ │ │ │ -0006e820: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0006e830: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0006e840: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0006e850: 6520 7375 6d54 776f 4d6f 6e6f 6d69 616c e sumTwoMonomial │ │ │ │ -0006e860: 733a 2073 756d 5477 6f4d 6f6e 6f6d 6961 s: sumTwoMonomia │ │ │ │ -0006e870: 6c73 2c20 6973 2061 202a 6e6f 7465 206d ls, is a *note m │ │ │ │ -0006e880: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -0006e890: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0006e8a0: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0006e6f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ +0006e700: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006e710: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006e720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a53 ------------+..S │ │ │ │ +0006e730: 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d 3d3d ee also.======== │ │ │ │ +0006e740: 0a0a 2020 2a20 2a6e 6f74 6520 7477 6f4d .. * *note twoM │ │ │ │ +0006e750: 6f6e 6f6d 6961 6c73 3a20 7477 6f4d 6f6e onomials: twoMon │ │ │ │ +0006e760: 6f6d 6961 6c73 2c20 2d2d 2074 616c 6c79 omials, -- tally │ │ │ │ +0006e770: 2074 6865 2073 6571 7565 6e63 6573 206f the sequences o │ │ │ │ +0006e780: 6620 4252 616e 6b73 2066 6f72 0a20 2020 f BRanks for. │ │ │ │ +0006e790: 2063 6572 7461 696e 2065 7861 6d70 6c65 certain example │ │ │ │ +0006e7a0: 730a 0a57 6179 7320 746f 2075 7365 2073 s..Ways to use s │ │ │ │ +0006e7b0: 756d 5477 6f4d 6f6e 6f6d 6961 6c73 3a0a umTwoMonomials:. │ │ │ │ +0006e7c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006e7d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 ============.. │ │ │ │ +0006e7e0: 2a20 2273 756d 5477 6f4d 6f6e 6f6d 6961 * "sumTwoMonomia │ │ │ │ +0006e7f0: 6c73 285a 5a2c 5a5a 2922 0a0a 466f 7220 ls(ZZ,ZZ)"..For │ │ │ │ +0006e800: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +0006e810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +0006e820: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +0006e830: 6f74 6520 7375 6d54 776f 4d6f 6e6f 6d69 ote sumTwoMonomi │ │ │ │ +0006e840: 616c 733a 2073 756d 5477 6f4d 6f6e 6f6d als: sumTwoMonom │ │ │ │ +0006e850: 6961 6c73 2c20 6973 2061 202a 6e6f 7465 ials, is a *note │ │ │ │ +0006e860: 206d 6574 686f 6420 6675 6e63 7469 6f6e method function │ │ │ │ +0006e870: 3a0a 284d 6163 6175 6c61 7932 446f 6329 :.(Macaulay2Doc) │ │ │ │ +0006e880: 4d65 7468 6f64 4675 6e63 7469 6f6e 2c2e MethodFunction,. │ │ │ │ +0006e890: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +0006e8a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e8b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e8c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006e8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006e8f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0006e900: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0006e910: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0006e920: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0006e930: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0006e940: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0006e950: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0006e960: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ -0006e970: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -0006e980: 6c75 7469 6f6e 732e 6d32 3a34 3531 323a lutions.m2:4512: │ │ │ │ -0006e990: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ -0006e9a0: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -0006e9b0: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ -0006e9c0: 204e 6f64 653a 2054 6174 6552 6573 6f6c Node: TateResol │ │ │ │ -0006e9d0: 7574 696f 6e2c 204e 6578 743a 2074 656e ution, Next: ten │ │ │ │ -0006e9e0: 736f 7257 6974 6843 6f6d 706f 6e65 6e74 sorWithComponent │ │ │ │ -0006e9f0: 732c 2050 7265 763a 2073 756d 5477 6f4d s, Prev: sumTwoM │ │ │ │ -0006ea00: 6f6e 6f6d 6961 6c73 2c20 5570 3a20 546f onomials, Up: To │ │ │ │ -0006ea10: 700a 0a54 6174 6552 6573 6f6c 7574 696f p..TateResolutio │ │ │ │ -0006ea20: 6e20 2d2d 2054 6174 6552 6573 6f6c 7574 n -- TateResolut │ │ │ │ -0006ea30: 696f 6e20 6f66 2061 206d 6f64 756c 6520 ion of a module │ │ │ │ -0006ea40: 6f76 6572 2061 6e20 6578 7465 7269 6f72 over an exterior │ │ │ │ -0006ea50: 2061 6c67 6562 7261 0a2a 2a2a 2a2a 2a2a algebra.******* │ │ │ │ +0006e8e0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0006e8f0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0006e900: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0006e910: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0006e920: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +0006e930: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +0006e940: 7061 636b 6167 6573 2f0a 436f 6d70 6c65 packages/.Comple │ │ │ │ +0006e950: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +0006e960: 736f 6c75 7469 6f6e 732e 6d32 3a34 3531 solutions.m2:451 │ │ │ │ +0006e970: 323a 302e 0a1f 0a46 696c 653a 2043 6f6d 2:0....File: Com │ │ │ │ +0006e980: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +0006e990: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66 nResolutions.inf │ │ │ │ +0006e9a0: 6f2c 204e 6f64 653a 2054 6174 6552 6573 o, Node: TateRes │ │ │ │ +0006e9b0: 6f6c 7574 696f 6e2c 204e 6578 743a 2074 olution, Next: t │ │ │ │ +0006e9c0: 656e 736f 7257 6974 6843 6f6d 706f 6e65 ensorWithCompone │ │ │ │ +0006e9d0: 6e74 732c 2050 7265 763a 2073 756d 5477 nts, Prev: sumTw │ │ │ │ +0006e9e0: 6f4d 6f6e 6f6d 6961 6c73 2c20 5570 3a20 oMonomials, Up: │ │ │ │ +0006e9f0: 546f 700a 0a54 6174 6552 6573 6f6c 7574 Top..TateResolut │ │ │ │ +0006ea00: 696f 6e20 2d2d 2054 6174 6552 6573 6f6c ion -- TateResol │ │ │ │ +0006ea10: 7574 696f 6e20 6f66 2061 206d 6f64 756c ution of a modul │ │ │ │ +0006ea20: 6520 6f76 6572 2061 6e20 6578 7465 7269 e over an exteri │ │ │ │ +0006ea30: 6f72 2061 6c67 6562 7261 0a2a 2a2a 2a2a or algebra.***** │ │ │ │ +0006ea40: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +0006ea50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006ea60: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0006ea70: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006ea80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -0006ea90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -0006eaa0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ -0006eab0: 2020 2020 4620 3d20 5461 7465 5265 736f F = TateReso │ │ │ │ -0006eac0: 6c75 7469 6f6e 284d 2c6c 6f77 6572 2c75 lution(M,lower,u │ │ │ │ -0006ead0: 7070 6572 290a 2020 2a20 496e 7075 7473 pper). * Inputs │ │ │ │ -0006eae0: 3a0a 2020 2020 2020 2a20 4d2c 2061 202a :. * M, a * │ │ │ │ -0006eaf0: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ -0006eb00: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ -0006eb10: 652c 2c20 0a20 2020 2020 202a 206c 6f77 e,, . * low │ │ │ │ -0006eb20: 6572 2c20 616e 202a 6e6f 7465 2069 6e74 er, an *note int │ │ │ │ -0006eb30: 6567 6572 3a20 284d 6163 6175 6c61 7932 eger: (Macaulay2 │ │ │ │ -0006eb40: 446f 6329 5a5a 2c2c 200a 2020 2020 2020 Doc)ZZ,, . │ │ │ │ -0006eb50: 2a20 7570 7065 722c 2061 6e20 2a6e 6f74 * upper, an *not │ │ │ │ -0006eb60: 6520 696e 7465 6765 723a 2028 4d61 6361 e integer: (Maca │ │ │ │ -0006eb70: 756c 6179 3244 6f63 295a 5a2c 2c20 6c6f ulay2Doc)ZZ,, lo │ │ │ │ -0006eb80: 7765 7220 616e 6420 7570 7065 7220 626f wer and upper bo │ │ │ │ -0006eb90: 756e 6473 2066 6f72 0a20 2020 2020 2020 unds for. │ │ │ │ -0006eba0: 2074 6865 2072 6573 6f6c 7574 696f 6e0a the resolution. │ │ │ │ -0006ebb0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -0006ebc0: 2020 202a 2046 2c20 6120 2a6e 6f74 6520 * F, a *note │ │ │ │ -0006ebd0: 636f 6d70 6c65 783a 2028 436f 6d70 6c65 complex: (Comple │ │ │ │ -0006ebe0: 7865 7329 436f 6d70 6c65 782c 2c20 0a0a xes)Complex,, .. │ │ │ │ -0006ebf0: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -0006ec00: 3d3d 3d3d 3d3d 3d0a 0a46 6f72 6d73 2061 =======..Forms a │ │ │ │ -0006ec10: 6e20 696e 7465 7276 616c 2c20 6c6f 7765 n interval, lowe │ │ │ │ -0006ec20: 722e 2e75 7070 6572 2c20 6f66 2061 2064 r..upper, of a d │ │ │ │ -0006ec30: 6f75 626c 7920 696e 6669 6e69 7465 2066 oubly infinite f │ │ │ │ -0006ec40: 7265 6520 7265 736f 6c75 7469 6f6e 206f ree resolution o │ │ │ │ -0006ec50: 6620 6120 610a 436f 6865 6e2d 4d61 6361 f a a.Cohen-Maca │ │ │ │ -0006ec60: 756c 6179 206d 6f64 756c 6520 6f76 6572 ulay module over │ │ │ │ -0006ec70: 2061 2047 6f72 656e 7374 6569 6e20 7269 a Gorenstein ri │ │ │ │ -0006ec80: 6e67 2c20 7375 6368 2061 7320 616e 7920 ng, such as any │ │ │ │ -0006ec90: 6d6f 6475 6c65 206f 7665 7220 616e 0a65 module over an.e │ │ │ │ -0006eca0: 7874 6572 696f 7220 616c 6765 6272 6120 xterior algebra │ │ │ │ -0006ecb0: 2861 6374 7561 6c6c 792c 2061 6e79 206d (actually, any m │ │ │ │ -0006ecc0: 6f64 756c 6520 6f76 6572 2061 6e79 2072 odule over any r │ │ │ │ -0006ecd0: 696e 672e 290a 0a2b 2d2d 2d2d 2d2d 2d2d ing.)..+-------- │ │ │ │ +0006ea80: 0a0a 2020 2a20 5573 6167 653a 200a 2020 .. * Usage: . │ │ │ │ +0006ea90: 2020 2020 2020 4620 3d20 5461 7465 5265 F = TateRe │ │ │ │ +0006eaa0: 736f 6c75 7469 6f6e 284d 2c6c 6f77 6572 solution(M,lower │ │ │ │ +0006eab0: 2c75 7070 6572 290a 2020 2a20 496e 7075 ,upper). * Inpu │ │ │ │ +0006eac0: 7473 3a0a 2020 2020 2020 2a20 4d2c 2061 ts:. * M, a │ │ │ │ +0006ead0: 202a 6e6f 7465 206d 6f64 756c 653a 2028 *note module: ( │ │ │ │ +0006eae0: 4d61 6361 756c 6179 3244 6f63 294d 6f64 Macaulay2Doc)Mod │ │ │ │ +0006eaf0: 756c 652c 2c20 0a20 2020 2020 202a 206c ule,, . * l │ │ │ │ +0006eb00: 6f77 6572 2c20 616e 202a 6e6f 7465 2069 ower, an *note i │ │ │ │ +0006eb10: 6e74 6567 6572 3a20 284d 6163 6175 6c61 nteger: (Macaula │ │ │ │ +0006eb20: 7932 446f 6329 5a5a 2c2c 200a 2020 2020 y2Doc)ZZ,, . │ │ │ │ +0006eb30: 2020 2a20 7570 7065 722c 2061 6e20 2a6e * upper, an *n │ │ │ │ +0006eb40: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +0006eb50: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +0006eb60: 6c6f 7765 7220 616e 6420 7570 7065 7220 lower and upper │ │ │ │ +0006eb70: 626f 756e 6473 2066 6f72 0a20 2020 2020 bounds for. │ │ │ │ +0006eb80: 2020 2074 6865 2072 6573 6f6c 7574 696f the resolutio │ │ │ │ +0006eb90: 6e0a 2020 2a20 4f75 7470 7574 733a 0a20 n. * Outputs:. │ │ │ │ +0006eba0: 2020 2020 202a 2046 2c20 6120 2a6e 6f74 * F, a *not │ │ │ │ +0006ebb0: 6520 636f 6d70 6c65 783a 2028 436f 6d70 e complex: (Comp │ │ │ │ +0006ebc0: 6c65 7865 7329 436f 6d70 6c65 782c 2c20 lexes)Complex,, │ │ │ │ +0006ebd0: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0006ebe0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a46 6f72 6d73 =========..Forms │ │ │ │ +0006ebf0: 2061 6e20 696e 7465 7276 616c 2c20 6c6f an interval, lo │ │ │ │ +0006ec00: 7765 722e 2e75 7070 6572 2c20 6f66 2061 wer..upper, of a │ │ │ │ +0006ec10: 2064 6f75 626c 7920 696e 6669 6e69 7465 doubly infinite │ │ │ │ +0006ec20: 2066 7265 6520 7265 736f 6c75 7469 6f6e free resolution │ │ │ │ +0006ec30: 206f 6620 6120 610a 436f 6865 6e2d 4d61 of a a.Cohen-Ma │ │ │ │ +0006ec40: 6361 756c 6179 206d 6f64 756c 6520 6f76 caulay module ov │ │ │ │ +0006ec50: 6572 2061 2047 6f72 656e 7374 6569 6e20 er a Gorenstein │ │ │ │ +0006ec60: 7269 6e67 2c20 7375 6368 2061 7320 616e ring, such as an │ │ │ │ +0006ec70: 7920 6d6f 6475 6c65 206f 7665 7220 616e y module over an │ │ │ │ +0006ec80: 0a65 7874 6572 696f 7220 616c 6765 6272 .exterior algebr │ │ │ │ +0006ec90: 6120 2861 6374 7561 6c6c 792c 2061 6e79 a (actually, any │ │ │ │ +0006eca0: 206d 6f64 756c 6520 6f76 6572 2061 6e79 module over any │ │ │ │ +0006ecb0: 2072 696e 672e 290a 0a2b 2d2d 2d2d 2d2d ring.)..+------ │ │ │ │ +0006ecc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006ecd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ece0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006ecf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ed20: 2d2d 2d2d 2d2b 0a7c 6931 203a 2045 203d -----+.|i1 : E = │ │ │ │ -0006ed30: 205a 5a2f 3130 315b 612c 622c 632c 2053 ZZ/101[a,b,c, S │ │ │ │ -0006ed40: 6b65 7743 6f6d 6d75 7461 7469 7665 3d3e kewCommutative=> │ │ │ │ -0006ed50: 7472 7565 5d20 2020 2020 2020 2020 2020 true] │ │ │ │ +0006ed00: 2d2d 2d2d 2d2d 2d2b 0a7c 6931 203a 2045 -------+.|i1 : E │ │ │ │ +0006ed10: 203d 205a 5a2f 3130 315b 612c 622c 632c = ZZ/101[a,b,c, │ │ │ │ +0006ed20: 2053 6b65 7743 6f6d 6d75 7461 7469 7665 SkewCommutative │ │ │ │ +0006ed30: 3d3e 7472 7565 5d20 2020 2020 2020 2020 =>true] │ │ │ │ +0006ed40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ed50: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0006ed60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ed70: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006ed70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ed80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ed90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eda0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006eda0: 2020 2020 2020 207c 0a7c 6f31 203d 2045 |.|o1 = E │ │ │ │ 0006edb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006edc0: 2020 2020 207c 0a7c 6f31 203d 2045 2020 |.|o1 = E │ │ │ │ +0006edc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006edd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ede0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006edf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006edf0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0006ee00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee10: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006ee10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ee30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ee60: 2020 2020 207c 0a7c 6f31 203a 2050 6f6c |.|o1 : Pol │ │ │ │ -0006ee70: 796e 6f6d 6961 6c52 696e 672c 2033 2073 ynomialRing, 3 s │ │ │ │ -0006ee80: 6b65 7720 636f 6d6d 7574 6174 6976 6520 kew commutative │ │ │ │ -0006ee90: 7661 7269 6162 6c65 2873 2920 2020 2020 variable(s) │ │ │ │ -0006eea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eeb0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006ee40: 2020 2020 2020 207c 0a7c 6f31 203a 2050 |.|o1 : P │ │ │ │ +0006ee50: 6f6c 796e 6f6d 6961 6c52 696e 672c 2033 olynomialRing, 3 │ │ │ │ +0006ee60: 2073 6b65 7720 636f 6d6d 7574 6174 6976 skew commutativ │ │ │ │ +0006ee70: 6520 7661 7269 6162 6c65 2873 2920 2020 e variable(s) │ │ │ │ +0006ee80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ee90: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006eea0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006eeb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006eec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006eed0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006eee0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006eef0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ef00: 2d2d 2d2d 2d2b 0a7c 6932 203a 204d 203d -----+.|i2 : M = │ │ │ │ -0006ef10: 2063 6f6b 6572 206d 6170 2845 5e32 2c20 coker map(E^2, │ │ │ │ -0006ef20: 455e 7b2d 317d 2c20 6d61 7472 6978 2261 E^{-1}, matrix"a │ │ │ │ -0006ef30: 623b 6263 2229 2020 2020 2020 2020 2020 b;bc") │ │ │ │ +0006eee0: 2d2d 2d2d 2d2d 2d2b 0a7c 6932 203a 204d -------+.|i2 : M │ │ │ │ +0006eef0: 203d 2063 6f6b 6572 206d 6170 2845 5e32 = coker map(E^2 │ │ │ │ +0006ef00: 2c20 455e 7b2d 317d 2c20 6d61 7472 6978 , E^{-1}, matrix │ │ │ │ +0006ef10: 2261 623b 6263 2229 2020 2020 2020 2020 "ab;bc") │ │ │ │ +0006ef20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006ef30: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0006ef40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ef50: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006ef50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ef60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006ef70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ef80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006ef90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efa0: 2020 2020 207c 0a7c 6f32 203d 2063 6f6b |.|o2 = cok │ │ │ │ -0006efb0: 6572 6e65 6c20 7c20 6162 207c 2020 2020 ernel | ab | │ │ │ │ +0006ef80: 2020 2020 2020 207c 0a7c 6f32 203d 2063 |.|o2 = c │ │ │ │ +0006ef90: 6f6b 6572 6e65 6c20 7c20 6162 207c 2020 okernel | ab | │ │ │ │ +0006efa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006efb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006efc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006efe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006eff0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -0006f000: 2020 2020 2020 7c20 6263 207c 2020 2020 | bc | │ │ │ │ +0006efd0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +0006efe0: 2020 2020 2020 2020 7c20 6263 207c 2020 | bc | │ │ │ │ +0006eff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f020: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0006f030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f040: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006f040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0006f060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f070: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0006f080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f090: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +0006f090: 2020 2020 2020 3220 2020 2020 2020 2020 2 │ │ │ │ 0006f0a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0b0: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ -0006f0c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f0e0: 2020 2020 207c 0a7c 6f32 203a 2045 2d6d |.|o2 : E-m │ │ │ │ -0006f0f0: 6f64 756c 652c 2071 756f 7469 656e 7420 odule, quotient │ │ │ │ -0006f100: 6f66 2045 2020 2020 2020 2020 2020 2020 of E │ │ │ │ -0006f110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0006f130: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +0006f0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f0c0: 2020 2020 2020 207c 0a7c 6f32 203a 2045 |.|o2 : E │ │ │ │ +0006f0d0: 2d6d 6f64 756c 652c 2071 756f 7469 656e -module, quotien │ │ │ │ +0006f0e0: 7420 6f66 2045 2020 2020 2020 2020 2020 t of E │ │ │ │ +0006f0f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0006f110: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +0006f120: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0006f130: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f140: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0006f150: 2d2d 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info │ │ │ │ -0006fb70: 726d 6174 696f 6e0a 2a2a 2a2a 2a2a 2a2a rmation.******** │ │ │ │ +0006f9f0: 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 ----------..The │ │ │ │ +0006fa00: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ +0006fa10: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ +0006fa20: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ +0006fa30: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ +0006fa40: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ +0006fa50: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ +0006fa60: 732f 0a43 6f6d 706c 6574 6549 6e74 6572 s/.CompleteInter │ │ │ │ +0006fa70: 7365 6374 696f 6e52 6573 6f6c 7574 696f sectionResolutio │ │ │ │ +0006fa80: 6e73 2e6d 323a 3334 3939 3a30 2e0a 1f0a ns.m2:3499:0.... │ │ │ │ +0006fa90: 4669 6c65 3a20 436f 6d70 6c65 7465 496e File: CompleteIn │ │ │ │ +0006faa0: 7465 7273 6563 7469 6f6e 5265 736f 6c75 tersectionResolu │ │ │ │ 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652c ulay2Doc)Module, │ │ │ │ -0006fc60: 2c20 0a20 202a 204f 7574 7075 7473 3a0a , . * Outputs:. │ │ │ │ -0006fc70: 2020 2020 2020 2a20 542c 2061 202a 6e6f * T, a *no │ │ │ │ -0006fc80: 7465 206d 6f64 756c 653a 2028 4d61 6361 te module: (Maca │ │ │ │ -0006fc90: 756c 6179 3244 6f63 294d 6f64 756c 652c ulay2Doc)Module, │ │ │ │ -0006fca0: 2c20 0a0a 4465 7363 7269 7074 696f 6e0a , ..Description. │ │ │ │ -0006fcb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 6620 ===========..If │ │ │ │ -0006fcc0: 4d20 616e 642f 6f72 204e 2061 7265 2064 M and/or N are d │ │ │ │ -0006fcd0: 6972 6563 7420 7375 6d20 6d6f 6475 6c65 irect sum module │ │ │ │ -0006fce0: 7320 2869 7344 6972 6563 7453 756d 204d s (isDirectSum M │ │ │ │ -0006fcf0: 203d 3d20 7472 7565 2920 7468 656e 2054 == true) then T │ │ │ │ -0006fd00: 2069 7320 7468 650a 6469 7265 6374 2073 is the.direct s │ │ │ │ -0006fd10: 756d 206f 6620 7468 6520 7465 6e73 6f72 um of the tensor │ │ │ │ -0006fd20: 2070 726f 6475 6374 7320 6265 7477 6565 products betwee │ │ │ │ -0006fd30: 6e20 7468 6520 636f 6d70 6f6e 656e 7473 n the components │ │ │ │ -0006fd40: 2e20 5468 6973 2053 484f 554c 4420 6265 . This SHOULD be │ │ │ │ -0006fd50: 2062 7569 6c74 0a69 6e74 6f20 4d2a 2a4e built.into M**N │ │ │ │ -0006fd60: 2c20 6275 7420 6973 6e27 7420 6173 206f , but isn't as o │ │ │ │ -0006fd70: 6620 4d32 2c20 762e 2031 2e37 0a0a 5365 f M2, v. 1.7..Se │ │ │ │ -0006fd80: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -0006fd90: 0a20 202a 202a 6e6f 7465 2048 6f6d 5769 . * *note HomWi │ │ │ │ -0006fda0: 7468 436f 6d70 6f6e 656e 7473 3a20 486f thComponents: Ho │ │ │ │ -0006fdb0: 6d57 6974 6843 6f6d 706f 6e65 6e74 732c mWithComponents, │ │ │ │ -0006fdc0: 202d 2d20 636f 6d70 7574 6573 2048 6f6d -- computes Hom │ │ │ │ -0006fdd0: 2c20 7072 6573 6572 7669 6e67 0a20 2020 , preserving. │ │ │ │ -0006fde0: 2064 6972 6563 7420 7375 6d20 696e 666f direct sum info │ │ │ │ -0006fdf0: 726d 6174 696f 6e0a 2020 2a20 2a6e 6f74 rmation. * *not │ │ │ │ -0006fe00: 6520 6475 616c 5769 7468 436f 6d70 6f6e e dualWithCompon │ │ │ │ -0006fe10: 656e 7473 3a20 6475 616c 5769 7468 436f ents: dualWithCo │ │ │ │ -0006fe20: 6d70 6f6e 656e 7473 2c20 2d2d 2064 7561 mponents, -- dua │ │ │ │ -0006fe30: 6c20 6d6f 6475 6c65 2070 7265 7365 7276 l module preserv │ │ │ │ -0006fe40: 696e 670a 2020 2020 6469 7265 6374 2073 ing. direct s │ │ │ │ -0006fe50: 756d 2069 6e66 6f72 6d61 7469 6f6e 0a0a um information.. │ │ │ │ -0006fe60: 5761 7973 2074 6f20 7573 6520 7465 6e73 Ways to use tens │ │ │ │ -0006fe70: 6f72 5769 7468 436f 6d70 6f6e 656e 7473 orWithComponents │ │ │ │ -0006fe80: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ -0006fe90: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0006fea0: 3d3d 3d0a 0a20 202a 2022 7465 6e73 6f72 ===.. * "tensor │ │ │ │ -0006feb0: 5769 7468 436f 6d70 6f6e 656e 7473 284d WithComponents(M │ │ │ │ -0006fec0: 6f64 756c 652c 4d6f 6475 6c65 2922 0a0a odule,Module)".. │ │ │ │ -0006fed0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ -0006fee0: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ -0006fef0: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ -0006ff00: 7420 2a6e 6f74 6520 7465 6e73 6f72 5769 t *note tensorWi │ │ │ │ -0006ff10: 7468 436f 6d70 6f6e 656e 7473 3a20 7465 thComponents: te │ │ │ │ -0006ff20: 6e73 6f72 5769 7468 436f 6d70 6f6e 656e nsorWithComponen │ │ │ │ -0006ff30: 7473 2c20 6973 2061 202a 6e6f 7465 206d ts, is a *note m │ │ │ │ -0006ff40: 6574 686f 640a 6675 6e63 7469 6f6e 3a20 ethod.function: │ │ │ │ -0006ff50: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0006ff60: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ +0006fba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a 0a20 *************.. │ │ │ │ +0006fbb0: 202a 2055 7361 6765 3a20 0a20 2020 2020 * Usage: . │ │ │ │ +0006fbc0: 2020 2054 203d 2074 656e 736f 7228 4d2c T = tensor(M, │ │ │ │ +0006fbd0: 4e29 0a20 202a 2049 6e70 7574 733a 0a20 N). * Inputs:. │ │ │ │ +0006fbe0: 2020 2020 202a 204d 2c20 6120 2a6e 6f74 * M, a *not │ │ │ │ +0006fbf0: 6520 6d6f 6475 6c65 3a20 284d 6163 6175 e module: (Macau │ │ │ │ +0006fc00: 6c61 7932 446f 6329 4d6f 6475 6c65 2c2c lay2Doc)Module,, │ │ │ │ +0006fc10: 200a 2020 2020 2020 2a20 4e2c 2061 202a . * N, a * │ │ │ │ +0006fc20: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ +0006fc30: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ +0006fc40: 652c 2c20 0a20 202a 204f 7574 7075 7473 e,, . * Outputs │ │ │ │ +0006fc50: 3a0a 2020 2020 2020 2a20 542c 2061 202a :. * T, a * │ │ │ │ +0006fc60: 6e6f 7465 206d 6f64 756c 653a 2028 4d61 note module: (Ma │ │ │ │ +0006fc70: 6361 756c 6179 3244 6f63 294d 6f64 756c caulay2Doc)Modul │ │ │ │ +0006fc80: 652c 2c20 0a0a 4465 7363 7269 7074 696f e,, ..Descriptio │ │ │ │ +0006fc90: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a49 n.===========..I │ │ │ │ +0006fca0: 6620 4d20 616e 642f 6f72 204e 2061 7265 f M and/or N are │ │ │ │ +0006fcb0: 2064 6972 6563 7420 7375 6d20 6d6f 6475 direct sum modu │ │ │ │ +0006fcc0: 6c65 7320 2869 7344 6972 6563 7453 756d les (isDirectSum │ │ │ │ +0006fcd0: 204d 203d 3d20 7472 7565 2920 7468 656e M == true) then │ │ │ │ +0006fce0: 2054 2069 7320 7468 650a 6469 7265 6374 T is the.direct │ │ │ │ +0006fcf0: 2073 756d 206f 6620 7468 6520 7465 6e73 sum of the tens │ │ │ │ +0006fd00: 6f72 2070 726f 6475 6374 7320 6265 7477 or products betw │ │ │ │ +0006fd10: 6565 6e20 7468 6520 636f 6d70 6f6e 656e een the componen │ │ │ │ +0006fd20: 7473 2e20 5468 6973 2053 484f 554c 4420 ts. This SHOULD │ │ │ │ +0006fd30: 6265 2062 7569 6c74 0a69 6e74 6f20 4d2a be built.into M* │ │ │ │ +0006fd40: 2a4e 2c20 6275 7420 6973 6e27 7420 6173 *N, but isn't as │ │ │ │ +0006fd50: 206f 6620 4d32 2c20 762e 2031 2e37 0a0a of M2, v. 1.7.. │ │ │ │ +0006fd60: 5365 6520 616c 736f 0a3d 3d3d 3d3d 3d3d See also.======= │ │ │ │ +0006fd70: 3d0a 0a20 202a 202a 6e6f 7465 2048 6f6d =.. * *note Hom │ │ │ │ +0006fd80: 5769 7468 436f 6d70 6f6e 656e 7473 3a20 WithComponents: │ │ │ │ +0006fd90: 486f 6d57 6974 6843 6f6d 706f 6e65 6e74 HomWithComponent │ │ │ │ +0006fda0: 732c 202d 2d20 636f 6d70 7574 6573 2048 s, -- computes H │ │ │ │ +0006fdb0: 6f6d 2c20 7072 6573 6572 7669 6e67 0a20 om, preserving. │ │ │ │ +0006fdc0: 2020 2064 6972 6563 7420 7375 6d20 696e direct sum in │ │ │ │ +0006fdd0: 666f 726d 6174 696f 6e0a 2020 2a20 2a6e formation. * *n │ │ │ │ +0006fde0: 6f74 6520 6475 616c 5769 7468 436f 6d70 ote dualWithComp │ │ │ │ +0006fdf0: 6f6e 656e 7473 3a20 6475 616c 5769 7468 onents: dualWith │ │ │ │ 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2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0006ffb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0006ffc0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0006ffd0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0006ffe0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0006fff0: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -00070000: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -00070010: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -00070020: 636b 6167 6573 2f0a 436f 6d70 6c65 7465 ckages/.Complete │ │ │ │ -00070030: 496e 7465 7273 6563 7469 6f6e 5265 736f IntersectionReso │ │ │ │ -00070040: 6c75 7469 6f6e 732e 6d32 3a32 3636 373a lutions.m2:2667: │ │ │ │ -00070050: 302e 0a1f 0a46 696c 653a 2043 6f6d 706c 0....File: Compl │ │ │ │ -00070060: 6574 6549 6e74 6572 7365 6374 696f 6e52 eteIntersectionR │ │ │ │ -00070070: 6573 6f6c 7574 696f 6e73 2e69 6e66 6f2c esolutions.info, │ │ │ │ -00070080: 204e 6f64 653a 2074 6f41 7272 6179 2c20 Node: toArray, │ │ │ │ -00070090: 4e65 7874 3a20 7477 6f4d 6f6e 6f6d 6961 Next: twoMonomia │ │ │ │ -000700a0: 6c73 2c20 5072 6576 3a20 7465 6e73 6f72 ls, Prev: tensor │ │ │ │ -000700b0: 5769 7468 436f 6d70 6f6e 656e 7473 2c20 WithComponents, │ │ │ │ -000700c0: 5570 3a20 546f 700a 0a74 6f41 7272 6179 Up: Top..toArray │ │ │ │ -000700d0: 202d 2d20 6d61 6b65 7320 616e 2061 7272 -- makes an arr │ │ │ │ -000700e0: 6179 2066 726f 6d20 6120 4c69 7374 206f ay from a List o │ │ │ │ -000700f0: 7220 6672 6f6d 2061 2073 696e 676c 6520 r from a single │ │ │ │ -00070100: 696e 7465 6765 720a 2a2a 2a2a 2a2a 2a2a integer.******** │ │ │ │ +0006ffa0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +0006ffb0: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +0006ffc0: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +0006ffd0: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +0006ffe0: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +0006fff0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00070000: 7061 636b 6167 6573 2f0a 436f 6d70 6c65 packages/.Comple │ │ │ │ +00070010: 7465 496e 7465 7273 6563 7469 6f6e 5265 teIntersectionRe │ │ │ │ +00070020: 736f 6c75 7469 6f6e 732e 6d32 3a32 3636 solutions.m2:266 │ │ │ │ +00070030: 373a 302e 0a1f 0a46 696c 653a 2043 6f6d 7:0....File: Com │ │ │ │ +00070040: 706c 6574 6549 6e74 6572 7365 6374 696f pleteIntersectio │ │ │ │ +00070050: 6e52 6573 6f6c 7574 696f 6e73 2e69 6e66 nResolutions.inf │ │ │ │ +00070060: 6f2c 204e 6f64 653a 2074 6f41 7272 6179 o, Node: toArray │ │ │ │ +00070070: 2c20 4e65 7874 3a20 7477 6f4d 6f6e 6f6d , Next: twoMonom │ │ │ │ +00070080: 6961 6c73 2c20 5072 6576 3a20 7465 6e73 ials, Prev: tens │ │ │ │ +00070090: 6f72 5769 7468 436f 6d70 6f6e 656e 7473 orWithComponents │ │ │ │ +000700a0: 2c20 5570 3a20 546f 700a 0a74 6f41 7272 , Up: Top..toArr │ │ │ │ +000700b0: 6179 202d 2d20 6d61 6b65 7320 616e 2061 ay -- makes an a │ │ │ │ +000700c0: 7272 6179 2066 726f 6d20 6120 4c69 7374 rray from a List │ │ │ │ +000700d0: 206f 7220 6672 6f6d 2061 2073 696e 676c or from a singl │ │ │ │ +000700e0: 6520 696e 7465 6765 720a 2a2a 2a2a 2a2a e integer.****** │ │ │ │ +000700f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00070110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070120: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070130: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00070140: 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 6167 ******.. * Usag │ │ │ │ -00070150: 653a 200a 2020 2020 2020 2020 6172 7220 e: . arr │ │ │ │ -00070160: 3d20 746f 4172 7261 7920 4c0a 2020 2020 = toArray L. │ │ │ │ -00070170: 2020 2020 6172 7220 3d20 746f 4172 7261 arr = toArra │ │ │ │ -00070180: 7920 6e0a 2020 2a20 496e 7075 7473 3a0a y n. * Inputs:. │ │ │ │ -00070190: 2020 2020 2020 2a20 4c2c 2061 202a 6e6f * L, a *no │ │ │ │ -000701a0: 7465 206c 6973 743a 2028 4d61 6361 756c te list: (Macaul │ │ │ │ -000701b0: 6179 3244 6f63 294c 6973 742c 2c20 0a20 ay2Doc)List,, . │ │ │ │ -000701c0: 2020 2020 202a 206e 2c20 616e 202a 6e6f * n, an *no │ │ │ │ -000701d0: 7465 2069 6e74 6567 6572 3a20 284d 6163 te integer: (Mac │ │ │ │ -000701e0: 6175 6c61 7932 446f 6329 5a5a 2c2c 200a aulay2Doc)ZZ,, . │ │ │ │ -000701f0: 2020 2a20 4f75 7470 7574 733a 0a20 2020 * Outputs:. │ │ │ │ -00070200: 2020 202a 2061 7272 2c20 616e 202a 6e6f * arr, an *no │ │ │ │ -00070210: 7465 2061 7272 6179 3a20 284d 6163 6175 te array: (Macau │ │ │ │ -00070220: 6c61 7932 446f 6329 4172 7261 792c 2c20 lay2Doc)Array,, │ │ │ │ -00070230: 0a0a 5761 7973 2074 6f20 7573 6520 746f ..Ways to use to │ │ │ │ -00070240: 4172 7261 793a 0a3d 3d3d 3d3d 3d3d 3d3d Array:.========= │ │ │ │ -00070250: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00070260: 2022 746f 4172 7261 7928 4c69 7374 2922 "toArray(List)" │ │ │ │ -00070270: 0a20 202a 2022 746f 4172 7261 7928 5a5a . * "toArray(ZZ │ │ │ │ -00070280: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ -00070290: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ -000702a0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ -000702b0: 626a 6563 7420 2a6e 6f74 6520 746f 4172 bject *note toAr │ │ │ │ -000702c0: 7261 793a 2074 6f41 7272 6179 2c20 6973 ray: toArray, is │ │ │ │ -000702d0: 2061 202a 6e6f 7465 206d 6574 686f 6420 a *note method │ │ │ │ -000702e0: 6675 6e63 7469 6f6e 3a0a 284d 6163 6175 function:.(Macau │ │ │ │ -000702f0: 6c61 7932 446f 6329 4d65 7468 6f64 4675 lay2Doc)MethodFu │ │ │ │ -00070300: 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d nction,...------ │ │ │ │ +00070120: 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 5573 ********.. * Us │ │ │ │ +00070130: 6167 653a 200a 2020 2020 2020 2020 6172 age: . ar │ │ │ │ +00070140: 7220 3d20 746f 4172 7261 7920 4c0a 2020 r = toArray L. │ │ │ │ +00070150: 2020 2020 2020 6172 7220 3d20 746f 4172 arr = toAr │ │ │ │ +00070160: 7261 7920 6e0a 2020 2a20 496e 7075 7473 ray n. * Inputs │ │ │ │ +00070170: 3a0a 2020 2020 2020 2a20 4c2c 2061 202a :. * L, a * │ │ │ │ +00070180: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +00070190: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +000701a0: 0a20 2020 2020 202a 206e 2c20 616e 202a . * n, an * │ │ │ │ +000701b0: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ +000701c0: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ +000701d0: 200a 2020 2a20 4f75 7470 7574 733a 0a20 . * Outputs:. │ │ │ │ +000701e0: 2020 2020 202a 2061 7272 2c20 616e 202a * arr, an * │ │ │ │ +000701f0: 6e6f 7465 2061 7272 6179 3a20 284d 6163 note array: (Mac │ │ │ │ +00070200: 6175 6c61 7932 446f 6329 4172 7261 792c aulay2Doc)Array, │ │ │ │ +00070210: 2c20 0a0a 5761 7973 2074 6f20 7573 6520 , ..Ways to use │ │ │ │ +00070220: 746f 4172 7261 793a 0a3d 3d3d 3d3d 3d3d toArray:.======= │ │ │ │ +00070230: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 =============.. │ │ │ │ +00070240: 202a 2022 746f 4172 7261 7928 4c69 7374 * "toArray(List │ │ │ │ +00070250: 2922 0a20 202a 2022 746f 4172 7261 7928 )". * "toArray( │ │ │ │ +00070260: 5a5a 2922 0a0a 466f 7220 7468 6520 7072 ZZ)"..For the pr │ │ │ │ +00070270: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ +00070280: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ +00070290: 206f 626a 6563 7420 2a6e 6f74 6520 746f object *note to │ │ │ │ +000702a0: 4172 7261 793a 2074 6f41 7272 6179 2c20 Array: toArray, │ │ │ │ +000702b0: 6973 2061 202a 6e6f 7465 206d 6574 686f is a *note metho │ │ │ │ +000702c0: 6420 6675 6e63 7469 6f6e 3a0a 284d 6163 d function:.(Mac │ │ │ │ +000702d0: 6175 6c61 7932 446f 6329 4d65 7468 6f64 aulay2Doc)Method │ │ │ │ +000702e0: 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d 2d2d Function,...---- │ │ │ │ +000702f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070320: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070350: 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 ---------..The s │ │ │ │ -00070360: 6f75 7263 6520 6f66 2074 6869 7320 646f ource of this do │ │ │ │ -00070370: 6375 6d65 6e74 2069 7320 696e 0a2f 6275 cument is in./bu │ │ │ │ -00070380: 696c 642f 7265 7072 6f64 7563 6962 6c65 ild/reproducible │ │ │ │ -00070390: 2d70 6174 682f 6d61 6361 756c 6179 322d -path/macaulay2- │ │ │ │ -000703a0: 312e 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+ds/M2/Ma │ │ │ │ -000703b0: 6361 756c 6179 322f 7061 636b 6167 6573 caulay2/packages │ │ │ │ -000703c0: 2f0a 436f 6d70 6c65 7465 496e 7465 7273 /.CompleteInters │ │ │ │ -000703d0: 6563 7469 6f6e 5265 736f 6c75 7469 6f6e ectionResolution │ │ │ │ -000703e0: 732e 6d32 3a33 3934 333a 302e 0a1f 0a46 s.m2:3943:0....F │ │ │ │ -000703f0: 696c 653a 2043 6f6d 706c 6574 6549 6e74 ile: CompleteInt │ │ │ │ -00070400: 6572 7365 6374 696f 6e52 6573 6f6c 7574 ersectionResolut │ │ │ │ -00070410: 696f 6e73 2e69 6e66 6f2c 204e 6f64 653a ions.info, Node: │ │ │ │ -00070420: 2074 776f 4d6f 6e6f 6d69 616c 732c 2050 twoMonomials, P │ │ │ │ -00070430: 7265 763a 2074 6f41 7272 6179 2c20 5570 rev: toArray, Up │ │ │ │ -00070440: 3a20 546f 700a 0a74 776f 4d6f 6e6f 6d69 : Top..twoMonomi │ │ │ │ -00070450: 616c 7320 2d2d 2074 616c 6c79 2074 6865 als -- tally the │ │ │ │ -00070460: 2073 6571 7565 6e63 6573 206f 6620 4252 sequences of BR │ │ │ │ -00070470: 616e 6b73 2066 6f72 2063 6572 7461 696e anks for certain │ │ │ │ -00070480: 2065 7861 6d70 6c65 730a 2a2a 2a2a 2a2a examples.****** │ │ │ │ +00070330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00070340: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00070350: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00070360: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00070370: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00070380: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +00070390: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +000703a0: 6573 2f0a 436f 6d70 6c65 7465 496e 7465 es/.CompleteInte │ │ │ │ +000703b0: 7273 6563 7469 6f6e 5265 736f 6c75 7469 rsectionResoluti │ │ │ │ +000703c0: 6f6e 732e 6d32 3a33 3934 333a 302e 0a1f ons.m2:3943:0... │ │ │ │ +000703d0: 0a46 696c 653a 2043 6f6d 706c 6574 6549 .File: CompleteI │ │ │ │ +000703e0: 6e74 6572 7365 6374 696f 6e52 6573 6f6c ntersectionResol │ │ │ │ +000703f0: 7574 696f 6e73 2e69 6e66 6f2c 204e 6f64 utions.info, Nod │ │ │ │ +00070400: 653a 2074 776f 4d6f 6e6f 6d69 616c 732c e: twoMonomials, │ │ │ │ +00070410: 2050 7265 763a 2074 6f41 7272 6179 2c20 Prev: toArray, │ │ │ │ +00070420: 5570 3a20 546f 700a 0a74 776f 4d6f 6e6f Up: Top..twoMono │ │ │ │ +00070430: 6d69 616c 7320 2d2d 2074 616c 6c79 2074 mials -- tally t │ │ │ │ +00070440: 6865 2073 6571 7565 6e63 6573 206f 6620 he sequences of │ │ │ │ +00070450: 4252 616e 6b73 2066 6f72 2063 6572 7461 BRanks for certa │ │ │ │ +00070460: 696e 2065 7861 6d70 6c65 730a 2a2a 2a2a in examples.**** │ │ │ │ +00070470: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00070480: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00070490: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000704a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000704b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000704c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ -000704d0: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -000704e0: 2020 5420 3d20 5477 6f4d 6f6e 6f6d 6961 T = TwoMonomia │ │ │ │ -000704f0: 6c73 2863 2c64 290a 2020 2a20 496e 7075 ls(c,d). * Inpu │ │ │ │ -00070500: 7473 3a0a 2020 2020 2020 2a20 632c 2061 ts:. * c, a │ │ │ │ -00070510: 6e20 2a6e 6f74 6520 696e 7465 6765 723a n *note integer: │ │ │ │ -00070520: 2028 4d61 6361 756c 6179 3244 6f63 295a (Macaulay2Doc)Z │ │ │ │ -00070530: 5a2c 2c20 636f 6469 6d65 6e73 696f 6e20 Z,, codimension │ │ │ │ -00070540: 696e 2077 6869 6368 2074 6f20 776f 726b in which to work │ │ │ │ -00070550: 0a20 2020 2020 202a 2064 2c20 616e 202a . * d, an * │ │ │ │ -00070560: 6e6f 7465 2069 6e74 6567 6572 3a20 284d note integer: (M │ │ │ │ -00070570: 6163 6175 6c61 7932 446f 6329 5a5a 2c2c acaulay2Doc)ZZ,, │ │ │ │ -00070580: 2064 6567 7265 6520 6f66 2074 6865 206d degree of the m │ │ │ │ -00070590: 6f6e 6f6d 6961 6c73 2074 6f20 7461 6b65 onomials to take │ │ │ │ -000705a0: 0a20 202a 202a 6e6f 7465 204f 7074 696f . * *note Optio │ │ │ │ -000705b0: 6e61 6c20 696e 7075 7473 3a20 284d 6163 nal inputs: (Mac │ │ │ │ -000705c0: 6175 6c61 7932 446f 6329 7573 696e 6720 aulay2Doc)using │ │ │ │ -000705d0: 6675 6e63 7469 6f6e 7320 7769 7468 206f functions with o │ │ │ │ -000705e0: 7074 696f 6e61 6c20 696e 7075 7473 2c3a ptional inputs,: │ │ │ │ -000705f0: 0a20 2020 2020 202a 204f 7074 696d 6973 . * Optimis │ │ │ │ -00070600: 6d20 3d3e 202e 2e2e 2c20 6465 6661 756c m => ..., defaul │ │ │ │ -00070610: 7420 7661 6c75 6520 300a 2020 2a20 4f75 t value 0. * Ou │ │ │ │ -00070620: 7470 7574 733a 0a20 2020 2020 202a 2054 tputs:. * T │ │ │ │ -00070630: 2c20 6120 2a6e 6f74 6520 7461 6c6c 793a , a *note tally: │ │ │ │ -00070640: 2028 4d61 6361 756c 6179 3244 6f63 2954 (Macaulay2Doc)T │ │ │ │ -00070650: 616c 6c79 2c2c 200a 0a44 6573 6372 6970 ally,, ..Descrip │ │ │ │ -00070660: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00070670: 0a0a 7461 6c6c 6965 7320 7468 6520 7365 ..tallies the se │ │ │ │ -00070680: 7175 656e 6365 7320 6f66 2042 2d72 616e quences of B-ran │ │ │ │ -00070690: 6b73 2074 6861 7420 6f63 6375 7220 666f ks that occur fo │ │ │ │ -000706a0: 7220 6964 6561 6c73 2067 656e 6572 6174 r ideals generat │ │ │ │ -000706b0: 6564 2062 7920 7061 6972 7320 6f66 0a6d ed by pairs of.m │ │ │ │ -000706c0: 6f6e 6f6d 6961 6c73 2069 6e20 5220 3d20 onomials in R = │ │ │ │ -000706d0: 532f 2864 2d74 6820 706f 7765 7273 206f S/(d-th powers o │ │ │ │ -000706e0: 6620 7468 6520 7661 7269 6162 6c65 7329 f the variables) │ │ │ │ -000706f0: 2c20 7769 7468 2066 756c 6c20 636f 6d70 , with full comp │ │ │ │ -00070700: 6c65 7869 7479 2028 3d63 293b 0a74 6861 lexity (=c);.tha │ │ │ │ -00070710: 7420 6973 2c20 666f 7220 616e 2061 7070 t is, for an app │ │ │ │ -00070720: 726f 7072 6961 7465 2073 797a 7967 7920 ropriate syzygy │ │ │ │ -00070730: 4d20 6f66 204d 3020 3d20 522f 286d 312c M of M0 = R/(m1, │ │ │ │ -00070740: 206d 3229 2077 6865 7265 206d 3120 616e m2) where m1 an │ │ │ │ -00070750: 6420 6d32 2061 7265 0a6d 6f6e 6f6d 6961 d m2 are.monomia │ │ │ │ -00070760: 6c73 206f 6620 7468 6520 7361 6d65 2064 ls of the same d │ │ │ │ -00070770: 6567 7265 652e 0a0a 2b2d 2d2d 2d2d 2d2d egree...+------- │ │ │ │ +000704a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ +000704b0: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +000704c0: 2020 2020 5420 3d20 5477 6f4d 6f6e 6f6d T = TwoMonom │ │ │ │ +000704d0: 6961 6c73 2863 2c64 290a 2020 2a20 496e ials(c,d). * In │ │ │ │ +000704e0: 7075 7473 3a0a 2020 2020 2020 2a20 632c puts:. * c, │ │ │ │ +000704f0: 2061 6e20 2a6e 6f74 6520 696e 7465 6765 an *note intege │ │ │ │ +00070500: 723a 2028 4d61 6361 756c 6179 3244 6f63 r: (Macaulay2Doc │ │ │ │ +00070510: 295a 5a2c 2c20 636f 6469 6d65 6e73 696f )ZZ,, codimensio │ │ │ │ +00070520: 6e20 696e 2077 6869 6368 2074 6f20 776f n in which to wo │ │ │ │ +00070530: 726b 0a20 2020 2020 202a 2064 2c20 616e rk. * d, an │ │ │ │ +00070540: 202a 6e6f 7465 2069 6e74 6567 6572 3a20 *note integer: │ │ │ │ +00070550: 284d 6163 6175 6c61 7932 446f 6329 5a5a (Macaulay2Doc)ZZ │ │ │ │ +00070560: 2c2c 2064 6567 7265 6520 6f66 2074 6865 ,, degree of the │ │ │ │ +00070570: 206d 6f6e 6f6d 6961 6c73 2074 6f20 7461 monomials to ta │ │ │ │ +00070580: 6b65 0a20 202a 202a 6e6f 7465 204f 7074 ke. * *note Opt │ │ │ │ +00070590: 696f 6e61 6c20 696e 7075 7473 3a20 284d ional inputs: (M │ │ │ │ +000705a0: 6163 6175 6c61 7932 446f 6329 7573 696e acaulay2Doc)usin │ │ │ │ +000705b0: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ +000705c0: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ +000705d0: 2c3a 0a20 2020 2020 202a 204f 7074 696d ,:. * Optim │ │ │ │ +000705e0: 6973 6d20 3d3e 202e 2e2e 2c20 6465 6661 ism => ..., defa │ │ │ │ +000705f0: 756c 7420 7661 6c75 6520 300a 2020 2a20 ult value 0. * │ │ │ │ +00070600: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00070610: 2054 2c20 6120 2a6e 6f74 6520 7461 6c6c T, a *note tall │ │ │ │ +00070620: 793a 2028 4d61 6361 756c 6179 3244 6f63 y: (Macaulay2Doc │ │ │ │ +00070630: 2954 616c 6c79 2c2c 200a 0a44 6573 6372 )Tally,, ..Descr │ │ │ │ +00070640: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ +00070650: 3d3d 0a0a 7461 6c6c 6965 7320 7468 6520 ==..tallies the │ │ │ │ +00070660: 7365 7175 656e 6365 7320 6f66 2042 2d72 sequences of B-r │ │ │ │ +00070670: 616e 6b73 2074 6861 7420 6f63 6375 7220 anks that occur │ │ │ │ +00070680: 666f 7220 6964 6561 6c73 2067 656e 6572 for ideals gener │ │ │ │ +00070690: 6174 6564 2062 7920 7061 6972 7320 6f66 ated by pairs of │ │ │ │ +000706a0: 0a6d 6f6e 6f6d 6961 6c73 2069 6e20 5220 .monomials in R │ │ │ │ +000706b0: 3d20 532f 2864 2d74 6820 706f 7765 7273 = S/(d-th powers │ │ │ │ +000706c0: 206f 6620 7468 6520 7661 7269 6162 6c65 of the variable │ │ │ │ +000706d0: 7329 2c20 7769 7468 2066 756c 6c20 636f s), with full co │ │ │ │ +000706e0: 6d70 6c65 7869 7479 2028 3d63 293b 0a74 mplexity (=c);.t │ │ │ │ +000706f0: 6861 7420 6973 2c20 666f 7220 616e 2061 hat is, for an a │ │ │ │ +00070700: 7070 726f 7072 6961 7465 2073 797a 7967 ppropriate syzyg │ │ │ │ +00070710: 7920 4d20 6f66 204d 3020 3d20 522f 286d y M of M0 = R/(m │ │ │ │ +00070720: 312c 206d 3229 2077 6865 7265 206d 3120 1, m2) where m1 │ │ │ │ +00070730: 616e 6420 6d32 2061 7265 0a6d 6f6e 6f6d and m2 are.monom │ │ │ │ +00070740: 6961 6c73 206f 6620 7468 6520 7361 6d65 ials of the same │ │ │ │ +00070750: 2064 6567 7265 652e 0a0a 2b2d 2d2d 2d2d degree...+----- │ │ │ │ +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 2b0a --------------+. │ │ │ │ -000707b0: 7c69 3120 3a20 7365 7452 616e 646f 6d53 |i1 : setRandomS │ │ │ │ -000707c0: 6565 6420 3020 2020 2020 2020 2020 2020 eed 0 │ │ │ │ -000707d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000707e0: 2020 2020 2020 7c0a 7c20 2d2d 2073 6574 |.| -- set │ │ │ │ -000707f0: 7469 6e67 2072 616e 646f 6d20 7365 6564 ting random seed │ │ │ │ -00070800: 2074 6f20 3020 2020 2020 2020 2020 2020 to 0 │ │ │ │ -00070810: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070820: 7c20 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: 2020 2020 2020 7c0a 7c6f 3120 3d20 3020 |.|o1 = 0 │ │ │ │ +00070790: 2b0a 7c69 3120 3a20 7365 7452 616e 646f +.|i1 : setRando │ │ │ │ +000707a0: 6d53 6565 6420 3020 2020 2020 2020 2020 mSeed 0 │ │ │ │ +000707b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000707c0: 2020 2020 2020 2020 7c0a 7c20 2d2d 2073 |.| -- s │ │ │ │ +000707d0: 6574 7469 6e67 2072 616e 646f 6d20 7365 etting random se │ │ │ │ +000707e0: 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6420 302e 3932 |.| -- used 0.92 │ │ │ │ +000708f0: 3436 3737 7320 2863 7075 293b 2030 2e36 4677s (cpu); 0.6 │ │ │ │ +00070900: 3238 3533 3473 2028 7468 7265 6164 293b 28534s (thread); │ │ │ │ +00070910: 2030 7320 2867 6329 7c0a 7c32 2020 2020 0s (gc)|.|2 │ │ │ │ +00070920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070960: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070970: 7c54 616c 6c79 7b7b 7b31 2c20 317d 7d20 |Tally{{{1, 1}} │ │ │ │ -00070980: 3d3e 2032 2020 2020 2020 2020 7d20 2020 => 2 } │ │ │ │ -00070990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000709a0: 2020 2020 2020 7c0a 7c20 2020 2020 207b |.| { │ │ │ │ -000709b0: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ -000709c0: 3d3e 2034 2020 2020 2020 2020 2020 2020 => 4 │ │ │ │ -000709d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000709e0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ -000709f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a10: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -00070a20: 6420 302e 3535 3236 3739 7320 2863 7075 d 0.552679s (cpu │ │ │ │ -00070a30: 293b 2030 2e33 3935 3239 3373 2028 7468 ); 0.395293s (th │ │ │ │ -00070a40: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ -00070a50: 7c33 2020 2020 2020 2020 2020 2020 2020 |3 │ │ │ │ -00070a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070a80: 2020 2020 2020 7c0a 7c54 616c 6c79 7b7b |.|Tally{{ │ │ │ │ -00070a90: 7b32 2c20 327d 2c20 7b31 2c20 327d 7d20 {2, 2}, {1, 2}} │ │ │ │ -00070aa0: 3d3e 2032 7d20 2020 2020 2020 2020 2020 => 2} │ │ │ │ -00070ab0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00070ac0: 7c20 2020 2020 207b 7b33 2c20 337d 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7320 2867 6329 thread); 0s (gc) │ │ │ │ +00070a30: 7c0a 7c33 2020 2020 2020 2020 2020 2020 |.|3 │ │ │ │ +00070a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070a60: 2020 2020 2020 2020 7c0a 7c54 616c 6c79 |.|Tally │ │ │ │ +00070a70: 7b7b 7b32 2c20 327d 2c20 7b31 2c20 327d {{{2, 2}, {1, 2} │ │ │ │ +00070a80: 7d20 3d3e 2032 7d20 2020 2020 2020 2020 } => 2} │ │ │ │ +00070a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070aa0: 7c0a 7c20 2020 2020 207b 7b33 2c20 337d |.| {{3, 3} │ │ │ │ +00070ab0: 2c20 7b32 2c20 337d 7d20 3d3e 2031 2020 , {2, 3}} => 1 │ │ │ │ +00070ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00070ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070af0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +00070af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00070b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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7c0a |. │ │ │ │ -00070ba0: 7c54 616c 6c79 7b7b 7b32 2c20 327d 2c20 |Tally{{{2, 2}, │ │ │ │ -00070bb0: 7b31 2c20 327d 7d20 3d3e 2031 7d20 2020 {1, 2}} => 1} │ │ │ │ -00070bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00070bd0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00070b80: 7c0a 7c54 616c 6c79 7b7b 7b32 2c20 327d |.|Tally{{{2, 2} │ │ │ │ +00070b90: 2c20 7b31 2c20 327d 7d20 3d3e 2031 7d20 , {1, 2}} => 1} │ │ │ │ +00070ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00070bb0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ +00070bc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00070bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00070be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070bf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00070c00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00070c10: 0a53 6565 2061 6c73 6f0a 3d3d 3d3d 3d3d .See also.====== │ │ │ │ -00070c20: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 7477 ==.. * *note tw │ │ │ │ -00070c30: 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 6f4d oMonomials: twoM │ │ │ │ -00070c40: 6f6e 6f6d 6961 6c73 2c20 2d2d 2074 616c onomials, -- tal │ │ │ │ -00070c50: 6c79 2074 6865 2073 6571 7565 6e63 6573 ly the sequences │ │ │ │ -00070c60: 206f 6620 4252 616e 6b73 2066 6f72 0a20 of BRanks for. │ │ │ │ -00070c70: 2020 2063 6572 7461 696e 2065 7861 6d70 certain examp │ │ │ │ -00070c80: 6c65 730a 0a57 6179 7320 746f 2075 7365 les..Ways to use │ │ │ │ -00070c90: 2074 776f 4d6f 6e6f 6d69 616c 733a 0a3d twoMonomials:.= │ │ │ │ -00070ca0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00070cb0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2274 ========.. * "t │ │ │ │ -00070cc0: 776f 4d6f 6e6f 6d69 616c 7328 5a5a 2c5a woMonomials(ZZ,Z │ │ │ │ -00070cd0: 5a29 220a 0a46 6f72 2074 6865 2070 726f Z)"..For the pro │ │ │ │ -00070ce0: 6772 616d 6d65 720a 3d3d 3d3d 3d3d 3d3d grammer.======== │ │ │ │ -00070cf0: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6520 ==========..The │ │ │ │ -00070d00: 6f62 6a65 6374 202a 6e6f 7465 2074 776f object *note two │ │ │ │ -00070d10: 4d6f 6e6f 6d69 616c 733a 2074 776f 4d6f Monomials: twoMo │ │ │ │ -00070d20: 6e6f 6d69 616c 732c 2069 7320 6120 2a6e nomials, is a *n │ │ │ │ -00070d30: 6f74 6520 6d65 7468 6f64 2066 756e 6374 ote method funct │ │ │ │ -00070d40: 696f 6e20 7769 7468 0a6f 7074 696f 6e73 ion with.options │ │ │ │ -00070d50: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ -00070d60: 4d65 7468 6f64 4675 6e63 7469 6f6e 5769 MethodFunctionWi │ │ │ │ -00070d70: 7468 4f70 7469 6f6e 732c 2e0a 0a2d 2d2d thOptions,...--- │ │ │ │ +00070bf0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d +..See also.==== │ │ │ │ +00070c00: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00070c10: 7477 6f4d 6f6e 6f6d 6961 6c73 3a20 7477 twoMonomials: tw │ │ │ │ +00070c20: 6f4d 6f6e 6f6d 6961 6c73 2c20 2d2d 2074 oMonomials, -- t │ │ │ │ +00070c30: 616c 6c79 2074 6865 2073 6571 7565 6e63 ally the sequenc │ │ │ │ +00070c40: 6573 206f 6620 4252 616e 6b73 2066 6f72 es of BRanks for │ │ │ │ +00070c50: 0a20 2020 2063 6572 7461 696e 2065 7861 . certain exa │ │ │ │ +00070c60: 6d70 6c65 730a 0a57 6179 7320 746f 2075 mples..Ways to u │ │ │ │ +00070c70: 7365 2074 776f 4d6f 6e6f 6d69 616c 733a se twoMonomials: │ │ │ │ +00070c80: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ +00070c90: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00070ca0: 2274 776f 4d6f 6e6f 6d69 616c 7328 5a5a "twoMonomials(ZZ │ │ │ │ +00070cb0: 2c5a 5a29 220a 0a46 6f72 2074 6865 2070 ,ZZ)"..For the p │ │ │ │ +00070cc0: 726f 6772 616d 6d65 720a 3d3d 3d3d 3d3d rogrammer.====== │ │ │ │ +00070cd0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 5468 ============..Th │ │ │ │ +00070ce0: 6520 6f62 6a65 6374 202a 6e6f 7465 2074 e object *note t │ │ │ │ +00070cf0: 776f 4d6f 6e6f 6d69 616c 733a 2074 776f woMonomials: two │ │ │ │ +00070d00: 4d6f 6e6f 6d69 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ization.365396.N │ │ │ │ +00071300: 6f64 653a 206d 6642 6f75 6e64 7f33 3734 ode: mfBound.374 │ │ │ │ +00071310: 3234 380a 4e6f 6465 3a20 6d6f 6475 6c65 248.Node: module │ │ │ │ +00071320: 4173 4578 747f 3337 3631 3430 0a4e 6f64 AsExt.376140.Nod │ │ │ │ +00071330: 653a 206e 6577 4578 747f 3338 3233 3838 e: newExt.382388 │ │ │ │ +00071340: 0a4e 6f64 653a 206f 6464 4578 744d 6f64 .Node: oddExtMod │ │ │ │ +00071350: 756c 657f 3431 3039 3734 0a4e 6f64 653a ule.410974.Node: │ │ │ │ +00071360: 204f 7074 696d 6973 6d7f 3431 3732 3634 Optimism.417264 │ │ │ │ +00071370: 0a4e 6f64 653a 204f 7574 5269 6e67 7f34 .Node: OutRing.4 │ │ │ │ +00071380: 3138 3739 300a 4e6f 6465 3a20 7073 694d 18790.Node: psiM │ │ │ │ +00071390: 6170 737f 3432 3031 3337 0a4e 6f64 653a aps.420137.Node: │ │ │ │ +000713a0: 2072 6567 756c 6172 6974 7953 6571 7565 regularitySeque │ │ │ │ +000713b0: 6e63 657f 3432 3136 3737 0a4e 6f64 653a nce.421677.Node: │ │ │ │ +000713c0: 2053 327f 3432 3532 3332 0a4e 6f64 653a S2.425232.Node: │ │ │ │ +000713d0: 2053 6861 6d61 7368 7f34 3336 3331 350a Shamash.436315. │ │ │ │ +000713e0: 4e6f 6465 3a20 7370 6c69 7474 696e 6773 Node: splittings │ │ │ │ +000713f0: 7f34 3434 3035 340a 4e6f 6465 3a20 7374 .444054.Node: st │ │ │ │ +00071400: 6162 6c65 486f 6d7f 3434 3932 3733 0a4e ableHom.449273.N │ │ │ │ +00071410: 6f64 653a 2073 756d 5477 6f4d 6f6e 6f6d ode: sumTwoMonom │ │ │ │ +00071420: 6961 6c73 7f34 3530 3531 300a 4e6f 6465 ials.450510.Node │ │ │ │ +00071430: 3a20 5461 7465 5265 736f 6c75 7469 6f6e : TateResolution │ │ │ │ +00071440: 7f34 3532 3938 310a 4e6f 6465 3a20 7465 .452981.Node: te │ │ │ │ +00071450: 6e73 6f72 5769 7468 436f 6d70 6f6e 656e nsorWithComponen │ │ │ │ +00071460: 7473 7f34 3537 3335 380a 4e6f 6465 3a20 ts.457358.Node: │ │ │ │ +00071470: 746f 4172 7261 797f 3435 3838 3035 0a4e toArray.458805.N │ │ │ │ +00071480: 6f64 653a 2074 776f 4d6f 6e6f 6d69 616c ode: twoMonomial │ │ │ │ +00071490: 737f 3435 3937 3237 0a1f 0a45 6e64 2054 s.459727...End T │ │ │ │ +000714a0: 6167 2054 6162 6c65 0a ag Table. │ │ ├── ./usr/share/info/ConnectionMatrices.info.gz │ │ │ ├── ConnectionMatrices.info │ │ │ │ @@ -2993,30 +2993,30 @@ │ │ │ │ 0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000bb20: 3920 3a20 656c 6170 7365 6454 696d 6520 9 : elapsedTime │ │ │ │ 0000bb30: 4120 3d20 7066 6166 6669 616e 5379 7374 A = pfaffianSyst │ │ │ │ 0000bb40: 656d 2049 3b20 2020 2020 2020 2020 2020 em I; │ │ │ │ 0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0000bb70: 2d2d 2034 2e33 3936 3431 7320 656c 6170 -- 4.39641s elap │ │ │ │ +0000bb70: 2d2d 2032 2e35 3139 3639 7320 656c 6170 -- 2.51969s elap │ │ │ │ 0000bb80: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0000bb90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000bbc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bbd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bbe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bbf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bc00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ 0000bc10: 3130 203a 2065 6c61 7073 6564 5469 6d65 10 : elapsedTime │ │ │ │ 0000bc20: 2061 7373 6572 7420 6973 496e 7465 6772 assert isIntegr │ │ │ │ 0000bc30: 6162 6c65 2041 2020 2020 2020 2020 2020 able A │ │ │ │ 0000bc40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc50: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -0000bc60: 2d2d 2035 2e34 3638 3832 7320 656c 6170 -- 5.46882s elap │ │ │ │ +0000bc60: 2d2d 2033 2e34 3430 3639 7320 656c 6170 -- 3.44069s elap │ │ │ │ 0000bc70: 7365 6420 2020 2020 2020 2020 2020 2020 sed │ │ │ │ 0000bc80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bca0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0000bcb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bcc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bcd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5137,15 +5137,15 @@ │ │ │ │ 00014100: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014110: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00014120: 7c69 3134 203a 2065 6c61 7073 6564 5469 |i14 : elapsedTi │ │ │ │ 00014130: 6d65 2067 203d 2067 6175 6765 4d61 7472 me g = gaugeMatr │ │ │ │ 00014140: 6978 2849 2c20 4229 3b20 2020 2020 2020 ix(I, B); │ │ │ │ 00014150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014160: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014170: 7c20 2d2d 202e 3630 3639 3734 7320 656c | -- .606974s el │ │ │ │ +00014170: 7c20 2d2d 202e 3434 3532 3834 7320 656c | -- .445284s el │ │ │ │ 00014180: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00014190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141b0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000141c0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000141d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000141e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -5167,30 +5167,30 @@ │ │ │ │ 000142e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000142f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 00014300: 7c69 3135 203a 2065 6c61 7073 6564 5469 |i15 : elapsedTi │ │ │ │ 00014310: 6d65 2041 3120 3d20 6761 7567 6554 7261 me A1 = gaugeTra │ │ │ │ 00014320: 6e73 666f 726d 2867 2c20 4129 3b20 2020 nsform(g, A); │ │ │ │ 00014330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014340: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014350: 7c20 2d2d 2031 2e33 3030 3333 7320 656c | -- 1.30033s el │ │ │ │ -00014360: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ +00014350: 7c20 2d2d 202e 3937 3332 3573 2065 6c61 | -- .97325s ela │ │ │ │ +00014360: 7073 6564 2020 2020 2020 2020 2020 2020 psed │ │ │ │ 00014370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014390: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 000143a0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000143b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000143e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ 000143f0: 7c69 3136 203a 2065 6c61 7073 6564 5469 |i16 : elapsedTi │ │ │ │ 00014400: 6d65 2061 7373 6572 7420 6973 496e 7465 me assert isInte │ │ │ │ 00014410: 6772 6162 6c65 2041 3120 2020 2020 2020 grable A1 │ │ │ │ 00014420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014430: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00014440: 7c20 2d2d 202e 3835 3739 3433 7320 656c | -- .857943s el │ │ │ │ +00014440: 7c20 2d2d 202e 3731 3233 3037 7320 656c | -- .712307s el │ │ │ │ 00014450: 6170 7365 6420 2020 2020 2020 2020 2020 apsed │ │ │ │ 00014460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014480: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ 00014490: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 000144a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000144b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -5608,31 +5608,31 @@ │ │ │ │ 00015e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015e80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015e90: 2d2d 2d2b 0a7c 6931 3920 3a20 656c 6170 ---+.|i19 : elap │ │ │ │ 00015ea0: 7365 6454 696d 6520 4132 203d 2067 6175 sedTime A2 = gau │ │ │ │ 00015eb0: 6765 5472 616e 7366 6f72 6d28 6368 616e geTransform(chan │ │ │ │ 00015ec0: 6765 4570 732c 2041 3129 3b20 2020 2020 geEps, A1); │ │ │ │ 00015ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015ee0: 2020 207c 0a7c 202d 2d20 2e33 3933 3538 |.| -- .39358 │ │ │ │ -00015ef0: 3773 2065 6c61 7073 6564 2020 2020 2020 7s elapsed │ │ │ │ +00015ee0: 2020 207c 0a7c 202d 2d20 2e32 3932 3131 |.| -- .29211 │ │ │ │ +00015ef0: 3173 2065 6c61 7073 6564 2020 2020 2020 1s elapsed │ │ │ │ 00015f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015f10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015f30: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00015f40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00015f50: 2d2d 2d2d 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 2d2b 0a7c 6932 3020 3a20 656c 6170 ---+.|i20 : elap │ │ │ │ 00015f90: 7365 6454 696d 6520 6173 7365 7274 2069 sedTime assert i │ │ │ │ 00015fa0: 7349 6e74 6567 7261 626c 6520 4132 2020 sIntegrable A2 │ │ │ │ 00015fb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015fd0: 2020 207c 0a7c 202d 2d20 312e 3036 3132 |.| -- 1.0612 │ │ │ │ -00015fe0: 3473 2065 6c61 7073 6564 2020 2020 2020 4s elapsed │ │ │ │ +00015fd0: 2020 207c 0a7c 202d 2d20 2e36 3034 3139 |.| -- .60419 │ │ │ │ +00015fe0: 3573 2065 6c61 7073 6564 2020 2020 2020 5s elapsed │ │ │ │ 00015ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00016020: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00016030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00016050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ @@ -6018,30 +6018,30 @@ │ │ │ │ 00017810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6937 -----------+.|i7 │ │ │ │ 00017830: 203a 2065 6c61 7073 6564 5469 6d65 2041 : elapsedTime A │ │ │ │ 00017840: 203d 2070 6661 6666 6961 6e53 7973 7465 = pfaffianSyste │ │ │ │ 00017850: 6d20 493b 2020 2020 2020 2020 2020 2020 m I; │ │ │ │ 00017860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017870: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00017880: 2d20 2e33 3239 3332 7320 656c 6170 7365 - .32932s elapse │ │ │ │ -00017890: 6420 2020 2020 2020 2020 2020 2020 2020 d │ │ │ │ +00017880: 2d20 2e31 3934 3773 2065 6c61 7073 6564 - .1947s elapsed │ │ │ │ +00017890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000178a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000178b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000178c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000178d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000178e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000178f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00017910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6938 -----------+.|i8 │ │ │ │ 00017920: 203a 2065 6c61 7073 6564 5469 6d65 2061 : elapsedTime a │ │ │ │ 00017930: 7373 6572 7420 6973 496e 7465 6772 6162 ssert isIntegrab │ │ │ │ 00017940: 6c65 2041 2020 2020 2020 2020 2020 2020 le A │ │ │ │ 00017950: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00017960: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00017970: 2d20 2e31 3638 3736 3873 2065 6c61 7073 - .168768s elaps │ │ │ │ +00017970: 2d20 2e31 3530 3536 3973 2065 6c61 7073 - .150569s elaps │ │ │ │ 00017980: 6564 2020 2020 2020 2020 2020 2020 2020 ed │ │ │ │ 00017990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000179a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000179b0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000179c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000179d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000179e0: 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 3035 3538 3539 3373 2028 ed 0.00558593s ( │ │ │ │ -000009a0: 6370 7529 3b20 302e 3030 3535 3831 3132 cpu); 0.00558112 │ │ │ │ +00000990: 6564 2030 2e30 3034 3539 3639 3973 2028 ed 0.00459699s ( │ │ │ │ +000009a0: 6370 7529 3b20 302e 3030 3435 3935 3535 cpu); 0.00459555 │ │ │ │ 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 2e30 3539 3136 3433 7320 2863 ed 0.0591643s (c │ │ │ │ -00001490: 7075 293b 2030 2e30 3539 3137 3432 7320 pu); 0.0591742s │ │ │ │ +00001480: 6564 2030 2e30 3436 3635 3837 7320 2863 ed 0.0466587s (c │ │ │ │ +00001490: 7075 293b 2030 2e30 3436 3636 3238 7320 pu); 0.0466628s │ │ │ │ 000014a0: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ 000014b0: 2920 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 │ │ │ │ @@ -387,16 +387,16 @@ │ │ │ │ 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 3335 3431 3336 7320 2863 ed 0.0354136s (c │ │ │ │ -000018a0: 7075 293b 2030 2e30 3335 3432 3033 7320 pu); 0.0354203s │ │ │ │ +00001890: 6564 2030 2e30 3239 3833 3639 7320 2863 ed 0.0298369s (c │ │ │ │ +000018a0: 7075 293b 2030 2e30 3239 3833 3535 7320 pu); 0.0298355s │ │ │ │ 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 │ │ │ │ @@ -412,17 +412,17 @@ │ │ │ │ 000019b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000019c0: 2d2d 2d2d 2d2d 2d2b 0a7c 6935 203a 2074 -------+.|i5 : t │ │ │ │ 000019d0: 696d 6520 7072 6f6a 6563 7469 7665 4465 ime projectiveDe │ │ │ │ 000019e0: 6772 6565 7320 7068 6920 2020 2020 2020 grees phi │ │ │ │ 000019f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a10: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001a20: 6564 2030 2e36 3833 3933 3873 2028 6370 ed 0.683938s (cp │ │ │ │ -00001a30: 7529 3b20 302e 3533 3731 3236 7320 2874 u); 0.537126s (t │ │ │ │ -00001a40: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ +00001a20: 6564 2030 2e34 3634 3436 3373 2028 6370 ed 0.464463s (cp │ │ │ │ +00001a30: 7529 3b20 302e 3339 3233 7320 2874 6872 u); 0.3923s (thr │ │ │ │ +00001a40: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ 00001a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a60: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ab0: 2020 2020 2020 207c 0a7c 6f35 203d 207b |.|o5 = { │ │ │ │ @@ -447,18 +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 2e32 3233 3032 3373 2028 6370 ed 0.223023s (cp │ │ │ │ -00001c60: 7529 3b20 302e 3133 3734 3333 7320 2874 u); 0.137433s (t │ │ │ │ -00001c70: 6872 6561 6429 3b20 3073 2028 6763 2920 hread); 0s (gc) │ │ │ │ -00001c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00001c50: 6564 2030 2e30 3539 3031 3634 7320 2863 ed 0.0590164s (c │ │ │ │ +00001c60: 7075 293b 2030 2e30 3539 3032 3335 7320 pu); 0.0590235s │ │ │ │ +00001c70: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ +00001c80: 2920 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} │ │ │ │ @@ -482,15 +482,15 @@ │ │ │ │ 00001e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00001e20: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2074 -------+.|i7 : t │ │ │ │ 00001e30: 696d 6520 7068 6920 3d20 746f 4d61 7028 ime phi = toMap( │ │ │ │ 00001e40: 7068 6920 2020 2020 2020 2020 2020 2020 phi │ │ │ │ 00001e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001e70: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00001e80: 6564 2030 2e30 3033 3534 3130 3973 2028 ed 0.00354109s ( │ │ │ │ +00001e80: 6564 2030 2e30 3032 3138 3732 3673 2028 ed 0.00218726s ( │ │ │ │ 00001e90: 6370 7520 2020 2020 2020 2020 2020 2020 cpu │ │ │ │ 00001ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ec0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00001ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00001ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -567,15 +567,15 @@ │ │ │ │ 00002360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00002370: 2d2d 2d2d 2d2d 2d7c 0a7c 2c44 6f6d 696e -------|.|,Domin │ │ │ │ 00002380: 616e 743d 3e4a 2920 2020 2020 2020 2020 ant=>J) │ │ │ │ 00002390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000023c0: 2020 2020 2020 207c 0a7c 293b 2030 2e30 |.|); 0.0 │ │ │ │ -000023d0: 3033 3534 3233 3573 2028 7468 7265 6164 0354235s (thread │ │ │ │ +000023d0: 3032 3139 3036 3173 2028 7468 7265 6164 0219061s (thread │ │ │ │ 000023e0: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ 000023f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002410: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00002420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00002440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -832,16 +832,16 @@ │ │ │ │ 000033f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00003400: 2d2d 2d2d 2d2d 2d2b 0a7c 6938 203a 2074 -------+.|i8 : t │ │ │ │ 00003410: 696d 6520 7073 6920 3d20 696e 7665 7273 ime psi = invers │ │ │ │ 00003420: 654d 6170 2070 6869 2020 2020 2020 2020 eMap phi │ │ │ │ 00003430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00003450: 2020 2020 2020 207c 0a7c 202d 2d20 7573 |.| -- us │ │ │ │ -00003460: 6564 2030 2e37 3433 3138 3373 2028 6370 ed 0.743183s (cp │ │ │ │ -00003470: 7529 3b20 302e 3636 3231 3035 7320 2874 u); 0.662105s (t │ │ │ │ +00003460: 6564 2030 2e34 3832 3332 3273 2028 6370 ed 0.482322s (cp │ │ │ │ +00003470: 7529 3b20 302e 3339 3936 3634 7320 2874 u); 0.399664s (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 3136 3435 3673 2028 6370 ed 0.016456s (cp │ │ │ │ -00004640: 7529 3b20 302e 3031 3634 3537 3973 2028 u); 0.0164579s ( │ │ │ │ -00004650: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ -00004660: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00004630: 6564 2030 2e30 3038 3938 3635 3173 2028 ed 0.00898651s ( │ │ │ │ +00004640: 6370 7529 3b20 302e 3030 3839 3836 3936 cpu); 0.00898696 │ │ │ │ +00004650: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00004660: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ 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 2e33 3736 3530 3773 2028 6370 ed 0.376507s (cp │ │ │ │ -000047d0: 7529 3b20 302e 3237 3433 3137 7320 2874 u); 0.274317s (t │ │ │ │ +000047c0: 6564 2030 2e31 3439 3432 3573 2028 6370 ed 0.149425s (cp │ │ │ │ +000047d0: 7529 3b20 302e 3134 3934 3331 7320 2874 u); 0.149431s (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 2e36 3532 3231 7320 2863 7075 ed 5.65221s (cpu │ │ │ │ -00004960: 293b 2034 2e37 3931 3539 7320 2874 6872 ); 4.79159s (thr │ │ │ │ +00004950: 6564 2034 2e39 3231 3532 7320 2863 7075 ed 4.92152s (cpu │ │ │ │ +00004960: 293b 2034 2e34 3439 3132 7320 2874 6872 ); 4.44912s (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: 3234 3034 3973 2028 6370 7529 3b20 302e 24049s (cpu); 0. │ │ │ │ -00004c50: 3030 3232 3431 3536 7320 2874 6872 6561 00224156s (threa │ │ │ │ +00004c40: 3235 3739 3273 2028 6370 7529 3b20 302e 25792s (cpu); 0. │ │ │ │ +00004c50: 3030 3232 3631 3531 7320 2874 6872 6561 00226151s (threa │ │ │ │ 00004c60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 00004c70: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00004c80: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00004c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00004cc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -1493,17 +1493,17 @@ │ │ │ │ 00005d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00005d50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 00005d60: 0a7c 6931 3320 3a20 7469 6d65 2070 6869 .|i13 : time phi │ │ │ │ 00005d70: 203d 2072 6174 696f 6e61 6c4d 6170 2870 = rationalMap(p │ │ │ │ 00005d80: 6869 2c44 6f6d 696e 616e 743d 3e32 2920 hi,Dominant=>2) │ │ │ │ 00005d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005da0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3532 .| -- used 0.052 │ │ │ │ -00005dc0: 3338 3737 7320 2863 7075 293b 2030 2e30 3877s (cpu); 0.0 │ │ │ │ -00005dd0: 3532 3339 3733 7320 2874 6872 6561 6429 523973s (thread) │ │ │ │ +00005db0: 0a7c 202d 2d20 7573 6564 2030 2e30 3534 .| -- used 0.054 │ │ │ │ +00005dc0: 3538 3835 7320 2863 7075 293b 2030 2e30 5885s (cpu); 0.0 │ │ │ │ +00005dd0: 3534 3539 3238 7320 2874 6872 6561 6429 545928s (thread) │ │ │ │ 00005de0: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ 00005df0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 00005e00: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 00005e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00005e40: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -2153,17 +2153,17 @@ │ │ │ │ 00008680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00008690: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 000086a0: 0a7c 6931 3420 3a20 7469 6d65 2070 6869 .|i14 : time phi │ │ │ │ 000086b0: 5e28 2d31 2920 2020 2020 2020 2020 2020 ^(-1) │ │ │ │ 000086c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000086d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000086e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000086f0: 0a7c 202d 2d20 7573 6564 2030 2e35 3836 .| -- used 0.586 │ │ │ │ -00008700: 3234 3173 2028 6370 7529 3b20 302e 3434 241s (cpu); 0.44 │ │ │ │ -00008710: 3634 3031 7320 2874 6872 6561 6429 3b20 6401s (thread); │ │ │ │ +000086f0: 0a7c 202d 2d20 7573 6564 2030 2e34 3938 .| -- used 0.498 │ │ │ │ +00008700: 3936 3173 2028 6370 7529 3b20 302e 3432 961s (cpu); 0.42 │ │ │ │ +00008710: 3338 3537 7320 2874 6872 6561 6429 3b20 3857s (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 2e36 3234 .| -- used 0.624 │ │ │ │ -0000a9b0: 3830 3973 2028 6370 7529 3b20 302e 3436 809s (cpu); 0.46 │ │ │ │ -0000a9c0: 3030 3832 7320 2874 6872 6561 6429 3b20 0082s (thread); │ │ │ │ +0000a9a0: 0a7c 202d 2d20 7573 6564 2030 2e32 3830 .| -- used 0.280 │ │ │ │ +0000a9b0: 3530 3673 2028 6370 7529 3b20 302e 3236 506s (cpu); 0.26 │ │ │ │ +0000a9c0: 3130 3933 7320 2874 6872 6561 6429 3b20 1093s (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 3235 .| -- used 0.025 │ │ │ │ -0000abe0: 3737 3536 7320 2863 7075 293b 2030 2e30 7756s (cpu); 0.0 │ │ │ │ -0000abf0: 3235 3232 3638 7320 2874 6872 6561 6429 252268s (thread) │ │ │ │ +0000abd0: 0a7c 202d 2d20 7573 6564 2030 2e30 3239 .| -- used 0.029 │ │ │ │ +0000abe0: 3138 3934 7320 2863 7075 293b 2030 2e30 1894s (cpu); 0.0 │ │ │ │ +0000abf0: 3137 3039 3931 7320 2874 6872 6561 6429 170991s (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 3034 .| -- used 0.004 │ │ │ │ -0000ae10: 3539 3032 3373 2028 6370 7529 3b20 302e 59023s (cpu); 0. │ │ │ │ -0000ae20: 3030 3435 3930 3732 7320 2874 6872 6561 00459072s (threa │ │ │ │ +0000ae00: 0a7c 202d 2d20 7573 6564 2030 2e30 3033 .| -- used 0.003 │ │ │ │ +0000ae10: 3232 3832 3673 2028 6370 7529 3b20 302e 22826s (cpu); 0. │ │ │ │ +0000ae20: 3030 3332 3332 3631 7320 2874 6872 6561 00323261s (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 3133 .| -- used 0.013 │ │ │ │ -0000b220: 3830 3435 7320 2863 7075 293b 2030 2e30 8045s (cpu); 0.0 │ │ │ │ -0000b230: 3133 3830 3739 7320 2874 6872 6561 6429 138079s (thread) │ │ │ │ -0000b240: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0000b210: 0a7c 202d 2d20 7573 6564 2030 2e30 3039 .| -- used 0.009 │ │ │ │ +0000b220: 3736 3935 3273 2028 6370 7529 3b20 302e 76952s (cpu); 0. │ │ │ │ +0000b230: 3030 3937 3735 3731 7320 2874 6872 6561 00977571s (threa │ │ │ │ +0000b240: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 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 3133 .| -- used 0.013 │ │ │ │ -0000b720: 3231 3636 7320 2863 7075 293b 2030 2e30 2166s (cpu); 0.0 │ │ │ │ -0000b730: 3133 3139 3036 7320 2874 6872 6561 6429 131906s (thread) │ │ │ │ -0000b740: 3b20 3073 2028 6763 2920 2020 2020 2020 ; 0s (gc) │ │ │ │ +0000b710: 0a7c 202d 2d20 7573 6564 2030 2e30 3039 .| -- used 0.009 │ │ │ │ +0000b720: 3632 3333 3873 2028 6370 7529 3b20 302e 62338s (cpu); 0. │ │ │ │ +0000b730: 3030 3936 3239 3036 7320 2874 6872 6561 00962906s (threa │ │ │ │ +0000b740: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000b750: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b760: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b7a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b7b0: 0a7c 6f32 3020 3a20 4d75 6c74 6968 6f6d .|o20 : Multihom │ │ │ │ @@ -2958,18 +2958,18 @@ │ │ │ │ 0000b8d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000b8e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000b8f0: 0a7c 6932 3120 3a20 7469 6d65 2064 6567 .|i21 : time deg │ │ │ │ 0000b900: 7265 6573 2066 2020 2020 2020 2020 2020 rees f │ │ │ │ 0000b910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b930: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000b940: 0a7c 202d 2d20 7573 6564 2031 2e36 3033 .| -- used 1.603 │ │ │ │ -0000b950: 3732 7320 2863 7075 293b 2031 2e31 3831 72s (cpu); 1.181 │ │ │ │ -0000b960: 3134 7320 2874 6872 6561 6429 3b20 3073 14s (thread); 0s │ │ │ │ -0000b970: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0000b940: 0a7c 202d 2d20 7573 6564 2031 2e31 3431 .| -- used 1.141 │ │ │ │ +0000b950: 3737 7320 2863 7075 293b 2030 2e38 3530 77s (cpu); 0.850 │ │ │ │ +0000b960: 3932 3173 2028 7468 7265 6164 293b 2030 921s (thread); 0 │ │ │ │ +0000b970: 7320 2867 6329 2020 2020 2020 2020 2020 s (gc) │ │ │ │ 0000b980: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b990: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000b9a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000b9d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000b9e0: 0a7c 6f32 3120 3d20 7b39 3034 2c20 3530 .|o21 = {904, 50 │ │ │ │ @@ -2993,17 +2993,17 @@ │ │ │ │ 0000bb00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0000bb10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ 0000bb20: 0a7c 6932 3220 3a20 7469 6d65 2064 6567 .|i22 : time deg │ │ │ │ 0000bb30: 7265 6520 6620 2020 2020 2020 2020 2020 ree f │ │ │ │ 0000bb40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bb60: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0000bb70: 0a7c 202d 2d20 7573 6564 2032 2e30 3136 .| -- used 2.016 │ │ │ │ -0000bb80: 3865 2d30 3573 2028 6370 7529 3b20 312e 8e-05s (cpu); 1. │ │ │ │ -0000bb90: 3936 3036 652d 3035 7320 2874 6872 6561 9606e-05s (threa │ │ │ │ +0000bb70: 0a7c 202d 2d20 7573 6564 2031 2e34 3238 .| -- used 1.428 │ │ │ │ +0000bb80: 3365 2d30 3573 2028 6370 7529 3b20 312e 3e-05s (cpu); 1. │ │ │ │ +0000bb90: 3334 3131 652d 3035 7320 2874 6872 6561 3411e-05s (threa │ │ │ │ 0000bba0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ 0000bbb0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ 0000bbc0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ 0000bbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bbf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0000bc00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ @@ -3019,16 +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: 3731 3638 3673 2028 6370 7529 3b20 302e 71686s (cpu); 0. │ │ │ │ -0000bd20: 3030 3137 3137 3935 7320 2874 6872 6561 00171795s (threa │ │ │ │ +0000bd10: 3339 3536 3373 2028 6370 7529 3b20 302e 39563s (cpu); 0. │ │ │ │ +0000bd20: 3030 3134 3030 3432 7320 2874 6872 6561 00140042s (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 3736 3832 3573 2028 6370 7529 3b20 0476825s (cpu); │ │ │ │ -000124b0: 302e 3030 3034 3732 3530 3673 2028 7468 0.000472506s (th │ │ │ │ +000124a0: 3034 3139 3232 3973 2028 6370 7529 3b20 0419229s (cpu); │ │ │ │ +000124b0: 302e 3030 3034 3134 3934 3973 2028 7468 0.000414949s (th │ │ │ │ 000124c0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ 000124d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000124e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 000124f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00012520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ @@ -4746,18 +4746,18 @@ │ │ │ │ 00012890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000128b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ 000128c0: 6935 203a 2074 696d 6520 7072 6f6a 6563 i5 : time projec │ │ │ │ 000128d0: 7469 7665 4465 6772 6565 7328 7073 692c tiveDegrees(psi, │ │ │ │ 000128e0: 3329 2020 2020 2020 2020 2020 2020 2020 3) │ │ │ │ 000128f0: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ -00012900: 2d20 7573 6564 2030 2e31 3139 3473 2028 - used 0.1194s ( │ │ │ │ -00012910: 6370 7529 3b20 302e 3131 3934 3233 7320 cpu); 0.119423s │ │ │ │ -00012920: 2874 6872 6561 6429 3b20 3073 2028 6763 (thread); 0s (gc │ │ │ │ -00012930: 2920 2020 2020 2020 207c 0a7c 2020 2020 ) |.| │ │ │ │ +00012900: 2d20 7573 6564 2030 2e31 3233 3030 3273 - used 0.123002s │ │ │ │ +00012910: 2028 6370 7529 3b20 302e 3132 3330 3038 (cpu); 0.123008 │ │ │ │ +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: 3332 3736 3973 2028 6370 7529 3b20 302e 32769s (cpu); 0. │ │ │ │ -00012a50: 3430 3139 3233 7320 2874 6872 6561 6429 401923s (thread) │ │ │ │ +00012a30: 207c 0a7c 202d 2d20 7573 6564 2030 2e34 |.| -- used 0.4 │ │ │ │ +00012a40: 3131 3931 3773 2028 6370 7529 3b20 302e 11917s (cpu); 0. │ │ │ │ +00012a50: 3334 3033 3137 7320 2874 6872 6561 6429 340317s (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 -- │ │ │ │ @@ -5189,16 +5189,16 @@ │ │ │ │ 00014440: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014450: 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3134 203a --------+.|i14 : │ │ │ │ 00014460: 2074 696d 6520 5420 3d20 6162 7374 7261 time T = abstra │ │ │ │ 00014470: 6374 5261 7469 6f6e 616c 4d61 7028 492c ctRationalMap(I, │ │ │ │ 00014480: 224f 4144 5022 2920 2020 2020 2020 2020 "OADP") │ │ │ │ 00014490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000144a0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -000144b0: 6420 302e 3033 3833 3137 3673 2028 6370 d 0.0383176s (cp │ │ │ │ -000144c0: 7529 3b20 302e 3033 3833 3230 3273 2028 u); 0.0383202s ( │ │ │ │ +000144b0: 6420 302e 3034 3138 3635 3273 2028 6370 d 0.0418652s (cp │ │ │ │ +000144c0: 7529 3b20 302e 3034 3138 3637 3373 2028 u); 0.0418673s ( │ │ │ │ 000144d0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 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 │ │ │ │ @@ -5261,48252 +5261,48253 @@ │ │ │ │ 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: 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 322e 3138 3636 3373 2028 6370 7529 d 2.18663s (cpu) │ │ │ │ -00014980: 3b20 312e 3538 3437 7320 2874 6872 6561 ; 1.5847s (threa │ │ │ │ -00014990: 6429 3b20 3073 2028 6763 297c 0a7c 2020 d); 0s (gc)|.| │ │ │ │ -000149a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014930: 2d2d 2b0a 7c69 3135 203a 2074 696d 6520 --+.|i15 : time │ │ │ │ +00014940: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ +00014950: 7328 542c 3229 2020 2020 2020 2020 2020 s(T,2) │ │ │ │ +00014960: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ +00014970: 7365 6420 322e 3230 3237 3573 2028 6370 sed 2.20275s (cp │ │ │ │ +00014980: 7529 3b20 312e 3536 3931 3473 2028 7468 u); 1.56914s (th │ │ │ │ +00014990: 7265 6164 293b 2030 7320 2867 6329 7c0a read); 0s (gc)|. │ │ │ │ +000149a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000149b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000149d0: 7c0a 7c6f 3135 203d 2033 2020 2020 2020 |.|o15 = 3 │ │ │ │ +000149d0: 2020 2020 7c0a 7c6f 3135 203d 2033 2020 |.|o15 = 3 │ │ │ │ 000149e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000149f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014a00: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00014a00: 2020 2020 2020 2020 2020 7c0a 2b2d 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 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 ---------------- │ │ │ │ +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 +--------------- │ │ │ │ 00014aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 │ │ │ │ +00014ac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6931 -----------+.|i1 │ │ │ │ +00014ad0: 3620 3a20 7469 6d65 2054 3220 3d20 5420 6 : time T2 = T │ │ │ │ +00014ae0: 2a20 5420 2020 2020 2020 2020 2020 2020 * T │ │ │ │ 00014af0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b00: 2020 7c0a 7c20 2d2d 2075 7365 6420 332e |.| -- used 3. │ │ │ │ -00014b10: 3030 3037 652d 3035 7320 2863 7075 293b 0007e-05s (cpu); │ │ │ │ -00014b20: 2032 2e39 3730 3565 2d30 3573 2028 7468 2.9705e-05s (th │ │ │ │ -00014b30: 7265 6164 293b 2030 7320 2867 6329 207c read); 0s (gc) | │ │ │ │ -00014b40: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00014b00: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ +00014b10: 7365 6420 322e 3230 3431 652d 3035 7320 sed 2.2041e-05s │ │ │ │ +00014b20: 2863 7075 293b 2032 2e31 3039 3265 2d30 (cpu); 2.1092e-0 │ │ │ │ +00014b30: 3573 2028 7468 7265 6164 293b 2030 7320 5s (thread); 0s │ │ │ │ +00014b40: 2867 6329 207c 0a7c 2020 2020 2020 2020 (gc) |.| │ │ │ │ 00014b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014b70: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00014b80: 3136 203d 202d 2d20 7261 7469 6f6e 616c 16 = -- rational │ │ │ │ -00014b90: 206d 6170 202d 2d20 2020 2020 2020 2020 map -- │ │ │ │ +00014b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014b80: 2020 7c0a 7c6f 3136 203d 202d 2d20 7261 |.|o16 = -- ra │ │ │ │ +00014b90: 7469 6f6e 616c 206d 6170 202d 2d20 2020 tional map -- │ │ │ │ 00014ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bb0: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ -00014bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014bd0: 205a 5a20 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +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 │ │ │ │ 00014be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00014c90: 2020 2020 2020 2020 2020 2020 2020 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 |.| │ │ │ │ +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 |.| │ │ │ │ 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: 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 +--------------- │ │ │ │ +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)|.+--------- │ │ │ │ 00014df0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00014e00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 332e 3637 3932 3873 2028 6370 sed 3.67928s (cp │ │ │ │ -00014e70: 7529 3b20 322e 3630 3139 7320 2874 6872 u); 2.6019s (thr │ │ │ │ -00014e80: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ -00014e90: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00014e10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00014e20: 2d2b 0a7c 6931 3720 3a20 7469 6d65 2070 -+.|i17 : time p │ │ │ │ +00014e30: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ +00014e40: 2854 322c 3229 2020 2020 2020 2020 2020 (T2,2) │ │ │ │ +00014e50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00014e60: 7c20 2d2d 2075 7365 6420 332e 3339 3439 | -- used 3.3949 │ │ │ │ +00014e70: 7320 2863 7075 293b 2032 2e34 3735 3138 s (cpu); 2.47518 │ │ │ │ +00014e80: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +00014e90: 6763 2920 2020 2020 2020 207c 0a7c 2020 gc) |.| │ │ │ │ 00014ea0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014eb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00014ec0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014ed0: 2020 7c0a 7c6f 3137 203d 2031 2020 2020 |.|o17 = 1 │ │ │ │ -00014ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014ed0: 2020 2020 2020 2020 7c0a 7c6f 3137 203d |.|o17 = │ │ │ │ +00014ee0: 2031 2020 2020 2020 2020 2020 2020 2020 1 │ │ │ │ 00014ef0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00014f00: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00014f10: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00014f00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00014f10: 2020 2020 207c 0a2b 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 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 ---------------- │ │ │ │ +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 :..+------------ │ │ │ │ 00014fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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}|.| │ │ │ │ +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}|.| │ │ │ │ 00015000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015010: 2020 2020 2020 2020 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 .| │ │ │ │ +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 |.| │ │ │ │ 00015060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015070: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00015080: 3138 203a 204c 6973 7420 2020 2020 2020 18 : List │ │ │ │ +00015070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015080: 2020 7c0a 7c6f 3138 203a 204c 6973 7420 |.|o18 : List │ │ │ │ 00015090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000150a0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000150b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000150a0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000150b0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ 000150c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000150d0: 2d2d 2d2d 2d2d 2b0a 7c69 3139 203a 2071 ------+.|i19 : q │ │ │ │ -000150e0: 203d 2054 2070 2020 2020 2020 2020 2020 = T p │ │ │ │ +000150d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000150e0: 3139 203a 2071 203d 2054 2070 2020 2020 19 : q = T p │ │ │ │ 000150f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015100: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015100: 2020 2020 2020 2020 207c 0a7c 2020 2020 |.| │ │ │ │ 00015110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015120: 2020 2020 2020 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 │ │ │ │ +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 |.| │ │ │ │ 00015170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015180: 2020 2020 2020 2020 2020 7c0a 7c6f 3139 |.|o19 │ │ │ │ -00015190: 203a 204c 6973 7420 2020 2020 2020 2020 : List │ │ │ │ +00015180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00015190: 7c0a 7c6f 3139 203a 204c 6973 7420 2020 |.|o19 : List │ │ │ │ 000151a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000151b0: 2020 2020 2020 207c 0a2b 2d2d 2d2d 2d2d |.+------ │ │ │ │ +000151b0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ 000151c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000151d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000151e0: 2d2d 2d2d 2b0a 7c69 3230 203a 2054 2071 ----+.|i20 : T q │ │ │ │ -000151f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000151e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3230 ----------+.|i20 │ │ │ │ +000151f0: 203a 2054 2071 2020 2020 2020 2020 2020 : T q │ │ │ │ 00015200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015210: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00015210: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 00015220: 2020 2020 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 │ │ │ │ +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 |.| │ │ │ │ 00015280: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00015290: 2020 2020 2020 2020 7c0a 7c6f 3230 203a |.|o20 : │ │ │ │ -000152a0: 204c 6973 7420 2020 2020 2020 2020 2020 List │ │ │ │ +00015290: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +000152a0: 7c6f 3230 203a 204c 6973 7420 2020 2020 |o20 : List │ │ │ │ 000152b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000152c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000152c0: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 000152d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000152e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 :..+------------ │ │ │ │ +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:..+------ │ │ │ │ 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 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 │ │ │ │ +00015370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6932 -----------+.|i2 │ │ │ │ +00015380: 3120 3a20 7469 6d65 2066 203d 2072 6174 1 : time f = rat │ │ │ │ +00015390: 696f 6e61 6c4d 6170 2054 2020 2020 2020 ionalMap T │ │ │ │ 000153a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000153b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ -000153c0: 7573 6564 2033 2e31 3236 3536 7320 2863 used 3.12656s (c │ │ │ │ -000153d0: 7075 293b 2032 2e32 3137 7320 2874 6872 pu); 2.217s (thr │ │ │ │ -000153e0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ -000153f0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00015400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000153b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000153c0: 0a7c 202d 2d20 7573 6564 2032 2e39 3332 .| -- used 2.932 │ │ │ │ +000153d0: 3632 7320 2863 7075 293b 2032 2e31 3337 62s (cpu); 2.137 │ │ │ │ +000153e0: 3539 7320 2874 6872 6561 6429 3b20 3073 59s (thread); 0s │ │ │ │ +000153f0: 2028 6763 2920 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00015400: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 00015410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00015430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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2d2d ---------------- │ │ │ │ +00015e50: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00015e60: 0a7c 2020 2020 2020 636f 6566 6669 6369 .| coeffici │ │ │ │ +00015e70: 656e 7420 7269 6e67 3a20 5a5a 2f36 3535 ent ring: ZZ/655 │ │ │ │ +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 |.+---------- │ │ │ │ 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: 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:.============ │ │ │ │ +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:.====== │ │ │ │ 00015f40: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -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 .--------------- │ │ │ │ +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 │ │ │ │ +00015f80: 696e 672c 506f 6c79 6e6f 6d69 616c 5269 ing,PolynomialRi │ │ │ │ +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,...--------- │ │ │ │ 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: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ -00016110: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ -00016120: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ -00016130: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ -00016140: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+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/ │ │ │ │ -00016170: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d .documentation.m │ │ │ │ -00016180: 323a 3131 3032 3a30 2e0a 1f0a 4669 6c65 2:1102:0....File │ │ │ │ -00016190: 3a20 4372 656d 6f6e 612e 696e 666f 2c20 : Cremona.info, │ │ │ │ -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.********** │ │ │ │ +00016100: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00016110: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00016120: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +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: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+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 │ │ │ │ +00016180: 7469 6f6e 2e6d 323a 3131 3032 3a30 2e0a tion.m2:1102:0.. │ │ │ │ +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.**** │ │ │ │ 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 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...+------- │ │ │ │ +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...+- │ │ │ │ 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 ---------------- │ │ │ │ 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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 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 │ │ │ │ +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 │ │ │ │ 00019c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 |.|------- │ │ │ │ +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 |.|- │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 00019d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00019dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00019de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00019df0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00019df0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 00019e80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00019ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00019f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00019f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00019fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0001a010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0001a060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0001a0b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0001a100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0001a150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a160: 2020 2020 2020 7c0a 7c2d 2d20 6170 7072 |.|-- appr │ │ │ │ -0001a170: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ -0001a180: 703a 2073 7465 7020 3130 206f 6620 3130 p: step 10 of 10 │ │ │ │ -0001a190: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0001a160: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +0001a170: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ +0001a180: 6572 7365 4d61 703a 2073 7465 7020 3130 erseMap: step 10 │ │ │ │ +0001a190: 206f 6620 3130 2020 2020 2020 2020 2020 of 10 │ │ │ │ 0001a1a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a1b0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0001a1c0: 6420 302e 3535 3931 3734 7320 2863 7075 d 0.559174s (cpu │ │ │ │ -0001a1d0: 293b 2030 2e33 3734 3838 3773 2028 7468 ); 0.374887s (th │ │ │ │ -0001a1e0: 7265 6164 293b 2030 7320 2867 6329 2020 read); 0s (gc) │ │ │ │ -0001a1f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001a200: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0001a1b0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001a1c0: 2d2d 2075 7365 6420 302e 3337 3131 3533 -- used 0.371153 │ │ │ │ +0001a1d0: 7320 2863 7075 293b 2030 2e32 3337 3734 s (cpu); 0.23774 │ │ │ │ +0001a1e0: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +0001a1f0: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0001a200: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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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 |.|- │ │ │ │ 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 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ -0001ef40: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ +0001ef30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +0001ef40: 2020 2020 2020 2020 2020 2020 2020 3220 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 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 |.|------- │ │ │ │ +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 |.|- │ │ │ │ 0001f030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f070: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 2020 ------|.| │ │ │ │ +0001f070: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ 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 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 |.|------- │ │ │ │ +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 |.|- │ │ │ │ 0001f170: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0001f1a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f1b0: 2d2d 2d2d 2d2d 7c0a 7c20 2032 2032 2020 ------|.| 2 2 │ │ │ │ -0001f1c0: 2020 2020 2020 2020 3220 2020 2020 2032 2 2 │ │ │ │ -0001f1d0: 2032 2020 2020 2020 2020 2020 3220 2020 2 2 │ │ │ │ -0001f1e0: 2020 2032 2032 2020 2020 2020 2020 2032 2 2 2 │ │ │ │ -0001f1f0: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ -0001f200: 2020 2020 2032 7c0a 7c37 7820 7820 202d 2|.|7x x - │ │ │ │ -0001f210: 2032 3978 2078 2078 2020 2b20 3336 7820 29x x x + 36x │ │ │ │ -0001f220: 7820 202d 2031 3178 2078 2078 2020 2b20 x - 11x x x + │ │ │ │ -0001f230: 3438 7820 7820 202b 2032 7820 7820 7820 48x x + 2x x x │ │ │ │ -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 │ │ │ │ -0001f260: 2020 2020 3020 3120 3820 2020 2020 2031 0 1 8 1 │ │ │ │ -0001f270: 2038 2020 2020 2020 3020 3220 3820 2020 8 0 2 8 │ │ │ │ -0001f280: 2020 2032 2038 2020 2020 2030 2034 2038 2 8 0 4 8 │ │ │ │ -0001f290: 2020 2020 2020 3120 3420 3820 2020 2020 1 4 8 │ │ │ │ -0001f2a0: 2032 2034 2038 7c0a 7c2d 2d2d 2d2d 2d2d 2 4 8|.|------- │ │ │ │ +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 │ │ │ │ +0001f220: 2b20 3336 7820 7820 202d 2031 3178 2078 + 36x x - 11x x │ │ │ │ +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|.|- │ │ │ │ 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 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0001f2f0: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 3220 ------|.| 2 │ │ │ │ -0001f300: 3220 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +0001f2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c20 ------------|.| │ │ │ │ +0001f300: 2020 2020 3220 3220 2020 2020 2020 2020 2 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 7c0a 7c2d 2034 3878 2078 |.|- 48x x │ │ │ │ -0001f350: 2020 2b20 3438 7820 7820 7820 7820 202d + 48x x x x - │ │ │ │ -0001f360: 2034 3878 2078 2078 2078 2020 2d20 3438 48x x x x - 48 │ │ │ │ -0001f370: 7820 7820 7820 7820 202b 2034 3878 2078 x x x x + 48x x │ │ │ │ -0001f380: 2078 2078 2020 2b20 3438 7820 7820 7820 x x + 48x x x │ │ │ │ -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 |.|------- │ │ │ │ +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 |.|- │ │ │ │ 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 7c0a 7c34 3878 2078 2078 ------|.|48x x x │ │ │ │ -0001f440: 2078 2020 2020 2020 2020 2020 2020 2020 x │ │ │ │ +0001f430: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c34 ------------|.|4 │ │ │ │ +0001f440: 3878 2078 2078 2078 2020 2020 2020 2020 8x x x 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 7c0a 7c20 2020 3020 3420 |.| 0 4 │ │ │ │ -0001f490: 3820 3920 2020 2020 2020 2020 2020 2020 8 9 │ │ │ │ +0001f480: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f490: 2020 3020 3420 3820 3920 2020 2020 2020 0 4 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 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001f4d0: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 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 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) │ │ │ │ +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) │ │ │ │ 0001f560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f570: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0001f570: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 0001f650: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0001f6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +0001f6b0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ +0001f6c0: 2d20 6170 7072 6f78 696d 6174 6549 6e76 - approximateInv │ │ │ │ +0001f6d0: 6572 7365 4d61 703a 2073 7465 7020 3320 erseMap: step 3 │ │ │ │ +0001f6e0: 6f66 2033 2020 2020 2020 2020 2020 2020 of 3 │ │ │ │ 0001f6f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f700: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0001f710: 6420 302e 3336 3739 3339 7320 2863 7075 d 0.367939s (cpu │ │ │ │ -0001f720: 293b 2030 2e32 3830 3934 3273 2028 7468 ); 0.280942s (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 |.| │ │ │ │ +0001f700: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0001f710: 2d2d 2075 7365 6420 302e 3232 3739 3635 -- used 0.227965 │ │ │ │ +0001f720: 7320 2863 7075 293b 2030 2e31 3631 3634 s (cpu); 0.16164 │ │ │ │ +0001f730: 3273 2028 7468 7265 6164 293b 2030 7320 2s (thread); 0s │ │ │ │ +0001f740: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +0001f750: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0001f760: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f780: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f7a0: 2020 2020 2020 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) |.+------- │ │ │ │ +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) |.+- │ │ │ │ 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 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 ) │ │ │ │ +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') │ │ │ │ 0001f870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0001f880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0001f890: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 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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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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*|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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+|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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_|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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_ |.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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*|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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-|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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+ |.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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*|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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_|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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+|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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- |.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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_|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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^|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 ---------------- │ │ │ │ -00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00028190: 0a7c 2a74 5f30 2a74 5f38 2d34 362a 745f .|*t_0*t_8-46*t_ │ │ │ │ -000281a0: 312a 745f 382b 3437 2a74 5f32 2a74 5f38 1*t_8+47*t_2*t_8 │ │ │ │ -000281b0: 2d33 342a 745f 332a 745f 382d 3234 2a74 -34*t_3*t_8-24*t │ │ │ │ -000281c0: 5f34 2a74 5f38 2d31 322a 745f 352a 745f _4*t_8-12*t_5*t_ │ │ │ │ -000281d0: 382d 3437 2a74 5f36 2a74 5f38 2b34 377c 8-47*t_6*t_8+47| │ │ │ │ -000281e0: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +00028180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028190: 2d2d 2d2d 2d7c 0a7c 2a74 5f30 2a74 5f38 -----|.|*t_0*t_8 │ │ │ │ +000281a0: 2d34 362a 745f 312a 745f 382b 3437 2a74 -46*t_1*t_8+47*t │ │ │ │ +000281b0: 5f32 2a74 5f38 2d33 342a 745f 332a 745f _2*t_8-34*t_3*t_ │ │ │ │ +000281c0: 382d 3234 2a74 5f34 2a74 5f38 2d31 322a 8-24*t_4*t_8-12* │ │ │ │ +000281d0: 745f 352a 745f 382d 3437 2a74 5f36 2a74 t_5*t_8-47*t_6*t │ │ │ │ +000281e0: 5f38 2b34 377c 0a7c 2d2d 2d2d 2d2d 2d2d _8+47|.|-------- │ │ │ │ 000281f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00028230: 0a7c 2a74 5f37 2a74 5f38 2c74 5f31 2a74 .|*t_7*t_8,t_1*t │ │ │ │ -00028240: 5f32 2b36 2a74 5f31 2a74 5f35 2b35 2a74 _2+6*t_1*t_5+5*t │ │ │ │ -00028250: 5f30 2a74 5f36 2d32 2a74 5f31 2a74 5f36 _0*t_6-2*t_1*t_6 │ │ │ │ -00028260: 2d74 5f34 2a74 5f36 2d74 5f35 2a74 5f36 -t_4*t_6-t_5*t_6 │ │ │ │ -00028270: 2b35 2a74 5f30 2a74 5f37 2b74 5f31 2a7c +5*t_0*t_7+t_1*| │ │ │ │ -00028280: 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .|-------------- │ │ │ │ +00028220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00028230: 2d2d 2d2d 2d7c 0a7c 2a74 5f37 2a74 5f38 -----|.|*t_7*t_8 │ │ │ │ +00028240: 2c74 5f31 2a74 5f32 2b36 2a74 5f31 2a74 ,t_1*t_2+6*t_1*t │ │ │ │ +00028250: 5f35 2b35 2a74 5f30 2a74 5f36 2d32 2a74 _5+5*t_0*t_6-2*t │ │ │ │ +00028260: 5f31 2a74 5f36 2d74 5f34 2a74 5f36 2d74 _1*t_6-t_4*t_6-t │ │ │ │ +00028270: 5f35 2a74 5f36 2b35 2a74 5f30 2a74 5f37 _5*t_6+5*t_0*t_7 │ │ │ │ +00028280: 2b74 5f31 2a7c 0a7c 2d2d 2d2d 2d2d 2d2d +t_1*|.|-------- │ │ │ │ 00028290: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000282a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000282b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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* │ │ │ │ -00028300: 745f 372a 745f 382c 745f 302a 745f 322b t_7*t_8,t_0*t_2+ │ │ │ │ -00028310: 745f 312a 745f 342b 352a 745f 302a 747c t_1*t_4+5*t_0*t| │ │ │ │ -00028320: 0a7c 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_ │ │ │ │ +000282e0: 322a 745f 372d 372a 745f 352a 745f 372b 2*t_7-7*t_5*t_7+ │ │ │ │ +000282f0: 322a 745f 362a 745f 372d 322a 745f 312a 2*t_6*t_7-2*t_1* │ │ │ │ +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|.|-------- │ │ │ │ 00028330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028340: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028350: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00028360: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ -00028370: 0a7c 5f35 2b33 322a 745f 312a 745f 352d .|_5+32*t_1*t_5- │ │ │ │ -00028380: 3230 2a74 5f32 2a74 5f35 2d34 372a 745f 20*t_2*t_5-47*t_ │ │ │ │ -00028390: 332a 745f 352d 3337 2a74 5f34 2a74 5f35 3*t_5-37*t_4*t_5 │ │ │ │ -000283a0: 2d32 352a 745f 355e 322b 3139 2a74 5f30 -25*t_5^2+19*t_0 │ │ │ │ -000283b0: 2a74 5f36 2b32 322a 745f 312a 745f 367c *t_6+22*t_1*t_6| │ │ │ │ -000283c0: 0a7c 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_ │ │ │ │ +00028380: 312a 745f 352d 3230 2a74 5f32 2a74 5f35 1*t_5-20*t_2*t_5 │ │ │ │ +00028390: 2d34 372a 745f 332a 745f 352d 3337 2a74 -47*t_3*t_5-37*t │ │ │ │ +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|.|-------- │ │ │ │ 000283d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 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 │ │ │ │ -00028450: 2a74 5f33 2a74 5f37 2b32 382a 745f 347c *t_3*t_7+28*t_4| │ │ │ │ -00028460: 0a7c 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|.|-------- │ │ │ │ 00028470: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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-|.|-------- │ │ │ │ 00028510: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028520: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00028530: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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 .|-------------- │ │ │ │ +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*|.|-------- │ │ │ │ 000285b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 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 │ │ │ │ -00028630: 2b31 392a 745f 312a 745f 382b 3230 2a7c +19*t_1*t_8+20*| │ │ │ │ -00028640: 0a7c 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*|.|-------- │ │ │ │ 00028650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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_|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .|-------------- │ │ │ │ +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|.|-------- │ │ │ │ 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 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 .+-------------- │ │ │ │ +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 |.+-------- │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 000289e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00028a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +00028a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028a50: 2020 2020 207c 0a7c 2d2d 2061 7070 726f |.|-- appro │ │ │ │ +00028a60: 7869 6d61 7465 496e 7665 7273 654d 6170 ximateInverseMap │ │ │ │ +00028a70: 3a20 7374 6570 2033 206f 6620 3320 2020 : step 3 of 3 │ │ │ │ 00028a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028a90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028aa0: 0a7c 202d 2d20 7573 6564 2032 2e37 3631 .| -- used 2.761 │ │ │ │ -00028ab0: 3373 2028 6370 7529 3b20 312e 3839 3330 3s (cpu); 1.8930 │ │ │ │ -00028ac0: 3373 2028 7468 7265 6164 293b 2030 7320 3s (thread); 0s │ │ │ │ -00028ad0: 2867 6329 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 .| │ │ │ │ +00028a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028aa0: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ +00028ab0: 2031 2e39 3836 3333 7320 2863 7075 293b 1.98633s (cpu); │ │ │ │ +00028ac0: 2031 2e36 3333 3536 7320 2874 6872 6561 1.63356s (threa │ │ │ │ +00028ad0: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +00028ae0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028af0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00028b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00028b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028b30: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00028b40: 0a7c 6f38 203d 202d 2d20 7261 7469 6f6e .|o8 = -- ration │ │ │ │ -00028b50: 616c 206d 6170 202d 2d20 2020 2020 2020 al map -- │ │ │ │ +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 -- │ │ │ │ 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 207c | │ │ │ │ -00028b90: 0a7c 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 |.| │ │ │ │ 00028ba0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028bb0: 2020 5a5a 2020 2020 2020 2020 2020 2020 ZZ │ │ │ │ +00028bb0: 2020 2020 2020 2020 5a5a 2020 2020 2020 ZZ │ │ │ │ 00028bc0: 2020 2020 2020 2020 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 .| │ │ │ │ +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 |.| │ │ │ │ 00028c40: 2020 2020 2020 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 │ │ │ │ +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 { │ │ │ │ 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 207c | │ │ │ │ -00028cd0: 0a7c 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 |.| │ │ │ │ 00028ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00028cf0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00028cf0: 2020 2020 2020 2020 3220 2020 2020 2020 2 │ │ │ │ 00028d00: 2020 2020 2020 2020 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 .| │ │ │ │ +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 |.| │ │ │ │ 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 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 │ │ │ │ +00028e00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00028e10: 2020 2020 207c 0a7c 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2020 2020 .| │ │ │ │ +0002d180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d190: 2020 2020 207c 0a7c 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 207c | │ │ │ │ -0002d1e0: 0a7c 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 |.| │ │ │ │ 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 207c | │ │ │ │ -0002d230: 0a7c 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 |.| │ │ │ │ 0002d240: 2020 2020 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-0002d310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002d320: 0a7c 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 |.| │ │ │ │ 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 207c | │ │ │ │ -0002d370: 0a7c 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 |.| │ │ │ │ 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 207c | │ │ │ │ -0002d3c0: 0a7c 2b20 3135 7820 7820 2020 2d20 3137 .|+ 15x x - 17 │ │ │ │ -0002d3d0: 7820 7820 2020 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, │ │ │ │ 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 207c | │ │ │ │ -0002d500: 0a7c 2020 2032 2031 3120 2020 2020 3520 .| 2 11 5 │ │ │ │ -0002d510: 3131 2020 2020 2020 2020 2020 2020 2020 11 │ │ │ │ +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 │ │ │ │ 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 207c | │ │ │ │ -0002d550: 0a7c 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 |.| │ │ │ │ 0002d560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002d580: 2020 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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 .| │ │ │ │ +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 |.| │ │ │ │ 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 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 .|-------------- │ │ │ │ +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 |.|-------- │ │ │ │ 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 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 │ │ │ │ +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 , │ │ │ │ 0002d800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0002d850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002d860: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002d870: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002d860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002d870: 2020 2020 207c 0a2b 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 2d2b ---------------+ │ │ │ │ -0002d8c0: 0a7c 6939 203a 202d 2d20 6275 742e 2e2e .|i9 : -- but... │ │ │ │ -0002d8d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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... │ │ │ │ 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 207c | │ │ │ │ -0002d910: 0a7c 2020 2020 2070 6869 202a 2070 7369 .| phi * psi │ │ │ │ -0002d920: 203d 3d20 3120 2020 2020 2020 2020 2020 == 1 │ │ │ │ +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 │ │ │ │ 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 207c | │ │ │ │ -0002d960: 0a7c 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 |.| │ │ │ │ 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 207c | │ │ │ │ -0002d9b0: 0a7c 6f39 203d 2066 616c 7365 2020 2020 .|o9 = false │ │ │ │ -0002d9c0: 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 │ │ │ │ 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 207c | │ │ │ │ -0002da00: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +0002d9f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002da00: 2020 2020 207c 0a2b 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 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 │ │ │ │ +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 │ │ │ │ 0002db20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0002db70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 0002dbc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +0002dbd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dbe0: 2020 2020 207c 0a7c 4365 7274 6966 793a |.|Certify: │ │ │ │ +0002dbf0: 206f 7574 7075 7420 6365 7274 6966 6965 output certifie │ │ │ │ +0002dc00: 6421 2020 2020 2020 2020 2020 2020 2020 d! │ │ │ │ 0002dc10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dc20: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002dc30: 0a7c 202d 2d20 7573 6564 2033 2e38 3935 .| -- used 3.895 │ │ │ │ -0002dc40: 3535 7320 2863 7075 293b 2032 2e39 3232 55s (cpu); 2.922 │ │ │ │ -0002dc50: 3338 7320 2874 6872 6561 6429 3b20 3073 38s (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 .| │ │ │ │ +0002dc20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc30: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ +0002dc40: 2032 2e37 3037 3536 7320 2863 7075 293b 2.70756s (cpu); │ │ │ │ +0002dc50: 2032 2e32 3630 3237 7320 2874 6872 6561 2.26027s (threa │ │ │ │ +0002dc60: 6429 3b20 3073 2028 6763 2920 2020 2020 d); 0s (gc) │ │ │ │ +0002dc70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0002dc80: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 0002dc90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0002dcb0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0002dcc0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -0002dcd0: 0a7c 6f31 3020 3d20 2d2d 2072 6174 696f .|o10 = -- ratio │ │ │ │ -0002dce0: 6e61 6c20 6d61 7020 2d2d 2020 2020 2020 nal map -- │ │ │ │ +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 -- │ │ │ │ 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 207c | │ │ │ │ -0002dd20: 0a7c 2020 2020 2020 2020 2020 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2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031fa0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00031fb0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00031fa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00031fb0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00031fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00031fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00031ff0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00032000: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00031ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032000: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00032010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032040: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00032050: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00032040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032050: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00032060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032080: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032090: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000320a0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00032090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000320a0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000320b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000320c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000320d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000320e0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000320f0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000320e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000320f0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00032100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032130: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00032140: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00032130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032140: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00032150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032170: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032180: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00032190: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00032180: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032190: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000321a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000321b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000321c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000321d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000321e0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +000321d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000321e0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 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 │ │ │ │ -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 │ │ │ │ -00032270: 202d 2032 7820 7820 2020 2d20 2020 207c - 2x x - | │ │ │ │ -00032280: 0a7c 2031 3020 2020 2020 2031 2031 3020 .| 10 1 10 │ │ │ │ -00032290: 2020 2020 2032 2031 3020 2020 2020 3420 2 10 4 │ │ │ │ -000322a0: 3130 2020 2020 2020 3620 3130 2020 2020 10 6 10 │ │ │ │ -000322b0: 2020 3020 3131 2020 2020 2020 3220 3131 0 11 2 11 │ │ │ │ -000322c0: 2020 2020 2034 2031 3120 2020 2020 207c 4 11 | │ │ │ │ -000322d0: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00032220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 │ │ │ │ +00032260: 2020 202d 2032 3578 2078 2020 202d 2032 - 25x x - 2 │ │ │ │ +00032270: 3478 2078 2020 202d 2032 7820 7820 2020 4x x - 2x x │ │ │ │ +00032280: 2d20 2020 207c 0a7c 2031 3020 2020 2020 - |.| 10 │ │ │ │ +00032290: 2031 2031 3020 2020 2020 2032 2031 3020 1 10 2 10 │ │ │ │ +000322a0: 2020 2020 3420 3130 2020 2020 2020 3620 4 10 6 │ │ │ │ +000322b0: 3130 2020 2020 2020 3020 3131 2020 2020 10 0 11 │ │ │ │ +000322c0: 2020 3220 3131 2020 2020 2034 2031 3120 2 11 4 11 │ │ │ │ +000322d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 000322e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000322f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032310: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -00032320: 0a7c 2020 2020 2020 2020 2020 2020 2020 .| │ │ │ │ +00032310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032320: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ 00032330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00032350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00032360: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -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 .|-------------- │ │ │ │ +00032360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032370: 2020 2020 207c 0a7c 7820 2020 2b20 3334 |.|x + 34 │ │ │ │ +00032380: 7820 7820 2020 2b20 3132 7820 7820 2020 x x + 12x x │ │ │ │ +00032390: 2d20 3235 7820 7820 2020 2b20 3230 7820 - 25x x + 20x │ │ │ │ +000323a0: 7820 2020 2d20 3578 2078 2020 202b 2039 x - 5x x + 9 │ │ │ │ +000323b0: 7820 7820 2020 2d20 3435 7820 7820 2020 x x - 45x x │ │ │ │ +000323c0: 2b20 2020 207c 0a7c 2031 3020 2020 2020 + |.| 10 │ │ │ │ +000323d0: 2032 2031 3020 2020 2020 2034 2031 3020 2 10 4 10 │ │ │ │ +000323e0: 2020 2020 2036 2031 3020 2020 2020 2030 6 10 0 │ │ │ │ +000323f0: 2031 3120 2020 2020 3120 3131 2020 2020 11 1 11 │ │ │ │ +00032400: 2032 2031 3120 2020 2020 2034 2031 3120 2 11 4 11 │ │ │ │ +00032410: 2020 2020 207c 0a7c 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 2d7c ---------------| │ │ │ │ -00032460: 0a7c 3135 7820 7820 202c 2020 2020 2020 .|15x x , │ │ │ │ +00032450: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00032460: 2d2d 2d2d 2d7c 0a7c 3135 7820 7820 202c -----|.|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 207c | │ │ │ │ -000324b0: 0a7c 2020 2035 2031 3120 2020 2020 2020 .| 5 11 │ │ │ │ +000324a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000324b0: 2020 2020 207c 0a7c 2020 2035 2031 3120 |.| 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 207c | │ │ │ │ -00032500: 0a7c 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 |.| │ │ │ │ 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 207c | │ │ │ │ -00032550: 0a7c 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 |.| │ │ │ │ 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 207c | │ │ │ │ -000325a0: 0a7c 3235 7820 7820 202c 2020 2020 2020 .|25x x , │ │ │ │ +00032590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000325a0: 2020 2020 207c 0a7c 3235 7820 7820 202c |.|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 207c | │ │ │ │ -000325f0: 0a7c 2020 2035 2031 3120 2020 2020 2020 .| 5 11 │ │ │ │ +000325e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000325f0: 2020 2020 207c 0a7c 2020 2035 2031 3120 |.| 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 207c | │ │ │ │ -00032640: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +00032630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00032640: 2020 2020 207c 0a2b 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 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 │ │ │ │ +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) │ │ │ │ 000326c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000326d0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ -000326e0: 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d .+-------------- │ │ │ │ +000326d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000326e0: 2020 2020 207c 0a2b 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 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:.========= │ │ │ │ +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:.=== │ │ │ │ 00032990: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -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,...-------- │ │ │ │ +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,...-- │ │ │ │ 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 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00032b60: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00032b70: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00032b80: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00032b90: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00032ba0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+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 │ │ │ │ -00032bd0: 6174 696f 6e2e 6d32 3a32 3237 3a30 2e0a ation.m2:227:0.. │ │ │ │ -00032be0: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e ..File: Cremona. │ │ │ │ -00032bf0: 696e 666f 2c20 4e6f 6465 3a20 426c 6f77 info, Node: Blow │ │ │ │ -00032c00: 5570 5374 7261 7465 6779 2c20 4e65 7874 UpStrategy, Next │ │ │ │ -00032c10: 3a20 4365 7274 6966 792c 2050 7265 763a : Certify, Prev: │ │ │ │ -00032c20: 2061 7070 726f 7869 6d61 7465 496e 7665 approximateInve │ │ │ │ -00032c30: 7273 654d 6170 2c20 5570 3a20 546f 700a rseMap, Up: Top. │ │ │ │ -00032c40: 0a42 6c6f 7755 7053 7472 6174 6567 790a .BlowUpStrategy. │ │ │ │ -00032c50: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a **************.. │ │ │ │ -00032c60: 4465 7363 7269 7074 696f 6e0a 3d3d 3d3d Description.==== │ │ │ │ -00032c70: 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 6973 =======..This is │ │ │ │ -00032c80: 2061 6e20 6f70 7469 6f6e 616c 2061 7267 an optional arg │ │ │ │ -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 │ │ │ │ -00032ce0: 656e 746c 792c 2074 6865 2070 6f73 7369 ently, the possi │ │ │ │ -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 │ │ │ │ -00032d20: 2069 6e64 6963 6174 6520 7477 6f20 6469 indicate two di │ │ │ │ -00032d30: 6666 6572 656e 7420 7761 7973 206f 6620 fferent ways of │ │ │ │ -00032d40: 636f 6d70 7574 696e 6720 7468 6520 2863 computing the (c │ │ │ │ -00032d50: 6c6f 7375 7265 206f 6620 7468 6529 0a67 losure of the).g │ │ │ │ -00032d60: 7261 7068 206f 6620 6120 7261 7469 6f6e raph of a ration │ │ │ │ -00032d70: 616c 206d 6170 2e20 5468 6520 6465 6661 al map. The defa │ │ │ │ -00032d80: 756c 7420 6368 6f69 6365 2069 7320 2245 ult choice is "E │ │ │ │ -00032d90: 6c69 6d69 6e61 7465 2220 616e 6420 7468 liminate" and th │ │ │ │ -00032da0: 6973 2069 730a 6765 6e65 7261 6c6c 7920 is is.generally │ │ │ │ -00032db0: 7072 6566 6572 6162 6c65 2e0a 0a46 756e preferable...Fun │ │ │ │ -00032dc0: 6374 696f 6e73 2077 6974 6820 6f70 7469 ctions with opti │ │ │ │ -00032dd0: 6f6e 616c 2061 7267 756d 656e 7420 6e61 onal argument na │ │ │ │ -00032de0: 6d65 6420 426c 6f77 5570 5374 7261 7465 med BlowUpStrate │ │ │ │ -00032df0: 6779 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d gy:.============ │ │ │ │ +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 3236 2e30 362b 6473 2f4d ay2-1.26.06+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 │ │ │ │ +00032cd0: 656e 7475 616c 6c79 2063 616c 6c20 6974 entually call it │ │ │ │ +00032ce0: 2e20 4375 7272 656e 746c 792c 2074 6865 . Currently, the │ │ │ │ +00032cf0: 2070 6f73 7369 626c 6520 7661 6c75 6573 possible values │ │ │ │ +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 │ │ │ │ +00032d40: 7973 206f 6620 636f 6d70 7574 696e 6720 ys of computing │ │ │ │ +00032d50: 7468 6520 2863 6c6f 7375 7265 206f 6620 the (closure of │ │ │ │ +00032d60: 7468 6529 0a67 7261 7068 206f 6620 6120 the).graph of a │ │ │ │ +00032d70: 7261 7469 6f6e 616c 206d 6170 2e20 5468 rational map. Th │ │ │ │ +00032d80: 6520 6465 6661 756c 7420 6368 6f69 6365 e default choice │ │ │ │ +00032d90: 2069 7320 2245 6c69 6d69 6e61 7465 2220 is "Eliminate" │ │ │ │ +00032da0: 616e 6420 7468 6973 2069 730a 6765 6e65 and this is.gene │ │ │ │ +00032db0: 7261 6c6c 7920 7072 6566 6572 6162 6c65 rally preferable │ │ │ │ +00032dc0: 2e0a 0a46 756e 6374 696f 6e73 2077 6974 ...Functions wit │ │ │ │ +00032dd0: 6820 6f70 7469 6f6e 616c 2061 7267 756d h optional argum │ │ │ │ +00032de0: 656e 7420 6e61 6d65 6420 426c 6f77 5570 ent named BlowUp │ │ │ │ +00032df0: 5374 7261 7465 6779 3a0a 3d3d 3d3d 3d3d Strategy:.====== │ │ │ │ 00032e00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00032e10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00032e20: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00032e30: 2243 6865 726e 5363 6877 6172 747a 4d61 "ChernSchwartzMa │ │ │ │ -00032e40: 6350 6865 7273 6f6e 282e 2e2e 2c42 6c6f cPherson(...,Blo │ │ │ │ -00032e50: 7755 7053 7472 6174 6567 793d 3e2e 2e2e wUpStrategy=>... │ │ │ │ -00032e60: 2922 0a20 202a 2022 6465 6772 6565 4d61 )". * "degreeMa │ │ │ │ -00032e70: 7028 2e2e 2e2c 426c 6f77 5570 5374 7261 p(...,BlowUpStra │ │ │ │ -00032e80: 7465 6779 3d3e 2e2e 2e29 220a 2020 2a20 tegy=>...)". * │ │ │ │ -00032e90: 2245 756c 6572 4368 6172 6163 7465 7269 "EulerCharacteri │ │ │ │ -00032ea0: 7374 6963 282e 2e2e 2c42 6c6f 7755 7053 stic(...,BlowUpS │ │ │ │ -00032eb0: 7472 6174 6567 793d 3e2e 2e2e 2922 0a20 trategy=>...)". │ │ │ │ -00032ec0: 202a 2022 6772 6170 6828 2e2e 2e2c 426c * "graph(...,Bl │ │ │ │ -00032ed0: 6f77 5570 5374 7261 7465 6779 3d3e 2e2e owUpStrategy=>.. │ │ │ │ -00032ee0: 2e29 220a 2020 2a20 2269 6e76 6572 7365 .)". * "inverse │ │ │ │ -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( │ │ │ │ -00032f20: 2e2e 2e2c 426c 6f77 5570 5374 7261 7465 ...,BlowUpStrate │ │ │ │ -00032f30: 6779 3d3e 2e2e 2e29 220a 2020 2a20 2270 gy=>...)". * "p │ │ │ │ -00032f40: 726f 6a65 6374 6976 6544 6567 7265 6573 rojectiveDegrees │ │ │ │ -00032f50: 282e 2e2e 2c42 6c6f 7755 7053 7472 6174 (...,BlowUpStrat │ │ │ │ -00032f60: 6567 793d 3e2e 2e2e 2922 0a20 202a 2022 egy=>...)". * " │ │ │ │ -00032f70: 5365 6772 6543 6c61 7373 282e 2e2e 2c42 SegreClass(...,B │ │ │ │ -00032f80: 6c6f 7755 7053 7472 6174 6567 793d 3e2e lowUpStrategy=>. │ │ │ │ -00032f90: 2e2e 2922 0a0a 466f 7220 7468 6520 7072 ..)"..For the pr │ │ │ │ -00032fa0: 6f67 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d ogrammer.======= │ │ │ │ -00032fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 ===========..The │ │ │ │ -00032fc0: 206f 626a 6563 7420 2a6e 6f74 6520 426c object *note Bl │ │ │ │ -00032fd0: 6f77 5570 5374 7261 7465 6779 3a20 426c owUpStrategy: Bl │ │ │ │ -00032fe0: 6f77 5570 5374 7261 7465 6779 2c20 6973 owUpStrategy, is │ │ │ │ -00032ff0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ -00033000: 0a28 4d61 6361 756c 6179 3244 6f63 2953 .(Macaulay2Doc)S │ │ │ │ -00033010: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ +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 │ │ │ │ +00032e60: 793d 3e2e 2e2e 2922 0a20 202a 2022 6465 y=>...)". * "de │ │ │ │ +00032e70: 6772 6565 4d61 7028 2e2e 2e2c 426c 6f77 greeMap(...,Blow │ │ │ │ +00032e80: 5570 5374 7261 7465 6779 3d3e 2e2e 2e29 UpStrategy=>...) │ │ │ │ +00032e90: 220a 2020 2a20 2245 756c 6572 4368 6172 ". * "EulerChar │ │ │ │ +00032ea0: 6163 7465 7269 7374 6963 282e 2e2e 2c42 acteristic(...,B │ │ │ │ +00032eb0: 6c6f 7755 7053 7472 6174 6567 793d 3e2e lowUpStrategy=>. │ │ │ │ +00032ec0: 2e2e 2922 0a20 202a 2022 6772 6170 6828 ..)". * "graph( │ │ │ │ +00032ed0: 2e2e 2e2c 426c 6f77 5570 5374 7261 7465 ...,BlowUpStrate │ │ │ │ +00032ee0: 6779 3d3e 2e2e 2e29 220a 2020 2a20 2269 gy=>...)". * "i │ │ │ │ +00032ef0: 6e76 6572 7365 4d61 7028 2e2e 2e2c 426c nverseMap(...,Bl │ │ │ │ +00032f00: 6f77 5570 5374 7261 7465 6779 3d3e 2e2e owUpStrategy=>.. │ │ │ │ +00032f10: 2e29 220a 2020 2a20 2269 7342 6972 6174 .)". * "isBirat │ │ │ │ +00032f20: 696f 6e61 6c28 2e2e 2e2c 426c 6f77 5570 ional(...,BlowUp │ │ │ │ +00032f30: 5374 7261 7465 6779 3d3e 2e2e 2e29 220a Strategy=>...)". │ │ │ │ +00032f40: 2020 2a20 2270 726f 6a65 6374 6976 6544 * "projectiveD │ │ │ │ +00032f50: 6567 7265 6573 282e 2e2e 2c42 6c6f 7755 egrees(...,BlowU │ │ │ │ +00032f60: 7053 7472 6174 6567 793d 3e2e 2e2e 2922 pStrategy=>...)" │ │ │ │ +00032f70: 0a20 202a 2022 5365 6772 6543 6c61 7373 . * "SegreClass │ │ │ │ +00032f80: 282e 2e2e 2c42 6c6f 7755 7053 7472 6174 (...,BlowUpStrat │ │ │ │ +00032f90: 6567 793d 3e2e 2e2e 2922 0a0a 466f 7220 egy=>...)"..For │ │ │ │ +00032fa0: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ +00032fb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00032fc0: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ +00032fd0: 6f74 6520 426c 6f77 5570 5374 7261 7465 ote BlowUpStrate │ │ │ │ +00032fe0: 6779 3a20 426c 6f77 5570 5374 7261 7465 gy: BlowUpStrate │ │ │ │ +00032ff0: 6779 2c20 6973 2061 202a 6e6f 7465 2073 gy, is a *note s │ │ │ │ +00033000: 796d 626f 6c3a 0a28 4d61 6361 756c 6179 ymbol:.(Macaulay │ │ │ │ +00033010: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ 00033020: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033030: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033040: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033050: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00033060: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00033070: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00033080: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00033090: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -000330a0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -000330b0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -000330c0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -000330d0: 4372 656d 6f6e 612f 0a64 6f63 756d 656e Cremona/.documen │ │ │ │ -000330e0: 7461 7469 6f6e 2e6d 323a 3936 333a 302e tation.m2:963:0. │ │ │ │ -000330f0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61 ...File: Cremona │ │ │ │ -00033100: 2e69 6e66 6f2c 204e 6f64 653a 2043 6572 .info, Node: Cer │ │ │ │ -00033110: 7469 6679 2c20 4e65 7874 3a20 4368 6572 tify, Next: Cher │ │ │ │ -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 │ │ │ │ -00033170: 2063 6f72 7265 6374 6e65 7373 206f 6620 correctness of │ │ │ │ -00033180: 6f75 7470 7574 0a2a 2a2a 2a2a 2a2a 2a2a output.********* │ │ │ │ +00033060: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00033070: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00033080: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00033090: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +000330a0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +000330b0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +000330c0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +000330d0: 6b61 6765 732f 4372 656d 6f6e 612f 0a64 kages/Cremona/.d │ │ │ │ +000330e0: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a ocumentation.m2: │ │ │ │ +000330f0: 3936 333a 302e 0a1f 0a46 696c 653a 2043 963:0....File: C │ │ │ │ +00033100: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 remona.info, Nod │ │ │ │ +00033110: 653a 2043 6572 7469 6679 2c20 4e65 7874 e: Certify, Next │ │ │ │ +00033120: 3a20 4368 6572 6e53 6368 7761 7274 7a4d : ChernSchwartzM │ │ │ │ +00033130: 6163 5068 6572 736f 6e2c 2050 7265 763a acPherson, Prev: │ │ │ │ +00033140: 2042 6c6f 7755 7053 7472 6174 6567 792c BlowUpStrategy, │ │ │ │ +00033150: 2055 703a 2054 6f70 0a0a 4365 7274 6966 Up: Top..Certif │ │ │ │ +00033160: 7920 2d2d 2077 6865 7468 6572 2074 6f20 y -- whether to │ │ │ │ +00033170: 656e 7375 7265 2063 6f72 7265 6374 6e65 ensure correctne │ │ │ │ +00033180: 7373 206f 6620 6f75 7470 7574 0a2a 2a2a ss of output.*** │ │ │ │ 00033190: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000331a0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000331b0: 2a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 6372 *********..Descr │ │ │ │ -000331c0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -000331d0: 3d3d 0a0a 5468 6973 206f 7074 696f 6e20 ==..This option │ │ │ │ -000331e0: 6163 6365 7074 7320 6120 2a6e 6f74 6520 accepts a *note │ │ │ │ -000331f0: 426f 6f6c 6561 6e3a 2028 4d61 6361 756c Boolean: (Macaul │ │ │ │ -00033200: 6179 3244 6f63 2942 6f6f 6c65 616e 2c20 ay2Doc)Boolean, │ │ │ │ -00033210: 7661 6c75 652c 2064 6566 6175 6c74 0a76 value, default.v │ │ │ │ -00033220: 616c 7565 2066 616c 7365 2e0a 0a49 6620 alue false...If │ │ │ │ -00033230: 7475 726e 6564 206f 6e20 696e 2074 6865 turned on in the │ │ │ │ -00033240: 206d 6574 686f 6473 202a 6e6f 7465 2069 methods *note i │ │ │ │ -00033250: 6e76 6572 7365 4d61 703a 2069 6e76 6572 nverseMap: inver │ │ │ │ -00033260: 7365 4d61 702c 2061 6e64 202a 6e6f 7465 seMap, and *note │ │ │ │ -00033270: 0a61 7070 726f 7869 6d61 7465 496e 7665 .approximateInve │ │ │ │ -00033280: 7273 654d 6170 3a20 6170 7072 6f78 696d rseMap: approxim │ │ │ │ -00033290: 6174 6549 6e76 6572 7365 4d61 702c 2c20 ateInverseMap,, │ │ │ │ -000332a0: 7468 656e 2069 7420 7769 6c6c 2062 6520 then it will be │ │ │ │ -000332b0: 6368 6563 6b65 6420 7768 6574 6865 720a checked whether. │ │ │ │ -000332c0: 7468 6520 6d61 7073 2069 6e20 696e 7075 the maps in inpu │ │ │ │ -000332d0: 7420 616e 6420 6f75 7470 7574 2061 7265 t and output are │ │ │ │ -000332e0: 206f 6e65 2074 6865 2069 6e76 6572 7365 one the inverse │ │ │ │ -000332f0: 206f 6620 7468 6520 6f74 6865 722c 2074 of the other, t │ │ │ │ -00033300: 6872 6f77 696e 6720 616e 0a65 7272 6f72 hrowing an.error │ │ │ │ -00033310: 2069 6620 7468 6579 2061 7265 206e 6f74 if they are not │ │ │ │ -00033320: 2e20 4163 7475 616c 6c79 2c20 2a6e 6f74 . Actually, *not │ │ │ │ -00033330: 6520 6170 7072 6f78 696d 6174 6549 6e76 e approximateInv │ │ │ │ -00033340: 6572 7365 4d61 703a 0a61 7070 726f 7869 erseMap:.approxi │ │ │ │ -00033350: 6d61 7465 496e 7665 7273 654d 6170 2c20 mateInverseMap, │ │ │ │ -00033360: 7769 6c6c 2066 6972 7374 2074 7279 2074 will first try t │ │ │ │ -00033370: 6f20 6669 7820 7468 6520 6572 726f 7220 o fix the error │ │ │ │ -00033380: 6f66 2074 6865 2061 7070 726f 7869 6d61 of the approxima │ │ │ │ -00033390: 7469 6f6e 2e0a 5768 656e 2074 7572 6e65 tion..When turne │ │ │ │ -000333a0: 6420 6f6e 2069 6e20 7468 6520 6d65 7468 d on in the meth │ │ │ │ -000333b0: 6f64 7320 2a6e 6f74 6520 7072 6f6a 6563 ods *note projec │ │ │ │ -000333c0: 7469 7665 4465 6772 6565 733a 2070 726f tiveDegrees: pro │ │ │ │ -000333d0: 6a65 6374 6976 6544 6567 7265 6573 2c2c jectiveDegrees,, │ │ │ │ -000333e0: 0a2a 6e6f 7465 2064 6567 7265 654d 6170 .*note degreeMap │ │ │ │ -000333f0: 3a20 6465 6772 6565 4d61 702c 2c20 2a6e : degreeMap,, *n │ │ │ │ -00033400: 6f74 6520 6973 4269 7261 7469 6f6e 616c ote isBirational │ │ │ │ -00033410: 3a20 6973 4269 7261 7469 6f6e 616c 2c2c : isBirational,, │ │ │ │ -00033420: 202a 6e6f 7465 0a69 7344 6f6d 696e 616e *note.isDominan │ │ │ │ -00033430: 743a 2069 7344 6f6d 696e 616e 742c 2c20 t: isDominant,, │ │ │ │ -00033440: 2a6e 6f74 6520 5365 6772 6543 6c61 7373 *note SegreClass │ │ │ │ -00033450: 3a20 5365 6772 6543 6c61 7373 2c2c 202a : SegreClass,, * │ │ │ │ -00033460: 6e6f 7465 0a45 756c 6572 4368 6172 6163 note.EulerCharac │ │ │ │ -00033470: 7465 7269 7374 6963 3a20 4575 6c65 7243 teristic: EulerC │ │ │ │ -00033480: 6861 7261 6374 6572 6973 7469 632c 2061 haracteristic, a │ │ │ │ -00033490: 6e64 202a 6e6f 7465 2043 6865 726e 5363 nd *note ChernSc │ │ │ │ -000334a0: 6877 6172 747a 4d61 6350 6865 7273 6f6e hwartzMacPherson │ │ │ │ -000334b0: 3a0a 4368 6572 6e53 6368 7761 7274 7a4d :.ChernSchwartzM │ │ │ │ -000334c0: 6163 5068 6572 736f 6e2c 2c20 6974 206d acPherson,, it m │ │ │ │ -000334d0: 6561 6e73 2077 6865 7468 6572 2074 6f20 eans whether to │ │ │ │ -000334e0: 7573 6520 6120 6e6f 6e2d 7072 6f62 6162 use a non-probab │ │ │ │ -000334f0: 696c 6973 7469 630a 616c 676f 7269 7468 ilistic.algorith │ │ │ │ -00033500: 6d2e 0a0a 4675 6e63 7469 6f6e 7320 7769 m...Functions wi │ │ │ │ -00033510: 7468 206f 7074 696f 6e61 6c20 6172 6775 th optional argu │ │ │ │ -00033520: 6d65 6e74 206e 616d 6564 2043 6572 7469 ment named Certi │ │ │ │ -00033530: 6679 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d fy:.============ │ │ │ │ +000331b0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ +000331c0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +000331d0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 206f ========..This o │ │ │ │ +000331e0: 7074 696f 6e20 6163 6365 7074 7320 6120 ption accepts a │ │ │ │ +000331f0: 2a6e 6f74 6520 426f 6f6c 6561 6e3a 2028 *note Boolean: ( │ │ │ │ +00033200: 4d61 6361 756c 6179 3244 6f63 2942 6f6f Macaulay2Doc)Boo │ │ │ │ +00033210: 6c65 616e 2c20 7661 6c75 652c 2064 6566 lean, value, def │ │ │ │ +00033220: 6175 6c74 0a76 616c 7565 2066 616c 7365 ault.value false │ │ │ │ +00033230: 2e0a 0a49 6620 7475 726e 6564 206f 6e20 ...If turned on │ │ │ │ +00033240: 696e 2074 6865 206d 6574 686f 6473 202a in the methods * │ │ │ │ +00033250: 6e6f 7465 2069 6e76 6572 7365 4d61 703a note inverseMap: │ │ │ │ +00033260: 2069 6e76 6572 7365 4d61 702c 2061 6e64 inverseMap, and │ │ │ │ +00033270: 202a 6e6f 7465 0a61 7070 726f 7869 6d61 *note.approxima │ │ │ │ +00033280: 7465 496e 7665 7273 654d 6170 3a20 6170 teInverseMap: ap │ │ │ │ +00033290: 7072 6f78 696d 6174 6549 6e76 6572 7365 proximateInverse │ │ │ │ +000332a0: 4d61 702c 2c20 7468 656e 2069 7420 7769 Map,, then it wi │ │ │ │ +000332b0: 6c6c 2062 6520 6368 6563 6b65 6420 7768 ll be checked wh │ │ │ │ +000332c0: 6574 6865 720a 7468 6520 6d61 7073 2069 ether.the maps i │ │ │ │ +000332d0: 6e20 696e 7075 7420 616e 6420 6f75 7470 n input and outp │ │ │ │ +000332e0: 7574 2061 7265 206f 6e65 2074 6865 2069 ut are one the i │ │ │ │ +000332f0: 6e76 6572 7365 206f 6620 7468 6520 6f74 nverse of the ot │ │ │ │ +00033300: 6865 722c 2074 6872 6f77 696e 6720 616e her, throwing an │ │ │ │ +00033310: 0a65 7272 6f72 2069 6620 7468 6579 2061 .error if they a │ │ │ │ +00033320: 7265 206e 6f74 2e20 4163 7475 616c 6c79 re not. Actually │ │ │ │ +00033330: 2c20 2a6e 6f74 6520 6170 7072 6f78 696d , *note approxim │ │ │ │ +00033340: 6174 6549 6e76 6572 7365 4d61 703a 0a61 ateInverseMap:.a │ │ │ │ +00033350: 7070 726f 7869 6d61 7465 496e 7665 7273 pproximateInvers │ │ │ │ +00033360: 654d 6170 2c20 7769 6c6c 2066 6972 7374 eMap, will first │ │ │ │ +00033370: 2074 7279 2074 6f20 6669 7820 7468 6520 try to fix the │ │ │ │ +00033380: 6572 726f 7220 6f66 2074 6865 2061 7070 error of the app │ │ │ │ +00033390: 726f 7869 6d61 7469 6f6e 2e0a 5768 656e roximation..When │ │ │ │ +000333a0: 2074 7572 6e65 6420 6f6e 2069 6e20 7468 turned on in th │ │ │ │ +000333b0: 6520 6d65 7468 6f64 7320 2a6e 6f74 6520 e methods *note │ │ │ │ +000333c0: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ +000333d0: 733a 2070 726f 6a65 6374 6976 6544 6567 s: projectiveDeg │ │ │ │ +000333e0: 7265 6573 2c2c 0a2a 6e6f 7465 2064 6567 rees,,.*note deg │ │ │ │ +000333f0: 7265 654d 6170 3a20 6465 6772 6565 4d61 reeMap: degreeMa │ │ │ │ +00033400: 702c 2c20 2a6e 6f74 6520 6973 4269 7261 p,, *note isBira │ │ │ │ +00033410: 7469 6f6e 616c 3a20 6973 4269 7261 7469 tional: isBirati │ │ │ │ +00033420: 6f6e 616c 2c2c 202a 6e6f 7465 0a69 7344 onal,, *note.isD │ │ │ │ +00033430: 6f6d 696e 616e 743a 2069 7344 6f6d 696e ominant: isDomin │ │ │ │ +00033440: 616e 742c 2c20 2a6e 6f74 6520 5365 6772 ant,, *note Segr │ │ │ │ +00033450: 6543 6c61 7373 3a20 5365 6772 6543 6c61 eClass: SegreCla │ │ │ │ +00033460: 7373 2c2c 202a 6e6f 7465 0a45 756c 6572 ss,, *note.Euler │ │ │ │ +00033470: 4368 6172 6163 7465 7269 7374 6963 3a20 Characteristic: │ │ │ │ +00033480: 4575 6c65 7243 6861 7261 6374 6572 6973 EulerCharacteris │ │ │ │ +00033490: 7469 632c 2061 6e64 202a 6e6f 7465 2043 tic, and *note C │ │ │ │ +000334a0: 6865 726e 5363 6877 6172 747a 4d61 6350 hernSchwartzMacP │ │ │ │ +000334b0: 6865 7273 6f6e 3a0a 4368 6572 6e53 6368 herson:.ChernSch │ │ │ │ +000334c0: 7761 7274 7a4d 6163 5068 6572 736f 6e2c wartzMacPherson, │ │ │ │ +000334d0: 2c20 6974 206d 6561 6e73 2077 6865 7468 , it means wheth │ │ │ │ +000334e0: 6572 2074 6f20 7573 6520 6120 6e6f 6e2d er to use a non- │ │ │ │ +000334f0: 7072 6f62 6162 696c 6973 7469 630a 616c probabilistic.al │ │ │ │ +00033500: 676f 7269 7468 6d2e 0a0a 4675 6e63 7469 gorithm...Functi │ │ │ │ +00033510: 6f6e 7320 7769 7468 206f 7074 696f 6e61 ons with optiona │ │ │ │ +00033520: 6c20 6172 6775 6d65 6e74 206e 616d 6564 l argument named │ │ │ │ +00033530: 2043 6572 7469 6679 3a0a 3d3d 3d3d 3d3d Certify:.====== │ │ │ │ 00033540: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00033550: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00033560: 3d3d 3d0a 0a20 202a 2022 6170 7072 6f78 ===.. * "approx │ │ │ │ -00033570: 696d 6174 6549 6e76 6572 7365 4d61 7028 imateInverseMap( │ │ │ │ -00033580: 2e2e 2e2c 4365 7274 6966 793d 3e2e 2e2e ...,Certify=>... │ │ │ │ -00033590: 2922 0a20 202a 2022 4368 6572 6e53 6368 )". * "ChernSch │ │ │ │ -000335a0: 7761 7274 7a4d 6163 5068 6572 736f 6e28 wartzMacPherson( │ │ │ │ -000335b0: 2e2e 2e2c 4365 7274 6966 793d 3e2e 2e2e ...,Certify=>... │ │ │ │ -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 │ │ │ │ -000335f0: 6861 7261 6374 6572 6973 7469 6328 2e2e haracteristic(.. │ │ │ │ -00033600: 2e2c 4365 7274 6966 793d 3e2e 2e2e 2922 .,Certify=>...)" │ │ │ │ -00033610: 0a20 202a 2022 6578 6365 7074 696f 6e61 . * "exceptiona │ │ │ │ -00033620: 6c4c 6f63 7573 282e 2e2e 2c43 6572 7469 lLocus(...,Certi │ │ │ │ -00033630: 6679 3d3e 2e2e 2e29 220a 2020 2a20 2269 fy=>...)". * "i │ │ │ │ -00033640: 6e76 6572 7365 4d61 7028 2e2e 2e2c 4365 nverseMap(...,Ce │ │ │ │ -00033650: 7274 6966 793d 3e2e 2e2e 2922 0a20 202a rtify=>...)". * │ │ │ │ -00033660: 2022 6973 4269 7261 7469 6f6e 616c 282e "isBirational(. │ │ │ │ -00033670: 2e2e 2c43 6572 7469 6679 3d3e 2e2e 2e29 ..,Certify=>...) │ │ │ │ -00033680: 220a 2020 2a20 2269 7344 6f6d 696e 616e ". * "isDominan │ │ │ │ -00033690: 7428 2e2e 2e2c 4365 7274 6966 793d 3e2e t(...,Certify=>. │ │ │ │ -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 │ │ │ │ -00033720: 6a65 6374 202a 6e6f 7465 2043 6572 7469 ject *note Certi │ │ │ │ -00033730: 6679 3a20 4365 7274 6966 792c 2069 7320 fy: Certify, is │ │ │ │ -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,...-------- │ │ │ │ +00033560: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 2022 =========.. * " │ │ │ │ +00033570: 6170 7072 6f78 696d 6174 6549 6e76 6572 approximateInver │ │ │ │ +00033580: 7365 4d61 7028 2e2e 2e2c 4365 7274 6966 seMap(...,Certif │ │ │ │ +00033590: 793d 3e2e 2e2e 2922 0a20 202a 2022 4368 y=>...)". * "Ch │ │ │ │ +000335a0: 6572 6e53 6368 7761 7274 7a4d 6163 5068 ernSchwartzMacPh │ │ │ │ +000335b0: 6572 736f 6e28 2e2e 2e2c 4365 7274 6966 erson(...,Certif │ │ │ │ +000335c0: 793d 3e2e 2e2e 2922 0a20 202a 2022 6465 y=>...)". * "de │ │ │ │ +000335d0: 6772 6565 4d61 7028 2e2e 2e2c 4365 7274 greeMap(...,Cert │ │ │ │ +000335e0: 6966 793d 3e2e 2e2e 2922 0a20 202a 2022 ify=>...)". * " │ │ │ │ +000335f0: 4575 6c65 7243 6861 7261 6374 6572 6973 EulerCharacteris │ │ │ │ +00033600: 7469 6328 2e2e 2e2c 4365 7274 6966 793d tic(...,Certify= │ │ │ │ +00033610: 3e2e 2e2e 2922 0a20 202a 2022 6578 6365 >...)". * "exce │ │ │ │ +00033620: 7074 696f 6e61 6c4c 6f63 7573 282e 2e2e ptionalLocus(... │ │ │ │ +00033630: 2c43 6572 7469 6679 3d3e 2e2e 2e29 220a ,Certify=>...)". │ │ │ │ +00033640: 2020 2a20 2269 6e76 6572 7365 4d61 7028 * "inverseMap( │ │ │ │ +00033650: 2e2e 2e2c 4365 7274 6966 793d 3e2e 2e2e ...,Certify=>... │ │ │ │ +00033660: 2922 0a20 202a 2022 6973 4269 7261 7469 )". * "isBirati │ │ │ │ +00033670: 6f6e 616c 282e 2e2e 2c43 6572 7469 6679 onal(...,Certify │ │ │ │ +00033680: 3d3e 2e2e 2e29 220a 2020 2a20 2269 7344 =>...)". * "isD │ │ │ │ +00033690: 6f6d 696e 616e 7428 2e2e 2e2c 4365 7274 ominant(...,Cert │ │ │ │ +000336a0: 6966 793d 3e2e 2e2e 2922 0a20 202a 2022 ify=>...)". * " │ │ │ │ +000336b0: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ +000336c0: 7328 2e2e 2e2c 4365 7274 6966 793d 3e2e s(...,Certify=>. │ │ │ │ +000336d0: 2e2e 2922 0a20 202a 2022 5365 6772 6543 ..)". * "SegreC │ │ │ │ +000336e0: 6c61 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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 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Aluffi. See al │ │ │ │ -00033e00: 736f 2074 6865 2063 6f72 7265 7370 6f6e so the correspon │ │ │ │ -00033e10: 6469 6e67 206d 6574 686f 6473 2069 6e20 ding methods in │ │ │ │ -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 │ │ │ │ -00033e40: 7777 2e6d 6174 682e 6673 752e 6564 752f ww.math.fsu.edu/ │ │ │ │ -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 │ │ │ │ -00033ed0: 6e20 7468 6520 6578 616d 706c 6520 6265 n the example be │ │ │ │ -00033ee0: 6c6f 772c 2077 6520 636f 6d70 7574 6520 low, we compute │ │ │ │ -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 │ │ │ │ -00033f10: 6720 6f66 0a24 5c6d 6174 6862 627b 507d g of.$\mathbb{P} │ │ │ │ -00033f20: 5e34 2420 6f66 2074 6865 2043 6865 726e ^4$ 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 ...+------------ │ │ │ │ +00033940: 2a2a 2a2a 2a2a 2a0a 0a20 202a 2055 7361 *******.. * Usa │ │ │ │ +00033950: 6765 3a20 0a20 2020 2020 2020 2043 6865 ge: . Che │ │ │ │ +00033960: 726e 5363 6877 6172 747a 4d61 6350 6865 rnSchwartzMacPhe │ │ │ │ +00033970: 7273 6f6e 2049 0a20 202a 2049 6e70 7574 rson I. * Input │ │ │ │ +00033980: 733a 0a20 2020 2020 202a 2049 2c20 616e s:. * I, an │ │ │ │ +00033990: 202a 6e6f 7465 2069 6465 616c 3a20 284d *note ideal: (M │ │ │ │ +000339a0: 6163 6175 6c61 7932 446f 6329 4964 6561 acaulay2Doc)Idea │ │ │ │ +000339b0: 6c2c 2c20 6120 686f 6d6f 6765 6e65 6f75 l,, a homogeneou │ │ │ │ +000339c0: 7320 6964 6561 6c20 6465 6669 6e69 6e67 s ideal defining │ │ │ │ +000339d0: 2061 0a20 2020 2020 2020 2063 6c6f 7365 a. close │ │ │ │ +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 │ │ │ │ +00033a30: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ +00033a40: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ +00033a50: 2c3a 0a20 2020 2020 202a 202a 6e6f 7465 ,:. * *note │ │ │ │ +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 │ │ │ │ +00033ac0: 793a 2043 6572 7469 6679 2c20 3d3e 202e y: Certify, => . │ │ │ │ +00033ad0: 2e2e 2c20 6465 6661 756c 7420 7661 6c75 .., default valu │ │ │ │ +00033ae0: 6520 6661 6c73 652c 2077 6865 7468 6572 e false, whether │ │ │ │ +00033af0: 2074 6f20 656e 7375 7265 0a20 2020 2020 to ensure. │ │ │ │ +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 │ │ │ │ +00033b80: 7565 2c0a 2020 2a20 4f75 7470 7574 733a ue,. * Outputs: │ │ │ │ +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 │ │ │ │ +00033c90: 7465 7269 7374 6963 206f 6620 7468 6520 teristic of the │ │ │ │ +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.. │ │ │ │ +00033ce0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +00033cf0: 3d3d 3d3d 3d3d 3d3d 0a0a 5468 6973 2069 ========..This i │ │ │ │ +00033d00: 7320 616e 2065 7861 6d70 6c65 206f 6620 s an example of │ │ │ │ +00033d10: 6170 706c 6963 6174 696f 6e20 6f66 2074 application of t │ │ │ │ +00033d20: 6865 206d 6574 686f 6420 2a6e 6f74 6520 he method *note │ │ │ │ +00033d30: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ +00033d40: 733a 0a70 726f 6a65 6374 6976 6544 6567 s:.projectiveDeg │ │ │ │ +00033d50: 7265 6573 2c2c 2064 7565 2074 6f20 7265 rees,, due to re │ │ │ │ +00033d60: 7375 6c74 7320 7368 6f77 6e20 696e 2043 sults shown in C │ │ │ │ +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 │ │ │ │ +00033da0: 6865 6d65 7320 2873 6565 2068 7474 703a hemes (see http: │ │ │ │ +00033db0: 2f2f 7777 772e 7363 6965 6e63 6564 6972 //www.sciencedir │ │ │ │ +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 ) │ │ │ │ +00033df0: 2c20 6279 2050 2e20 416c 7566 6669 2e20 , by P. Aluffi. │ │ │ │ +00033e00: 5365 6520 616c 736f 2074 6865 2063 6f72 See also the cor │ │ │ │ +00033e10: 7265 7370 6f6e 6469 6e67 206d 6574 686f responding metho │ │ │ │ +00033e20: 6473 2069 6e20 7468 650a 7061 636b 6167 ds in the.packag │ │ │ │ +00033e30: 6573 2043 534d 2d41 2028 7365 6520 6874 es CSM-A (see ht │ │ │ │ +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 │ │ │ │ +00033e90: 7374 6963 436c 6173 7365 733a 2028 4368 sticClasses: (Ch │ │ │ │ +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 │ │ │ │ +00033ef0: 6d70 7574 6520 7468 6520 7075 7368 2d66 mpute the push-f │ │ │ │ +00033f00: 6f72 7761 7264 2074 6f20 7468 6520 4368 orward to the Ch │ │ │ │ +00033f10: 6f77 2072 696e 6720 6f66 0a24 5c6d 6174 ow ring of.$\mat │ │ │ │ +00033f20: 6862 627b 507d 5e34 2420 6f66 2074 6865 hbb{P}^4$ of the │ │ │ │ +00033f30: 2043 6865 726e 2d53 6368 7761 7274 7a2d Chern-Schwartz- │ │ │ │ +00033f40: 4d61 6350 6865 7273 6f6e 2063 6c61 7373 MacPherson class │ │ │ │ +00033f50: 206f 6620 7468 6520 636f 6e65 206f 7665 of the cone ove │ │ │ │ +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...+------ │ │ │ │ 00033fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00033fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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] │ │ │ │ +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] │ │ │ │ 00034010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034020: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00034020: 2020 2020 2020 2020 207c 0a7c 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 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 ] │ │ │ │ +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 ] │ │ │ │ 00034080: 2020 2020 2020 2020 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 | │ │ │ │ +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 |.| │ │ │ │ 000340e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000340f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00034130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034140: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00034140: 2020 2020 2020 7c0a 2b2d 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 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 })|.| │ │ │ │ +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}})|.| │ │ │ │ 000341c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000341d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 |.| │ │ │ │ +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 |.| │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 00034330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034340: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00034340: 2020 2020 2020 207c 0a2b 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 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 2031 |.| -- used 1 │ │ │ │ -000343c0: 2e32 3639 3536 7320 2863 7075 293b 2030 .26956s (cpu); 0 │ │ │ │ -000343d0: 2e38 3934 3033 3173 2028 7468 7265 6164 .894031s (thread │ │ │ │ -000343e0: 293b 2030 7320 2867 6329 2020 7c0a 7c20 ); 0s (gc) |.| │ │ │ │ -000343f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00034370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00034380: 2b0a 7c69 3320 3a20 7469 6d65 2043 6865 +.|i3 : time Che │ │ │ │ +00034390: 726e 5363 6877 6172 747a 4d61 6350 6865 rnSchwartzMacPhe │ │ │ │ +000343a0: 7273 6f6e 2043 2020 2020 2020 2020 2020 rson C │ │ │ │ +000343b0: 2020 2020 2020 2020 207c 0a7c 202d 2d20 |.| -- │ │ │ │ +000343c0: 7573 6564 2031 2e30 3931 3439 7320 2863 used 1.09149s (c │ │ │ │ +000343d0: 7075 293b 2030 2e38 3530 3034 3973 2028 pu); 0.850049s ( │ │ │ │ +000343e0: 7468 7265 6164 293b 2030 7320 2867 6329 thread); 0s (gc) │ │ │ │ +000343f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00034400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034420: 2020 2020 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 │ │ │ │ +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 │ │ │ │ 00034480: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034490: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +00034490: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ 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: 7c0a 7c20 2020 2020 5a5a 5b48 5d20 2020 |.| ZZ[H] │ │ │ │ -000344e0: 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] │ │ │ │ 000344f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034500: 2020 2020 2020 2020 207c 0a7c 6f33 203a |.|o3 : │ │ │ │ -00034510: 202d 2d2d 2d2d 2020 2020 2020 2020 2020 ----- │ │ │ │ +00034500: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00034510: 0a7c 6f33 203a 202d 2d2d 2d2d 2020 2020 .|o3 : ----- │ │ │ │ 00034520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034540: 2020 7c0a 7c20 2020 2020 2020 2035 2020 |.| 5 │ │ │ │ -00034550: 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 │ │ │ │ 00034560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034570: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00034580: 2020 2020 2048 2020 2020 2020 2020 2020 H │ │ │ │ +00034570: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00034580: 207c 0a7c 2020 2020 2020 2048 2020 2020 |.| H │ │ │ │ 00034590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000345a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000345b0: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +000345b0: 2020 2020 2020 2020 2020 7c0a 2b2d 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 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 2e34 3639 .| -- used 1.469 │ │ │ │ -00034670: 3032 7320 2863 7075 293b 2031 2e30 3033 02s (cpu); 1.003 │ │ │ │ -00034680: 3039 7320 2874 6872 6561 6429 3b20 3073 09s (thread); 0s │ │ │ │ -00034690: 2028 6763 2920 2020 7c0a 7c20 2020 2020 (gc) |.| │ │ │ │ -000346a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 2e33 3335 3531 7320 2863 7075 293b 1.33551s (cpu); │ │ │ │ +00034680: 2030 2e38 3431 3537 3373 2028 7468 7265 0.841573s (thre │ │ │ │ +00034690: 6164 293b 2030 7320 2867 6329 2020 7c0a ad); 0s (gc) |. │ │ │ │ +000346a0: 7c20 2020 2020 2020 2020 2020 2020 2020 | │ │ │ │ 000346b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000346c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000346d0: 207c 0a7c 2020 2020 2020 2034 2020 2020 |.| 4 │ │ │ │ -000346e0: 2033 2020 2020 2032 2020 2020 2020 2020 3 2 │ │ │ │ +000346d0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ +000346e0: 2034 2020 2020 2033 2020 2020 2032 2020 4 3 2 │ │ │ │ 000346f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034700: 2020 2020 2020 2020 2020 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 │ │ │ │ +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 │ │ │ │ 00034730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034740: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00034740: 2020 2020 2020 2020 207c 0a7c 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 7c0a 7c20 |.| │ │ │ │ -00034780: 2020 2020 5a5a 5b48 5d20 2020 2020 2020 ZZ[H] │ │ │ │ +00034770: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00034780: 2020 7c0a 7c20 2020 2020 5a5a 5b48 5d20 |.| ZZ[H] │ │ │ │ 00034790: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000347a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000347b0: 2020 2020 207c 0a7c 6f34 203a 202d 2d2d |.|o4 : --- │ │ │ │ -000347c0: 2d2d 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 : ----- │ │ │ │ 000347d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000347e0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000347f0: 7c20 2020 2020 2020 2035 2020 2020 2020 | 5 │ │ │ │ +000347e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000347f0: 2020 2020 7c0a 7c20 2020 2020 2020 2035 |.| 5 │ │ │ │ 00034800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034810: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034820: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ -00034830: 2048 2020 2020 2020 2020 2020 2020 2020 H │ │ │ │ +00034820: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ +00034830: 2020 2020 2020 2048 2020 2020 2020 2020 H │ │ │ │ 00034840: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034850: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034860: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +00034860: 2020 2020 2020 7c0a 2b2d 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 2d2b 0a7c 6935 203a ---------+.|i5 : │ │ │ │ -000348a0: 206f 6f20 3d3d 206f 6f6f 2020 2020 2020 oo == ooo │ │ │ │ +00034890: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b ---------------+ │ │ │ │ +000348a0: 0a7c 6935 203a 206f 6f20 3d3d 206f 6f6f .|i5 : oo == ooo │ │ │ │ 000348b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000348c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000348d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000348d0: 2020 2020 2020 2020 7c0a 7c20 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 207c 0a7c 6f35 |.|o5 │ │ │ │ -00034910: 203d 2074 7275 6520 2020 2020 2020 2020 = true │ │ │ │ +00034900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00034910: 207c 0a7c 6f35 203d 2074 7275 6520 2020 |.|o5 = true │ │ │ │ 00034920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00034930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034940: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00034940: 2020 2020 2020 2020 2020 7c0a 2b2d 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 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)...+-------- │ │ │ │ +00034970: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00034980: 2d2d 2d2b 0a0a 496e 2074 6865 2063 6173 ---+..In the cas │ │ │ │ +00034990: 6520 7768 656e 2074 6865 2069 6e70 7574 e when the input │ │ │ │ +000349a0: 2069 6465 616c 2049 2064 6566 696e 6573 ideal I defines │ │ │ │ +000349b0: 2061 2073 6d6f 6f74 6820 7072 6f6a 6563 a smooth projec │ │ │ │ +000349c0: 7469 7665 2076 6172 6965 7479 2024 5824 tive variety $X$ │ │ │ │ +000349d0: 2c20 7468 650a 7075 7368 2d66 6f72 7761 , the.push-forwa │ │ │ │ +000349e0: 7264 206f 6620 2463 5f7b 534d 7d28 5829 rd of $c_{SM}(X) │ │ │ │ +000349f0: 2420 6361 6e20 6265 2063 6f6d 7075 7465 $ can be compute │ │ │ │ +00034a00: 6420 6d75 6368 206d 6f72 6520 6566 6669 d much more effi │ │ │ │ +00034a10: 6369 656e 746c 7920 7573 696e 6720 2a6e ciently using *n │ │ │ │ +00034a20: 6f74 650a 5365 6772 6543 6c61 7373 3a20 ote.SegreClass: │ │ │ │ +00034a30: 5365 6772 6543 6c61 7373 2c2e 2049 6e64 SegreClass,. Ind │ │ │ │ +00034a40: 6565 642c 2069 6e20 7468 6973 2063 6173 eed, in this cas │ │ │ │ +00034a50: 652c 2024 635f 7b53 4d7d 2858 2924 2063 e, $c_{SM}(X)$ c │ │ │ │ +00034a60: 6f69 6e63 6964 6573 2077 6974 6820 7468 oincides with th │ │ │ │ +00034a70: 650a 2874 6f74 616c 2920 4368 6572 6e20 e.(total) Chern │ │ │ │ +00034a80: 636c 6173 7320 6f66 2074 6865 2074 616e class of the tan │ │ │ │ +00034a90: 6765 6e74 2062 756e 646c 6520 6f66 2024 gent bundle of $ │ │ │ │ +00034aa0: 5824 2061 6e64 2063 616e 2062 6520 6f62 X$ and can be ob │ │ │ │ +00034ab0: 7461 696e 6564 2061 7320 666f 6c6c 6f77 tained as follow │ │ │ │ +00034ac0: 730a 2869 6e20 6765 6e65 7261 6c20 7468 s.(in general th │ │ │ │ +00034ad0: 6520 6d65 7468 6f64 2062 656c 6f77 2067 e method below g │ │ │ │ +00034ae0: 6976 6573 2074 6865 2070 7573 682d 666f ives the push-fo │ │ │ │ +00034af0: 7277 6172 6420 6f66 2074 6865 2073 6f2d rward of the so- │ │ │ │ +00034b00: 6361 6c6c 6564 0a43 6865 726e 2d46 756c called.Chern-Ful │ │ │ │ +00034b10: 746f 6e20 636c 6173 7329 2e0a 0a2b 2d2d ton class)...+-- │ │ │ │ 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 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 |.+-------- │ │ │ │ +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 |.+-- │ │ │ │ 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 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 │ │ │ │ +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 ) -> ( │ │ │ │ 00034c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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); │ │ │ │ +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); │ │ │ │ 00034c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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)); │ │ │ │ +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)); │ │ │ │ 00034ce0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00034cf0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00034cf0: 2020 2020 2020 2020 2020 207c 0a2b 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 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 │ │ │ │ +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) │ │ │ │ 00034d80: 2020 2020 2020 2020 2020 2020 2020 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 |.| │ │ │ │ +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 |.| │ │ │ │ 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 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 ---------------- │ │ │ │ +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 ------------- │ │ │ │ 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 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 ---------------- │ │ │ │ +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 ------------- │ │ │ │ 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 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 ) │ │ │ │ +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 ) │ │ │ │ 00035040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00035090: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000350a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000350b0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000350b0: 2020 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ -00035110: 2020 2020 2020 2020 5a5a 2020 2020 2020 ZZ │ │ │ │ +00035100: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00035110: 2020 2020 2020 2020 2020 2020 2020 5a5a 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 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 │ │ │ │ +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 │ │ │ │ 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 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|.|-------- │ │ │ │ +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|.|-- │ │ │ │ 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 2d7c 0a7c 2020 5d20 2020 2020 -----|.| ] │ │ │ │ -00035250: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00035240: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ +00035250: 5d20 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 207c 0a7c 2c34 2020 2020 2020 |.|,4 │ │ │ │ +00035290: 2020 2020 2020 2020 2020 207c 0a7c 2c34 |.|,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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000352e0: 2020 2020 2020 2020 2020 207c 0a2b 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 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 │ │ │ │ +00035330: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6939 -----------+.|i9 │ │ │ │ +00035340: 203a 2074 696d 6520 4368 6572 6e43 6c61 : time ChernCla │ │ │ │ +00035350: 7373 2047 2020 2020 2020 2020 2020 2020 ss G │ │ │ │ 00035360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035380: 2020 2020 207c 0a7c 202d 2d20 7573 6564 |.| -- used │ │ │ │ -00035390: 2030 2e31 3130 3239 3873 2028 6370 7529 0.110298s (cpu) │ │ │ │ -000353a0: 3b20 302e 3131 3033 3035 7320 2874 6872 ; 0.110305s (thr │ │ │ │ -000353b0: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ -000353c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000353d0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00035380: 2020 2020 2020 2020 2020 207c 0a7c 202d |.| - │ │ │ │ +00035390: 2d20 7573 6564 2030 2e31 3139 3635 3773 - used 0.119657s │ │ │ │ +000353a0: 2028 6370 7529 3b20 302e 3131 3936 3539 (cpu); 0.119659 │ │ │ │ +000353b0: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ +000353c0: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ +000353d0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ 000353e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000353f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00035410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00035420: 2020 2020 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 │ │ │ │ +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 │ │ │ │ 00035460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 000354b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000354c0: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +000354c0: 2020 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c 2020 2020 205a 5a5b |.| ZZ[ │ │ │ │ -00035520: 485d 2020 2020 2020 2020 2020 2020 2020 H] │ │ │ │ +00035510: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00035520: 2020 205a 5a5b 485d 2020 2020 2020 2020 ZZ[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 207c 0a7c 6f39 203a 202d 2d2d |.|o9 : --- │ │ │ │ -00035570: 2d2d 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 : ----- │ │ │ │ 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 207c 0a7c 2020 2020 2020 2031 |.| 1 │ │ │ │ -000355c0: 3020 2020 2020 2020 2020 2020 2020 2020 0 │ │ │ │ +000355b0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +000355c0: 2020 2020 2031 3020 2020 2020 2020 2020 10 │ │ │ │ 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 207c 0a7c 2020 2020 2020 4820 |.| H │ │ │ │ -00035610: 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 │ │ │ │ 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 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00035650: 2020 2020 2020 2020 2020 207c 0a2b 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 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 │ │ │ │ +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) │ │ │ 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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:.=== │ │ │ │ 00035c20: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00035c30: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ -00035c40: 2022 4368 6572 6e53 6368 7761 7274 7a4d "ChernSchwartzM │ │ │ │ -00035c50: 6163 5068 6572 736f 6e28 4964 6561 6c29 acPherson(Ideal) │ │ │ │ -00035c60: 220a 0a46 6f72 2074 6865 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│ │ │ -00035de0: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d .documentation.m │ │ │ │ -00035df0: 323a 3332 363a 302e 0a1f 0a46 696c 653a 2:326:0....File: │ │ │ │ -00035e00: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e Cremona.info, N │ │ │ │ -00035e10: 6f64 653a 2043 6f64 696d 4273 496e 762c ode: CodimBsInv, │ │ │ │ -00035e20: 204e 6578 743a 2063 6f65 6666 6963 6965 Next: coefficie │ │ │ │ -00035e30: 6e74 5269 6e67 5f6c 7052 6174 696f 6e61 ntRing_lpRationa │ │ │ │ -00035e40: 6c4d 6170 5f72 702c 2050 7265 763a 2043 lMap_rp, Prev: C │ │ │ │ -00035e50: 6865 726e 5363 6877 6172 747a 4d61 6350 hernSchwartzMacP │ │ │ │ -00035e60: 6865 7273 6f6e 2c20 5570 3a20 546f 700a herson, Up: Top. │ │ │ │ -00035e70: 0a43 6f64 696d 4273 496e 760a 2a2a 2a2a .CodimBsInv.**** │ │ │ │ -00035e80: 2a2a 2a2a 2a2a 0a0a 4465 7363 7269 7074 ******..Descript │ │ │ │ -00035e90: 696f 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ion.===========. │ │ │ │ -00035ea0: 0a54 6869 7320 6973 2061 2074 6563 686e .This is a techn │ │ │ │ -00035eb0: 6963 616c 206f 7074 696f 6e20 666f 7220 ical option for │ │ │ │ -00035ec0: 2a6e 6f74 6520 6170 7072 6f78 696d 6174 *note approximat │ │ │ │ -00035ed0: 6549 6e76 6572 7365 4d61 703a 0a61 7070 eInverseMap:.app │ │ │ │ -00035ee0: 726f 7869 6d61 7465 496e 7665 7273 654d roximateInverseM │ │ │ │ -00035ef0: 6170 2c2e 2049 7420 6163 6365 7074 7320 ap,. It accepts │ │ │ │ -00035f00: 616e 2069 6e74 6567 6572 2077 6869 6368 an integer which │ │ │ │ -00035f10: 2073 686f 756c 6420 6265 2061 206c 6f77 should be a low │ │ │ │ -00035f20: 6572 2062 6f75 6e64 2066 6f72 0a74 6865 er bound for.the │ │ │ │ -00035f30: 2063 6f64 696d 656e 7369 6f6e 206f 6620 codimension of │ │ │ │ -00035f40: 7468 6520 6261 7365 206c 6f63 7573 206f the base locus o │ │ │ │ -00035f50: 6620 7468 6520 696e 7665 7273 6520 6d61 f the inverse ma │ │ │ │ -00035f60: 702e 2049 6e20 6d6f 7374 2063 6173 6573 p. In most cases │ │ │ │ -00035f70: 2c20 6f6e 6520 6361 6e0a 6f62 7461 696e , one can.obtain │ │ │ │ -00035f80: 2074 6865 206f 7074 696d 616c 2076 616c the optimal val │ │ │ │ -00035f90: 7565 2074 6f20 6265 2070 6173 7365 6420 ue to be passed │ │ │ │ -00035fa0: 6173 2069 6e20 7468 6520 666f 6c6c 6f77 as in the follow │ │ │ │ -00035fb0: 696e 6720 6578 616d 706c 652e 0a0a 0a2b ing example....+ │ │ │ │ -00035fc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00035d70: 2d2d 2d2d 2d2d 0a0a 5468 6520 736f 7572 ------..The sour │ │ │ │ +00035d80: 6365 206f 6620 7468 6973 2064 6f63 756d ce of this docum │ │ │ │ +00035d90: 656e 7420 6973 2069 6e0a 2f62 7569 6c64 ent is in./build │ │ │ │ +00035da0: 2f72 6570 726f 6475 6369 626c 652d 7061 /reproducible-pa │ │ │ │ +00035db0: 7468 2f6d 6163 6175 6c61 7932 2d31 2e32 th/macaulay2-1.2 │ │ │ │ +00035dc0: 362e 3036 2b64 732f 4d32 2f4d 6163 6175 6.06+ds/M2/Macau │ │ │ │ +00035dd0: 6c61 7932 2f70 6163 6b61 6765 732f 4372 lay2/packages/Cr │ │ │ │ +00035de0: 656d 6f6e 612f 0a64 6f63 756d 656e 7461 emona/.documenta │ │ │ │ +00035df0: 7469 6f6e 2e6d 323a 3332 363a 302e 0a1f tion.m2:326:0... │ │ │ │ +00035e00: 0a46 696c 653a 2043 7265 6d6f 6e61 2e69 .File: Cremona.i │ │ │ │ +00035e10: 6e66 6f2c 204e 6f64 653a 2043 6f64 696d nfo, Node: Codim │ │ │ │ +00035e20: 4273 496e 762c 204e 6578 743a 2063 6f65 BsInv, Next: coe │ │ │ │ +00035e30: 6666 6963 6965 6e74 5269 6e67 5f6c 7052 fficientRing_lpR │ │ │ │ +00035e40: 6174 696f 6e61 6c4d 6170 5f72 702c 2050 ationalMap_rp, P │ │ │ │ +00035e50: 7265 763a 2043 6865 726e 5363 6877 6172 rev: ChernSchwar │ │ │ │ +00035e60: 747a 4d61 6350 6865 7273 6f6e 2c20 5570 tzMacPherson, Up │ │ │ │ +00035e70: 3a20 546f 700a 0a43 6f64 696d 4273 496e : Top..CodimBsIn │ │ │ │ +00035e80: 760a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 4465 v.**********..De │ │ │ │ +00035e90: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ +00035ea0: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2061 =====..This is a │ │ │ │ +00035eb0: 2074 6563 686e 6963 616c 206f 7074 696f technical optio │ │ │ │ +00035ec0: 6e20 666f 7220 2a6e 6f74 6520 6170 7072 n for *note appr │ │ │ │ +00035ed0: 6f78 696d 6174 6549 6e76 6572 7365 4d61 oximateInverseMa │ │ │ │ +00035ee0: 703a 0a61 7070 726f 7869 6d61 7465 496e p:.approximateIn │ │ │ │ +00035ef0: 7665 7273 654d 6170 2c2e 2049 7420 6163 verseMap,. It ac │ │ │ │ +00035f00: 6365 7074 7320 616e 2069 6e74 6567 6572 cepts an integer │ │ │ │ +00035f10: 2077 6869 6368 2073 686f 756c 6420 6265 which should be │ │ │ │ +00035f20: 2061 206c 6f77 6572 2062 6f75 6e64 2066 a lower bound f │ │ │ │ +00035f30: 6f72 0a74 6865 2063 6f64 696d 656e 7369 or.the codimensi │ │ │ │ +00035f40: 6f6e 206f 6620 7468 6520 6261 7365 206c on of the base l │ │ │ │ +00035f50: 6f63 7573 206f 6620 7468 6520 696e 7665 ocus of the inve │ │ │ │ +00035f60: 7273 6520 6d61 702e 2049 6e20 6d6f 7374 rse map. In most │ │ │ │ +00035f70: 2063 6173 6573 2c20 6f6e 6520 6361 6e0a cases, one can. │ │ │ │ +00035f80: 6f62 7461 696e 2074 6865 206f 7074 696d obtain the optim │ │ │ │ +00035f90: 616c 2076 616c 7565 2074 6f20 6265 2070 al value to be p │ │ │ │ +00035fa0: 6173 7365 6420 6173 2069 6e20 7468 6520 assed as in the │ │ │ │ +00035fb0: 666f 6c6c 6f77 696e 6720 6578 616d 706c following exampl │ │ │ │ +00035fc0: 652e 0a0a 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d e....+---------- │ │ │ │ 00035fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035fe0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00035ff0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00036010: 6931 203a 2063 6f64 696d 4273 496e 7620 i1 : codimBsInv │ │ │ │ -00036020: 3d20 286d 2920 2d3e 2028 2020 2020 2020 = (m) -> ( │ │ │ │ +00036000: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036010: 2d2d 2d2b 0a7c 6931 203a 2063 6f64 696d ---+.|i1 : codim │ │ │ │ +00036020: 4273 496e 7620 3d20 286d 2920 2d3e 2028 BsInv = (m) -> ( │ │ │ │ 00036030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036050: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00036060: 2020 2020 2020 2020 2d2d 2069 6e70 7574 -- input │ │ │ │ -00036070: 3a20 6d2c 2074 6865 206c 6973 7420 6f66 : m, the list of │ │ │ │ -00036080: 2070 726f 6a65 6374 6976 6520 6465 6772 projective degr │ │ │ │ -00036090: 6565 7320 6f66 2061 2062 6972 6174 696f ees of a biratio │ │ │ │ -000360a0: 6e61 6c20 6d61 7020 2020 2020 207c 0a7c nal map |.| │ │ │ │ -000360b0: 2020 2020 2020 2020 2d2d 206f 7574 7075 -- outpu │ │ │ │ -000360c0: 743a 2074 6865 2063 6f64 696d 656e 7369 t: the codimensi │ │ │ │ -000360d0: 6f6e 206f 6620 7468 6520 6261 7365 206c on of the base l │ │ │ │ -000360e0: 6f63 7573 206f 6620 7468 6520 696e 7665 ocus of the inve │ │ │ │ -000360f0: 7273 6520 6d61 7020 2020 2020 207c 0a7c rse map |.| │ │ │ │ -00036100: 2020 2020 2020 2020 6b3a 3d23 6d20 2d31 k:=#m -1 │ │ │ │ -00036110: 3b20 7a3a 3d6d 5f6b 3b20 643a 3d66 6c6f ; z:=m_k; d:=flo │ │ │ │ -00036120: 6f72 286d 5f28 6b2d 3129 2f7a 293b 2020 or(m_(k-1)/z); │ │ │ │ -00036130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036140: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00036150: 2020 2020 2020 2020 666f 7220 6920 6672 for i fr │ │ │ │ -00036160: 6f6d 2032 2074 6f20 6b20 646f 2069 6620 om 2 to k do if │ │ │ │ -00036170: 7a2a 645e 6920 2d20 6d5f 286b 2d69 2920 z*d^i - m_(k-i) │ │ │ │ -00036180: 3e20 3020 7468 656e 2072 6574 7572 6e20 > 0 then return │ │ │ │ -00036190: 693b 2020 2020 2020 2020 2020 207c 0a7c i; |.| │ │ │ │ -000361a0: 2020 2020 2029 3b20 2020 2020 2020 2020 ); │ │ │ │ +00036050: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036060: 2020 207c 0a7c 2020 2020 2020 2020 2d2d |.| -- │ │ │ │ +00036070: 2069 6e70 7574 3a20 6d2c 2074 6865 206c input: m, the l │ │ │ │ +00036080: 6973 7420 6f66 2070 726f 6a65 6374 6976 ist of projectiv │ │ │ │ +00036090: 6520 6465 6772 6565 7320 6f66 2061 2062 e degrees of a b │ │ │ │ +000360a0: 6972 6174 696f 6e61 6c20 6d61 7020 2020 irational map │ │ │ │ +000360b0: 2020 207c 0a7c 2020 2020 2020 2020 2d2d |.| -- │ │ │ │ +000360c0: 206f 7574 7075 743a 2074 6865 2063 6f64 output: the cod │ │ │ │ +000360d0: 696d 656e 7369 6f6e 206f 6620 7468 6520 imension of the │ │ │ │ +000360e0: 6261 7365 206c 6f63 7573 206f 6620 7468 base locus of th │ │ │ │ +000360f0: 6520 696e 7665 7273 6520 6d61 7020 2020 e inverse map │ │ │ │ +00036100: 2020 207c 0a7c 2020 2020 2020 2020 6b3a |.| k: │ │ │ │ +00036110: 3d23 6d20 2d31 3b20 7a3a 3d6d 5f6b 3b20 =#m -1; z:=m_k; │ │ │ │ +00036120: 643a 3d66 6c6f 6f72 286d 5f28 6b2d 3129 d:=floor(m_(k-1) │ │ │ │ +00036130: 2f7a 293b 2020 2020 2020 2020 2020 2020 /z); │ │ │ │ +00036140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036150: 2020 207c 0a7c 2020 2020 2020 2020 666f |.| fo │ │ │ │ +00036160: 7220 6920 6672 6f6d 2032 2074 6f20 6b20 r i from 2 to k │ │ │ │ +00036170: 646f 2069 6620 7a2a 645e 6920 2d20 6d5f do if z*d^i - m_ │ │ │ │ +00036180: 286b 2d69 2920 3e20 3020 7468 656e 2072 (k-i) > 0 then r │ │ │ │ +00036190: 6574 7572 6e20 693b 2020 2020 2020 2020 eturn i; │ │ │ │ +000361a0: 2020 207c 0a7c 2020 2020 2029 3b20 2020 |.| ); │ │ │ │ 000361b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000361c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000361d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000361e0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -000361f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000361e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000361f0: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00036200: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036210: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00036240: 6932 203a 2070 6869 203d 2074 6f4d 6170 i2 : phi = toMap │ │ │ │ -00036250: 2074 7269 6d20 6d69 6e6f 7273 2832 2c67 trim minors(2,g │ │ │ │ -00036260: 656e 6572 6963 5379 6d6d 6574 7269 634d enericSymmetricM │ │ │ │ -00036270: 6174 7269 7828 5151 5b78 5f30 2e2e 785f atrix(QQ[x_0..x_ │ │ │ │ -00036280: 355d 2c33 2929 2020 2020 2020 207c 0a7c 5],3)) |.| │ │ │ │ -00036290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036230: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036240: 2d2d 2d2b 0a7c 6932 203a 2070 6869 203d ---+.|i2 : phi = │ │ │ │ +00036250: 2074 6f4d 6170 2074 7269 6d20 6d69 6e6f toMap trim mino │ │ │ │ +00036260: 7273 2832 2c67 656e 6572 6963 5379 6d6d rs(2,genericSymm │ │ │ │ +00036270: 6574 7269 634d 6174 7269 7828 5151 5b78 etricMatrix(QQ[x │ │ │ │ +00036280: 5f30 2e2e 785f 355d 2c33 2929 2020 2020 _0..x_5],3)) │ │ │ │ +00036290: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000362a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000362b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000362c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000362d0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000362e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000362d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000362e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000362f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036300: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00036300: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 00036310: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036320: 2020 2020 2020 2020 2032 2020 207c 0a7c 2 |.| │ │ │ │ -00036330: 6f32 203d 206d 6170 2028 5151 5b78 202e o2 = map (QQ[x . │ │ │ │ -00036340: 2e78 205d 2c20 5151 5b78 202e 2e78 205d .x ], QQ[x ..x ] │ │ │ │ -00036350: 2c20 7b78 2020 2d20 7820 7820 2c20 7820 , {x - x x , x │ │ │ │ -00036360: 7820 202d 2078 2078 202c 2078 2078 2020 x - x x , x x │ │ │ │ -00036370: 2d20 7820 7820 2c20 7820 202d 207c 0a7c - x x , x - |.| │ │ │ │ -00036380: 2020 2020 2020 2020 2020 2020 2020 3020 0 │ │ │ │ -00036390: 2020 3520 2020 2020 2020 3020 2020 3520 5 0 5 │ │ │ │ -000363a0: 2020 2020 3420 2020 2033 2035 2020 2032 4 3 5 2 │ │ │ │ -000363b0: 2034 2020 2020 3120 3520 2020 3220 3320 4 1 5 2 3 │ │ │ │ -000363c0: 2020 2031 2034 2020 2032 2020 207c 0a7c 1 4 2 |.| │ │ │ │ -000363d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036320: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ +00036330: 2020 207c 0a7c 6f32 203d 206d 6170 2028 |.|o2 = map ( │ │ │ │ +00036340: 5151 5b78 202e 2e78 205d 2c20 5151 5b78 QQ[x ..x ], QQ[x │ │ │ │ +00036350: 202e 2e78 205d 2c20 7b78 2020 2d20 7820 ..x ], {x - x │ │ │ │ +00036360: 7820 2c20 7820 7820 202d 2078 2078 202c x , x x - x x , │ │ │ │ +00036370: 2078 2078 2020 2d20 7820 7820 2c20 7820 x x - x x , x │ │ │ │ +00036380: 202d 207c 0a7c 2020 2020 2020 2020 2020 - |.| │ │ │ │ +00036390: 2020 2020 3020 2020 3520 2020 2020 2020 0 5 │ │ │ │ +000363a0: 3020 2020 3520 2020 2020 3420 2020 2033 0 5 4 3 │ │ │ │ +000363b0: 2035 2020 2032 2034 2020 2020 3120 3520 5 2 4 1 5 │ │ │ │ +000363c0: 2020 3220 3320 2020 2031 2034 2020 2032 2 3 1 4 2 │ │ │ │ +000363d0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000363e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000363f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036410: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00036420: 6f32 203a 2052 696e 674d 6170 2051 515b o2 : RingMap QQ[ │ │ │ │ -00036430: 7820 2e2e 7820 5d20 3c2d 2d20 5151 5b78 x ..x ] <-- QQ[x │ │ │ │ -00036440: 202e 2e78 205d 2020 2020 2020 2020 2020 ..x ] │ │ │ │ +00036410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036420: 2020 207c 0a7c 6f32 203a 2052 696e 674d |.|o2 : RingM │ │ │ │ +00036430: 6170 2051 515b 7820 2e2e 7820 5d20 3c2d ap QQ[x ..x ] <- │ │ │ │ +00036440: 2d20 5151 5b78 202e 2e78 205d 2020 2020 - QQ[x ..x ] │ │ │ │ 00036450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036460: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00036470: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036480: 2030 2020 2035 2020 2020 2020 2020 2020 0 5 │ │ │ │ -00036490: 3020 2020 3520 2020 2020 2020 2020 2020 0 5 │ │ │ │ +00036460: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036470: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +00036480: 2020 2020 2020 2030 2020 2035 2020 2020 0 5 │ │ │ │ +00036490: 2020 2020 2020 3020 2020 3520 2020 2020 0 5 │ │ │ │ 000364a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000364b0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000364c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000364b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000364c0: 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d |.|---------- │ │ │ │ 000364d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000364e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000364f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c -------------|.| │ │ │ │ -00036510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036520: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00036500: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036510: 2d2d 2d7c 0a7c 2020 2020 2020 2020 2020 ---|.| │ │ │ │ +00036520: 2020 2020 2020 2020 2020 3220 2020 2020 2 │ │ │ │ 00036530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036550: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -00036560: 7820 7820 2c20 7820 7820 202d 2078 2078 x x , x x - x x │ │ │ │ -00036570: 202c 2078 2020 2d20 7820 7820 7d29 2020 , x - x x }) │ │ │ │ -00036580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036560: 2020 207c 0a7c 7820 7820 2c20 7820 7820 |.|x x , x x │ │ │ │ +00036570: 202d 2078 2078 202c 2078 2020 2d20 7820 - x x , x - x │ │ │ │ +00036580: 7820 7d29 2020 2020 2020 2020 2020 2020 x }) │ │ │ │ 00036590: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000365a0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000365b0: 2030 2035 2020 2031 2032 2020 2020 3020 0 5 1 2 0 │ │ │ │ -000365c0: 3420 2020 3120 2020 2030 2033 2020 2020 4 1 0 3 │ │ │ │ -000365d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000365a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000365b0: 2020 207c 0a7c 2030 2035 2020 2031 2032 |.| 0 5 1 2 │ │ │ │ +000365c0: 2020 2020 3020 3420 2020 3120 2020 2030 0 4 1 0 │ │ │ │ +000365d0: 2033 2020 2020 2020 2020 2020 2020 2020 3 │ │ │ │ 000365e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000365f0: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00036600: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000365f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036600: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00036610: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036620: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c -------------+.| │ │ │ │ -00036650: 6933 203a 2063 6f64 696d 4273 496e 7620 i3 : codimBsInv │ │ │ │ -00036660: 7072 6f6a 6563 7469 7665 4465 6772 6565 projectiveDegree │ │ │ │ -00036670: 7320 7068 6920 2020 2020 2020 2020 2020 s phi │ │ │ │ +00036640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036650: 2d2d 2d2b 0a7c 6933 203a 2063 6f64 696d ---+.|i3 : codim │ │ │ │ +00036660: 4273 496e 7620 7072 6f6a 6563 7469 7665 BsInv projective │ │ │ │ +00036670: 4465 6772 6565 7320 7068 6920 2020 2020 Degrees phi │ │ │ │ 00036680: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036690: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000366a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036690: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000366a0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ 000366b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000366c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000366d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000366e0: 2020 2020 2020 2020 2020 2020 207c 0a7c |.| │ │ │ │ -000366f0: 6f33 203d 2033 2020 2020 2020 2020 2020 o3 = 3 │ │ │ │ +000366e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000366f0: 2020 207c 0a7c 6f33 203d 2033 2020 2020 |.|o3 = 3 │ │ │ │ 00036700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036710: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00036720: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00036730: 2020 2020 2020 2020 2020 2020 207c 0a2b |.+ │ │ │ │ -00036740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036730: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00036740: 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d |.+---------- │ │ │ │ 00036750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036760: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036770: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a -------------+.. │ │ │ │ -00036790: 486f 7765 7665 722c 2073 6f6d 6574 696d However, sometim │ │ │ │ -000367a0: 6573 206c 6172 6765 7220 7661 6c75 6573 es larger values │ │ │ │ -000367b0: 206d 6179 2062 6520 7072 6566 6572 6162 may be preferab │ │ │ │ -000367c0: 6c65 2e0a 0a46 756e 6374 696f 6e73 2077 le...Functions w │ │ │ │ -000367d0: 6974 6820 6f70 7469 6f6e 616c 2061 7267 ith optional arg │ │ │ │ -000367e0: 756d 656e 7420 6e61 6d65 6420 436f 6469 ument named Codi │ │ │ │ -000367f0: 6d42 7349 6e76 3a0a 3d3d 3d3d 3d3d 3d3d mBsInv:.======== │ │ │ │ +00036780: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036790: 2d2d 2d2b 0a0a 486f 7765 7665 722c 2073 ---+..However, s │ │ │ │ +000367a0: 6f6d 6574 696d 6573 206c 6172 6765 7220 ometimes larger │ │ │ │ +000367b0: 7661 6c75 6573 206d 6179 2062 6520 7072 values may be pr │ │ │ │ +000367c0: 6566 6572 6162 6c65 2e0a 0a46 756e 6374 eferable...Funct │ │ │ │ +000367d0: 696f 6e73 2077 6974 6820 6f70 7469 6f6e ions with option │ │ │ │ +000367e0: 616c 2061 7267 756d 656e 7420 6e61 6d65 al argument name │ │ │ │ +000367f0: 6420 436f 6469 6d42 7349 6e76 3a0a 3d3d d CodimBsInv:.== │ │ │ │ 00036800: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00036810: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00036820: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ -00036830: 2261 7070 726f 7869 6d61 7465 496e 7665 "approximateInve │ │ │ │ -00036840: 7273 654d 6170 282e 2e2e 2c43 6f64 696d rseMap(...,Codim │ │ │ │ -00036850: 4273 496e 763d 3e2e 2e2e 2922 0a0a 466f BsInv=>...)"..Fo │ │ │ │ -00036860: 7220 7468 6520 7072 6f67 7261 6d6d 6572 r the programmer │ │ │ │ -00036870: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00036880: 3d3d 3d0a 0a54 6865 206f 626a 6563 7420 ===..The object │ │ │ │ -00036890: 2a6e 6f74 6520 436f 6469 6d42 7349 6e76 *note CodimBsInv │ │ │ │ -000368a0: 3a20 436f 6469 6d42 7349 6e76 2c20 6973 : CodimBsInv, is │ │ │ │ -000368b0: 2061 202a 6e6f 7465 2073 796d 626f 6c3a a *note symbol: │ │ │ │ -000368c0: 0a28 4d61 6361 756c 6179 3244 6f63 2953 .(Macaulay2Doc)S │ │ │ │ -000368d0: 796d 626f 6c2c 2e0a 0a2d 2d2d 2d2d 2d2d ymbol,...------- │ │ │ │ +00036820: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00036830: 0a0a 2020 2a20 2261 7070 726f 7869 6d61 .. * "approxima │ │ │ │ +00036840: 7465 496e 7665 7273 654d 6170 282e 2e2e teInverseMap(... │ │ │ │ +00036850: 2c43 6f64 696d 4273 496e 763d 3e2e 2e2e ,CodimBsInv=>... │ │ │ │ +00036860: 2922 0a0a 466f 7220 7468 6520 7072 6f67 )"..For the prog │ │ │ │ +00036870: 7261 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d rammer.========= │ │ │ │ +00036880: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f =========..The o │ │ │ │ +00036890: 626a 6563 7420 2a6e 6f74 6520 436f 6469 bject *note Codi │ │ │ │ +000368a0: 6d42 7349 6e76 3a20 436f 6469 6d42 7349 mBsInv: CodimBsI │ │ │ │ +000368b0: 6e76 2c20 6973 2061 202a 6e6f 7465 2073 nv, is a *note s │ │ │ │ +000368c0: 796d 626f 6c3a 0a28 4d61 6361 756c 6179 ymbol:.(Macaulay │ │ │ │ +000368d0: 3244 6f63 2953 796d 626f 6c2c 2e0a 0a2d 2Doc)Symbol,...- │ │ │ │ 000368e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000368f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036900: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00036910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00036920: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00036930: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00036940: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00036950: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00036960: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00036970: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00036980: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00036990: 4372 656d 6f6e 612f 0a64 6f63 756d 656e Cremona/.documen │ │ │ │ -000369a0: 7461 7469 6f6e 2e6d 323a 3130 373a 302e tation.m2:107:0. │ │ │ │ -000369b0: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61 ...File: Cremona │ │ │ │ -000369c0: 2e69 6e66 6f2c 204e 6f64 653a 2063 6f65 .info, Node: coe │ │ │ │ -000369d0: 6666 6963 6965 6e74 5269 6e67 5f6c 7052 fficientRing_lpR │ │ │ │ -000369e0: 6174 696f 6e61 6c4d 6170 5f72 702c 204e ationalMap_rp, N │ │ │ │ -000369f0: 6578 743a 2063 6f65 6666 6963 6965 6e74 ext: coefficient │ │ │ │ -00036a00: 735f 6c70 5261 7469 6f6e 616c 4d61 705f s_lpRationalMap_ │ │ │ │ -00036a10: 7270 2c20 5072 6576 3a20 436f 6469 6d42 rp, Prev: CodimB │ │ │ │ -00036a20: 7349 6e76 2c20 5570 3a20 546f 700a 0a63 sInv, Up: Top..c │ │ │ │ -00036a30: 6f65 6666 6963 6965 6e74 5269 6e67 2852 oefficientRing(R │ │ │ │ -00036a40: 6174 696f 6e61 6c4d 6170 2920 2d2d 2063 ationalMap) -- c │ │ │ │ -00036a50: 6f65 6666 6963 6965 6e74 2072 696e 6720 oefficient ring │ │ │ │ -00036a60: 6f66 2061 2072 6174 696f 6e61 6c20 6d61 of a rational ma │ │ │ │ -00036a70: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a p.************** │ │ │ │ +00036920: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00036930: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00036940: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00036950: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00036960: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00036970: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +00036980: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00036990: 6b61 6765 732f 4372 656d 6f6e 612f 0a64 kages/Cremona/.d │ │ │ │ +000369a0: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a ocumentation.m2: │ │ │ │ +000369b0: 3130 373a 302e 0a1f 0a46 696c 653a 2043 107:0....File: C │ │ │ │ +000369c0: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 remona.info, Nod │ │ │ │ +000369d0: 653a 2063 6f65 6666 6963 6965 6e74 5269 e: coefficientRi │ │ │ │ +000369e0: 6e67 5f6c 7052 6174 696f 6e61 6c4d 6170 ng_lpRationalMap │ │ │ │ +000369f0: 5f72 702c 204e 6578 743a 2063 6f65 6666 _rp, Next: coeff │ │ │ │ +00036a00: 6963 6965 6e74 735f 6c70 5261 7469 6f6e icients_lpRation │ │ │ │ +00036a10: 616c 4d61 705f 7270 2c20 5072 6576 3a20 alMap_rp, Prev: │ │ │ │ +00036a20: 436f 6469 6d42 7349 6e76 2c20 5570 3a20 CodimBsInv, Up: │ │ │ │ +00036a30: 546f 700a 0a63 6f65 6666 6963 6965 6e74 Top..coefficient │ │ │ │ +00036a40: 5269 6e67 2852 6174 696f 6e61 6c4d 6170 Ring(RationalMap │ │ │ │ +00036a50: 2920 2d2d 2063 6f65 6666 6963 6965 6e74 ) -- coefficient │ │ │ │ +00036a60: 2072 696e 6720 6f66 2061 2072 6174 696f ring of a ratio │ │ │ │ +00036a70: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a nal map.******** │ │ │ │ 00036a80: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00036a90: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00036aa0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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 ---------------- │ │ │ │ +00036ab0: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +00036ac0: 4675 6e63 7469 6f6e 3a20 2a6e 6f74 6520 Function: *note │ │ │ │ +00036ad0: 636f 6566 6669 6369 656e 7452 696e 673a coefficientRing: │ │ │ │ +00036ae0: 2028 4d61 6361 756c 6179 3244 6f63 2963 (Macaulay2Doc)c │ │ │ │ +00036af0: 6f65 6666 6963 6965 6e74 5269 6e67 2c0a oefficientRing,. │ │ │ │ +00036b00: 2020 2a20 5573 6167 653a 200a 2020 2020 * Usage: . │ │ │ │ +00036b10: 2020 2020 636f 6566 6669 6369 656e 7452 coefficientR │ │ │ │ +00036b20: 696e 6720 7068 690a 2020 2a20 496e 7075 ing phi. * Inpu │ │ │ │ +00036b30: 7473 3a0a 2020 2020 2020 2a20 7068 692c ts:. * phi, │ │ │ │ +00036b40: 2061 202a 6e6f 7465 2072 6174 696f 6e61 a *note rationa │ │ │ │ +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 .--------------- │ │ │ │ 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 0a0a 5468 6520 ----------..The │ │ │ │ -00036d30: 736f 7572 6365 206f 6620 7468 6973 2064 source of this d │ │ │ │ -00036d40: 6f63 756d 656e 7420 6973 2069 6e0a 2f62 ocument is in./b │ │ │ │ -00036d50: 7569 6c64 2f72 6570 726f 6475 6369 626c uild/reproducibl │ │ │ │ -00036d60: 652d 7061 7468 2f6d 6163 6175 6c61 7932 e-path/macaulay2 │ │ │ │ -00036d70: 2d31 2e32 362e 3036 2b64 732f 4d32 2f4d -1.26.06+ds/M2/M │ │ │ │ -00036d80: 6163 6175 6c61 7932 2f70 6163 6b61 6765 acaulay2/package │ │ │ │ -00036d90: 732f 4372 656d 6f6e 612f 0a64 6f63 756d s/Cremona/.docum │ │ │ │ -00036da0: 656e 7461 7469 6f6e 2e6d 323a 3438 323a entation.m2:482: │ │ │ │ -00036db0: 302e 0a1f 0a46 696c 653a 2043 7265 6d6f 0....File: Cremo │ │ │ │ -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 **************** │ │ │ │ +00036d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00036d30: 0a0a 5468 6520 736f 7572 6365 206f 6620 ..The source of │ │ │ │ +00036d40: 7468 6973 2064 6f63 756d 656e 7420 6973 this document is │ │ │ │ +00036d50: 2069 6e0a 2f62 7569 6c64 2f72 6570 726f in./build/repro │ │ │ │ +00036d60: 6475 6369 626c 652d 7061 7468 2f6d 6163 ducible-path/mac │ │ │ │ +00036d70: 6175 6c61 7932 2d31 2e32 362e 3036 2b64 aulay2-1.26.06+d │ │ │ │ +00036d80: 732f 4d32 2f4d 6163 6175 6c61 7932 2f70 s/M2/Macaulay2/p │ │ │ │ +00036d90: 6163 6b61 6765 732f 4372 656d 6f6e 612f ackages/Cremona/ │ │ │ │ +00036da0: 0a64 6f63 756d 656e 7461 7469 6f6e 2e6d .documentation.m │ │ │ │ +00036db0: 323a 3438 323a 302e 0a1f 0a46 696c 653a 2:482:0....File: │ │ │ │ +00036dc0: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e Cremona.info, N │ │ │ │ +00036dd0: 6f64 653a 2063 6f65 6666 6963 6965 6e74 ode: coefficient │ │ │ │ +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.********** │ │ │ │ 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: 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 =======..+------ │ │ │ │ +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.===========..+ │ │ │ │ 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 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 ---------------- │ │ │ │ +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 |.+---------- │ │ │ │ 00037180: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00037190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -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) │ │ │ │ +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) │ │ │ │ 000371d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000371e0: 2020 207c 0a7c 2020 2020 2020 2020 2020 |.| │ │ │ │ +000371e0: 2020 2020 2020 2020 207c 0a7c 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 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 │ │ │ │ +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 |.| 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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/ │ │ │ │ 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-00037a30: 0a7c 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 |.| │ │ │ │ 00037a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037a60: 2020 2020 2020 2020 2020 7c0a 7c6f 3620 |.|o6 │ │ │ │ -00037a70: 3d20 7472 7565 2020 2020 2020 2020 2020 = true │ │ │ │ +00037a60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00037a70: 7c0a 7c6f 3620 3d20 7472 7565 2020 2020 |.|o6 = true │ │ │ │ 00037a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00037a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00037aa0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00037aa0: 2020 2020 2020 2020 2020 207c 0a2b 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: 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.---------- │ │ │ │ +00037ae0: 2d2d 2d2d 2d2d 2b0a 0a57 6179 7320 746f ------+..Ways to │ │ │ │ +00037af0: 2075 7365 2074 6869 7320 6d65 7468 6f64 use this method │ │ │ │ +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 │ │ │ │ +00037b30: 7473 2852 6174 696f 6e61 6c4d 6170 293a ts(RationalMap): │ │ │ │ +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 │ │ │ │ +00037b70: 6e74 206d 6174 7269 7820 6f66 2061 2072 nt matrix of a r │ │ │ │ +00037b80: 6174 696f 6e61 6c20 6d61 700a 2d2d 2d2d ational 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 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ -00037be0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ -00037bf0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ -00037c00: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ -00037c10: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ -00037c20: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ -00037c30: 6179 322f 7061 636b 6167 6573 2f43 7265 ay2/packages/Cre │ │ │ │ -00037c40: 6d6f 6e61 2f0a 646f 6375 6d65 6e74 6174 mona/.documentat │ │ │ │ -00037c50: 696f 6e2e 6d32 3a35 3338 3a30 2e0a 1f0a ion.m2:538:0.... │ │ │ │ -00037c60: 4669 6c65 3a20 4372 656d 6f6e 612e 696e File: Cremona.in │ │ │ │ -00037c70: 666f 2c20 4e6f 6465 3a20 6465 6772 6565 fo, Node: degree │ │ │ │ -00037c80: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72 _lpRationalMap_r │ │ │ │ -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.************** │ │ │ │ +00037bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00037be0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00037bf0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00037c00: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00037c10: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00037c20: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +00037c30: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ +00037c40: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375 es/Cremona/.docu │ │ │ │ +00037c50: 6d65 6e74 6174 696f 6e2e 6d32 3a35 3338 mentation.m2:538 │ │ │ │ +00037c60: 3a30 2e0a 1f0a 4669 6c65 3a20 4372 656d :0....File: Crem │ │ │ │ +00037c70: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20 ona.info, Node: │ │ │ │ +00037c80: 6465 6772 6565 5f6c 7052 6174 696f 6e61 degree_lpRationa │ │ │ │ +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.******** │ │ │ │ 00037d10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00037d20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00037d30: 2a0a 0a20 202a 2046 756e 6374 696f 6e3a *.. * Function: │ │ │ │ -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 │ │ │ │ -00037df0: 3244 6f63 295a 5a2c 2c20 7468 6520 6465 2Doc)ZZ,, the de │ │ │ │ -00037e00: 6772 6565 206f 6620 7068 692e 2053 6f20 gree of phi. So │ │ │ │ -00037e10: 7468 6973 2076 616c 7565 0a20 2020 2020 this value. │ │ │ │ -00037e20: 2020 2069 7320 3120 6966 2061 6e64 206f is 1 if and o │ │ │ │ -00037e30: 6e6c 7920 6966 2074 6865 206d 6170 2069 nly if the map i │ │ │ │ -00037e40: 7320 6269 7261 7469 6f6e 616c 206f 6e74 s birational ont │ │ │ │ -00037e50: 6f20 6974 7320 696d 6167 652e 0a0a 4465 o its image...De │ │ │ │ -00037e60: 7363 7269 7074 696f 6e0a 3d3d 3d3d 3d3d scription.====== │ │ │ │ -00037e70: 3d3d 3d3d 3d0a 0a54 6869 7320 6973 2061 =====..This is a │ │ │ │ -00037e80: 2073 686f 7274 6375 7420 666f 7220 6465 shortcut for de │ │ │ │ -00037e90: 6772 6565 4d61 7028 7068 692c 4365 7274 greeMap(phi,Cert │ │ │ │ -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 │ │ │ │ -00037ed0: 6174 696f 6e61 6c4d 6170 293a 2064 6567 ationalMap): deg │ │ │ │ -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 │ │ │ │ -00037f40: 6174 696f 6e61 6c4d 6170 293a 2064 6567 ationalMap): deg │ │ │ │ -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.---------- │ │ │ │ +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 │ │ │ │ +00037dd0: 733a 0a20 2020 2020 202a 2061 6e20 2a6e s:. * an *n │ │ │ │ +00037de0: 6f74 6520 696e 7465 6765 723a 2028 4d61 ote integer: (Ma │ │ │ │ +00037df0: 6361 756c 6179 3244 6f63 295a 5a2c 2c20 caulay2Doc)ZZ,, │ │ │ │ +00037e00: 7468 6520 6465 6772 6565 206f 6620 7068 the degree of ph │ │ │ │ +00037e10: 692e 2053 6f20 7468 6973 2076 616c 7565 i. So this value │ │ │ │ +00037e20: 0a20 2020 2020 2020 2069 7320 3120 6966 . is 1 if │ │ │ │ +00037e30: 2061 6e64 206f 6e6c 7920 6966 2074 6865 and only if the │ │ │ │ +00037e40: 206d 6170 2069 7320 6269 7261 7469 6f6e map is biration │ │ │ │ +00037e50: 616c 206f 6e74 6f20 6974 7320 696d 6167 al onto its imag │ │ │ │ +00037e60: 652e 0a0a 4465 7363 7269 7074 696f 6e0a e...Description. │ │ │ │ +00037e70: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 ===========..Thi │ │ │ │ +00037e80: 7320 6973 2061 2073 686f 7274 6375 7420 s is a shortcut │ │ │ │ +00037e90: 666f 7220 6465 6772 6565 4d61 7028 7068 for degreeMap(ph │ │ │ │ +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 ================ │ │ │ │ +00037f30: 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 6465 ==.. * *note de │ │ │ │ +00037f40: 6772 6565 2852 6174 696f 6e61 6c4d 6170 gree(RationalMap │ │ │ │ +00037f50: 293a 2064 6567 7265 655f 6c70 5261 7469 ): degree_lpRati │ │ │ │ +00037f60: 6f6e 616c 4d61 705f 7270 2c20 2d2d 2064 onalMap_rp, -- d │ │ │ │ +00037f70: 6567 7265 6520 6f66 2061 2072 6174 696f egree of a ratio │ │ │ │ +00037f80: 6e61 6c0a 2020 2020 6d61 700a 2d2d 2d2d nal. 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 ---------------- │ │ │ │ -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 3236 h/macaulay2-1.26 │ │ │ │ -00038020: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+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.************** │ │ │ │ +00037fd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 6865 -----------..The │ │ │ │ +00037fe0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ +00037ff0: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ +00038000: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ +00038010: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ +00038020: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ +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.******** │ │ │ │ 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 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 ---------------- │ │ │ │ +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 ...+------------ │ │ │ │ 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 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 ---------------- │ │ │ │ +00038720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00038730: 2d2b 0a7c 6931 203a 202d 2d20 5461 6b65 -+.|i1 : -- Take │ │ │ │ +00038740: 2061 2072 6174 696f 6e61 6c20 6d61 7020 a rational map │ │ │ │ +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 |.|------------ │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 00038890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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) │ │ │ │ +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) │ │ │ │ 000388e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000388f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038900: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ -00038910: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00038900: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038910: 207c 0a2b 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 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 │ │ │ │ +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 │ │ │ │ +00038990: 7472 6978 2070 6163 6b28 362c 666f 7220 trix pack(6,for │ │ │ │ +000389a0: 6920 746f 2020 2020 2020 2020 2020 2020 i to │ │ │ │ +000389b0: 207c 0a7c 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 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 │ │ │ │ -00038a50: 203d 206d 6170 2028 7269 6e67 5038 2c20 = map (ringP8, │ │ │ │ -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 │ │ │ │ +000389f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038a00: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 │ │ │ │ +00038a60: 6e67 5038 2c20 7269 6e67 5031 342c 207b ngP8, ringP14, { │ │ │ │ +00038a70: 2d20 3935 7820 202b 2031 3831 7820 7820 - 95x + 181x x │ │ │ │ +00038a80: 202b 2031 3032 3878 2020 2d20 3133 3834 + 1028x - 1384 │ │ │ │ +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 |.| │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 00038b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038b80: 2020 2020 2020 2020 2020 207c 0a7c 2d2d |.|-- │ │ │ │ -00038b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00038b80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038b90: 207c 0a7c 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 ---------------- │ │ │ │ -00038bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 3131 -----------|.|11 │ │ │ │ -00038be0: 206c 6973 7420 7261 6e64 6f6d 2831 2c72 list random(1,r │ │ │ │ -00038bf0: 696e 6750 3829 2929 2920 2020 2020 2020 ingP8)))) │ │ │ │ +00038bd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00038be0: 2d7c 0a7c 3131 206c 6973 7420 7261 6e64 -|.|11 list rand │ │ │ │ +00038bf0: 6f6d 2831 2c72 696e 6750 3829 2929 2920 om(1,ringP8)))) │ │ │ │ 00038c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00038c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038c20: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ -00038c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038c20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038c30: 207c 0a7c 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 207c 0a7c 2020 |.| │ │ │ │ -00038c80: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ -00038c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038c70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038c80: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00038c90: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ 00038ca0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038cb0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ -00038cc0: 2020 2020 2020 2020 2020 207c 0a7c 2d20 |.|- │ │ │ │ -00038cd0: 3134 3535 7820 7820 202b 2035 3539 7820 1455x x + 559x │ │ │ │ -00038ce0: 202d 2035 3032 7820 7820 202b 2031 3236 - 502x x + 126 │ │ │ │ -00038cf0: 3478 2078 2020 2d20 3136 3278 2078 2020 4x x - 162x x │ │ │ │ -00038d00: 2b20 3132 3039 7820 202d 2031 3830 7820 + 1209x - 180x │ │ │ │ -00038d10: 7820 202d 2020 2020 2020 207c 0a7c 2020 x - |.| │ │ │ │ -00038d20: 2020 2020 2031 2032 2020 2020 2020 2032 1 2 2 │ │ │ │ -00038d30: 2020 2020 2020 2030 2033 2020 2020 2020 0 3 │ │ │ │ -00038d40: 2020 3120 3320 2020 2020 2020 3220 3320 1 3 2 3 │ │ │ │ -00038d50: 2020 2020 2020 2033 2020 2020 2020 2030 3 0 │ │ │ │ -00038d60: 2034 2020 2020 2020 2020 207c 0a7c 2d2d 4 |.|-- │ │ │ │ -00038d70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00038cb0: 2020 2020 2020 2020 2020 2020 2032 2020 2 │ │ │ │ +00038cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038cd0: 207c 0a7c 2d20 3134 3535 7820 7820 202b |.|- 1455x x + │ │ │ │ +00038ce0: 2035 3539 7820 202d 2035 3032 7820 7820 559x - 502x x │ │ │ │ +00038cf0: 202b 2031 3236 3478 2078 2020 2d20 3136 + 1264x x - 16 │ │ │ │ +00038d00: 3278 2078 2020 2b20 3132 3039 7820 202d 2x x + 1209x - │ │ │ │ +00038d10: 2031 3830 7820 7820 202d 2020 2020 2020 180x x - │ │ │ │ +00038d20: 207c 0a7c 2020 2020 2020 2031 2032 2020 |.| 1 2 │ │ │ │ +00038d30: 2020 2020 2032 2020 2020 2020 2030 2033 2 0 3 │ │ │ │ +00038d40: 2020 2020 2020 2020 3120 3320 2020 2020 1 3 │ │ │ │ +00038d50: 2020 3220 3320 2020 2020 2020 2033 2020 2 3 3 │ │ │ │ +00038d60: 2020 2020 2030 2034 2020 2020 2020 2020 0 4 │ │ │ │ +00038d70: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ 00038d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00038da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00038db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c 0a7c 2020 -----------|.| │ │ │ │ -00038dc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00038db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00038dc0: 2d7c 0a7c 2020 2020 2020 2020 2020 2020 -|.| │ │ │ │ 00038dd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038de0: 2032 2020 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00038de0: 2020 2020 2020 2032 2020 2020 2020 2020 2 │ │ │ │ 00038df0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00038e00: 2020 2020 2020 2020 2020 207c 0a7c 3530 |.|50 │ │ │ │ -00038e10: 3478 2078 2020 2d20 3131 3638 7820 7820 4x x - 1168x x │ │ │ │ -00038e20: 202d 2036 3736 7820 7820 202b 2035 3031 - 676x x + 501 │ │ │ │ -00038e30: 7820 202b 2037 3378 2078 2020 2b20 3132 x + 73x x + 12 │ │ │ │ -00038e40: 3633 7820 7820 202b 2031 3033 3578 2078 63x x + 1035x x │ │ │ │ -00038e50: 2020 2b20 3834 3478 2078 207c 0a7c 2020 + 844x x |.| │ │ │ │ -00038e60: 2020 3120 3420 2020 2020 2020 2032 2034 1 4 2 4 │ │ │ │ -00038e70: 2020 2020 2020 2033 2034 2020 2020 2020 3 4 │ │ │ │ -00038e80: 2034 2020 2020 2020 3020 3520 2020 2020 4 0 5 │ │ │ │ -00038e90: 2020 2031 2035 2020 2020 2020 2020 3220 1 5 2 │ │ │ │ -00038ea0: 3520 2020 2020 2020 3320 357c 0a7c 2d2d 5 3 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2020 2020 2020 2020 │ │ │ │ +00040020: 207c 0a7c 6f36 203d 206d 6170 2028 7269 |.|o6 = map (ri │ │ │ │ +00040030: 6e67 5038 2c20 7269 6e67 5038 2c20 7b2d ngP8, ringP8, {- │ │ │ │ +00040040: 2037 3830 7820 202d 2035 3036 7820 7820 780x - 506x x │ │ │ │ +00040050: 202b 2031 3533 3778 2020 2d20 3133 3278 + 1537x - 132x │ │ │ │ +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 |.| │ │ │ │ 000400d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000400e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000400f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00040100: 2020 2020 2020 2020 2020 207c 0a7c 6f36 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---------------- │ │ │ │ +000447a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000447b0: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ 000447c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000447d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000447e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000447f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5365 -----------+..Se │ │ │ │ -00044800: 6520 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a e also.========. │ │ │ │ -00044810: 0a20 202a 202a 6e6f 7465 2064 6567 7265 . * *note degre │ │ │ │ -00044820: 6528 5261 7469 6f6e 616c 4d61 7029 3a20 e(RationalMap): │ │ │ │ -00044830: 6465 6772 6565 5f6c 7052 6174 696f 6e61 degree_lpRationa │ │ │ │ -00044840: 6c4d 6170 5f72 702c 202d 2d20 6465 6772 lMap_rp, -- degr │ │ │ │ -00044850: 6565 206f 6620 6120 7261 7469 6f6e 616c ee of a rational │ │ │ │ -00044860: 0a20 2020 206d 6170 0a20 202a 202a 6e6f . map. * *no │ │ │ │ -00044870: 7465 2070 726f 6a65 6374 6976 6544 6567 te projectiveDeg │ │ │ │ -00044880: 7265 6573 3a20 7072 6f6a 6563 7469 7665 rees: projective │ │ │ │ -00044890: 4465 6772 6565 732c 202d 2d20 7072 6f6a Degrees, -- proj │ │ │ │ -000448a0: 6563 7469 7665 2064 6567 7265 6573 206f ective degrees o │ │ │ │ -000448b0: 6620 610a 2020 2020 7261 7469 6f6e 616c f a. rational │ │ │ │ -000448c0: 206d 6170 2062 6574 7765 656e 2070 726f map between pro │ │ │ │ -000448d0: 6a65 6374 6976 6520 7661 7269 6574 6965 jective varietie │ │ │ │ -000448e0: 730a 0a57 6179 7320 746f 2075 7365 2064 s..Ways to use d │ │ │ │ -000448f0: 6567 7265 654d 6170 3a0a 3d3d 3d3d 3d3d egreeMap:.====== │ │ │ │ +000447f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00044800: 2d2b 0a0a 5365 6520 616c 736f 0a3d 3d3d -+..See also.=== │ │ │ │ +00044810: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +00044820: 2064 6567 7265 6528 5261 7469 6f6e 616c degree(Rational │ │ │ │ +00044830: 4d61 7029 3a20 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3d3d ================ │ │ │ │ -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 ---------------- │ │ │ │ +00044910: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2264 6567 ======.. * "deg │ │ │ │ +00044920: 7265 654d 6170 2852 696e 674d 6170 2922 reeMap(RingMap)" │ │ │ │ +00044930: 0a20 202a 202a 6e6f 7465 2064 6567 7265 . * *note degre │ │ │ │ +00044940: 654d 6170 2852 6174 696f 6e61 6c4d 6170 eMap(RationalMap │ │ │ │ +00044950: 293a 2064 6567 7265 654d 6170 5f6c 7052 ): degreeMap_lpR │ │ │ │ +00044960: 6174 696f 6e61 6c4d 6170 5f72 702c 202d ationalMap_rp, - │ │ │ │ +00044970: 2d20 6465 6772 6565 206f 6620 610a 2020 - degree of a. │ │ │ │ +00044980: 2020 7261 7469 6f6e 616c 206d 6170 0a0a rational map.. │ │ │ │ +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 ..-------------- │ │ │ │ 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 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 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+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.******** │ │ │ │ +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 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +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.** │ │ │ │ 00044bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00044bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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.------ │ │ │ │ +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. │ │ │ │ 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 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 3236 2e30 362b 6473 2f4d 322f 4d61 1.26.06+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 .*************** │ │ │ │ +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 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ +000451b0: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ +000451c0: 636b 6167 6573 2f43 7265 6d6f 6e61 2f0a ckages/Cremona/. │ │ │ │ +000451d0: 646f 6375 6d65 6e74 6174 696f 6e2e 6d32 documentation.m2 │ │ │ │ +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.********* │ │ │ │ 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 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 ---------------- │ │ │ │ +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.------------- │ │ │ │ 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 0a0a 5468 ------------..Th │ │ │ │ -00045590: 6520 736f 7572 6365 206f 6620 7468 6973 e source of this │ │ │ │ -000455a0: 2064 6f63 756d 656e 7420 6973 2069 6e0a document is in. │ │ │ │ -000455b0: 2f62 7569 6c64 2f72 6570 726f 6475 6369 /build/reproduci │ │ │ │ -000455c0: 626c 652d 7061 7468 2f6d 6163 6175 6c61 ble-path/macaula │ │ │ │ -000455d0: 7932 2d31 2e32 362e 3036 2b64 732f 4d32 y2-1.26.06+ds/M2 │ │ │ │ -000455e0: 2f4d 6163 6175 6c61 7932 2f70 6163 6b61 /Macaulay2/packa │ │ │ │ -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 │ │ │ │ -00045670: 6f6e 616c 4d61 705f 7270 2c20 5570 3a20 onalMap_rp, Up: │ │ │ │ -00045680: 546f 700a 0a64 6573 6372 6962 6528 5261 Top..describe(Ra │ │ │ │ -00045690: 7469 6f6e 616c 4d61 7029 202d 2d20 6465 tionalMap) -- de │ │ │ │ -000456a0: 7363 7269 6265 2061 2072 6174 696f 6e61 scribe a rationa │ │ │ │ -000456b0: 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a 2a2a l map.********** │ │ │ │ +00045580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045590: 2d2d 0a0a 5468 6520 736f 7572 6365 206f --..The source o │ │ │ │ +000455a0: 6620 7468 6973 2064 6f63 756d 656e 7420 f this document │ │ │ │ +000455b0: 6973 2069 6e0a 2f62 7569 6c64 2f72 6570 is in./build/rep │ │ │ │ +000455c0: 726f 6475 6369 626c 652d 7061 7468 2f6d roducible-path/m │ │ │ │ +000455d0: 6163 6175 6c61 7932 2d31 2e32 362e 3036 acaulay2-1.26.06 │ │ │ │ +000455e0: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +000455f0: 2f70 6163 6b61 6765 732f 4372 656d 6f6e /packages/Cremon │ │ │ │ +00045600: 612f 0a64 6f63 756d 656e 7461 7469 6f6e a/.documentation │ │ │ │ +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.**** │ │ │ │ 000456c0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000456d0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -000456e0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 ******.. * Func │ │ │ │ -000456f0: 7469 6f6e 3a20 2a6e 6f74 6520 6465 7363 tion: *note desc │ │ │ │ -00045700: 7269 6265 3a20 284d 6163 6175 6c61 7932 ribe: (Macaulay2 │ │ │ │ -00045710: 446f 6329 6465 7363 7269 6265 2c0a 2020 Doc)describe,. │ │ │ │ -00045720: 2a20 5573 6167 653a 200a 2020 2020 2020 * Usage: . │ │ │ │ -00045730: 2020 6465 7363 7269 6265 2070 6869 0a20 describe phi. │ │ │ │ -00045740: 202a 2049 6e70 7574 733a 0a20 2020 2020 * Inputs:. │ │ │ │ -00045750: 202a 2070 6869 2c20 6120 2a6e 6f74 6520 * phi, a *note │ │ │ │ -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 │ │ │ │ -000457a0: 2070 6869 2c20 6769 7669 6e67 2073 6f6d phi, giving som │ │ │ │ -000457b0: 6520 696e 6469 6361 7469 6f6e 206f 6620 e indication of │ │ │ │ -000457c0: 7768 6174 2068 6173 2061 6c72 6561 6479 what has already │ │ │ │ -000457d0: 2062 6565 6e0a 2020 2020 2020 2020 6361 been. ca │ │ │ │ -000457e0: 6c63 756c 6174 6564 2e0a 0a44 6573 6372 lculated...Descr │ │ │ │ -000457f0: 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d iption.========= │ │ │ │ -00045800: 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d ==..+----------- │ │ │ │ +000456e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 ************.. │ │ │ │ +000456f0: 2a20 4675 6e63 7469 6f6e 3a20 2a6e 6f74 * Function: *not │ │ │ │ +00045700: 6520 6465 7363 7269 6265 3a20 284d 6163 e describe: (Mac │ │ │ │ +00045710: 6175 6c61 7932 446f 6329 6465 7363 7269 aulay2Doc)descri │ │ │ │ +00045720: 6265 2c0a 2020 2a20 5573 6167 653a 200a be,. * Usage: . │ │ │ │ +00045730: 2020 2020 2020 2020 6465 7363 7269 6265 describe │ │ │ │ +00045740: 2070 6869 0a20 202a 2049 6e70 7574 733a phi. * Inputs: │ │ │ │ +00045750: 0a20 2020 2020 202a 2070 6869 2c20 6120 . * phi, a │ │ │ │ +00045760: 2a6e 6f74 6520 7261 7469 6f6e 616c 206d *note rational m │ │ │ │ +00045770: 6170 3a20 5261 7469 6f6e 616c 4d61 702c ap: RationalMap, │ │ │ │ +00045780: 0a20 202a 204f 7574 7075 7473 3a0a 2020 . * Outputs:. │ │ │ │ +00045790: 2020 2020 2a20 6120 6465 7363 7269 7074 * a descript │ │ │ │ +000457a0: 696f 6e20 6f66 2070 6869 2c20 6769 7669 ion of phi, givi │ │ │ │ +000457b0: 6e67 2073 6f6d 6520 696e 6469 6361 7469 ng some indicati │ │ │ │ +000457c0: 6f6e 206f 6620 7768 6174 2068 6173 2061 on of what has a │ │ │ │ +000457d0: 6c72 6561 6479 2062 6565 6e0a 2020 2020 lready been. │ │ │ │ +000457e0: 2020 2020 6361 6c63 756c 6174 6564 2e0a calculated.. │ │ │ │ +000457f0: 0a44 6573 6372 6970 7469 6f6e 0a3d 3d3d .Description.=== │ │ │ │ +00045800: 3d3d 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d ========..+----- │ │ │ │ 00045810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00045820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00045830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045840: 2d2d 2d2d 2b0a 7c69 3120 3a20 5a5a 2f33 ----+.|i1 : ZZ/3 │ │ │ │ -00045850: 3333 3331 5b74 5f30 2e2e 745f 345d 3b20 3331[t_0..t_4]; │ │ │ │ -00045860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045840: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3120 ----------+.|i1 │ │ │ │ +00045850: 3a20 5a5a 2f33 3333 3331 5b74 5f30 2e2e : ZZ/33331[t_0.. │ │ │ │ +00045860: 745f 345d 3b20 2020 2020 2020 2020 2020 t_4]; │ │ │ │ 00045870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045880: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +00045880: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 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 2b0a 7c69 3220 3a20 --------+.|i2 : │ │ │ │ -000458d0: 7068 6920 3d20 7261 7469 6f6e 616c 4d61 phi = rationalMa │ │ │ │ -000458e0: 7020 6d69 6e6f 7273 2832 2c6d 6174 7269 p minors(2,matri │ │ │ │ -000458f0: 787b 7b74 5f30 2e2e 745f 337d 2c7b 745f x{{t_0..t_3},{t_ │ │ │ │ -00045900: 312e 2e74 5f34 7d7d 293b 7c0a 7c20 2020 1..t_4}});|.| │ │ │ │ -00045910: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000458c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ +000458d0: 7c69 3220 3a20 7068 6920 3d20 7261 7469 |i2 : phi = rati │ │ │ │ +000458e0: 6f6e 616c 4d61 7020 6d69 6e6f 7273 2832 onalMap minors(2 │ │ │ │ +000458f0: 2c6d 6174 7269 787b 7b74 5f30 2e2e 745f ,matrix{{t_0..t_ │ │ │ │ +00045900: 337d 2c7b 745f 312e 2e74 5f34 7d7d 293b 3},{t_1..t_4}}); │ │ │ │ +00045910: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ 00045920: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045940: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00045950: 3220 3a20 5261 7469 6f6e 616c 4d61 7020 2 : RationalMap │ │ │ │ -00045960: 2871 7561 6472 6174 6963 2072 6174 696f (quadratic ratio │ │ │ │ -00045970: 6e61 6c20 6d61 7020 6672 6f6d 2050 505e nal map from PP^ │ │ │ │ -00045980: 3420 746f 2050 505e 3529 2020 2020 7c0a 4 to PP^5) |. │ │ │ │ -00045990: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00045940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045950: 2020 7c0a 7c6f 3220 3a20 5261 7469 6f6e |.|o2 : Ration │ │ │ │ +00045960: 616c 4d61 7020 2871 7561 6472 6174 6963 alMap (quadratic │ │ │ │ +00045970: 2072 6174 696f 6e61 6c20 6d61 7020 6672 rational map fr │ │ │ │ +00045980: 6f6d 2050 505e 3420 746f 2050 505e 3529 om PP^4 to PP^5) │ │ │ │ +00045990: 2020 2020 7c0a 2b2d 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 ---------------- │ │ │ │ -000459d0: 2b0a 7c69 3320 3a20 6465 7363 7269 6265 +.|i3 : describe │ │ │ │ -000459e0: 2070 6869 2020 2020 2020 2020 2020 2020 phi │ │ │ │ +000459d0: 2d2d 2d2d 2d2d 2b0a 7c69 3320 3a20 6465 ------+.|i3 : de │ │ │ │ +000459e0: 7363 7269 6265 2070 6869 2020 2020 2020 scribe phi │ │ │ │ 000459f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045a10: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +00045a10: 2020 2020 2020 2020 7c0a 7c20 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 │ │ │ │ -00045a50: 2020 2020 7c0a 7c6f 3320 3d20 7261 7469 |.|o3 = rati │ │ │ │ -00045a60: 6f6e 616c 206d 6170 2064 6566 696e 6564 onal map defined │ │ │ │ -00045a70: 2062 7920 666f 726d 7320 6f66 2064 6567 by forms of deg │ │ │ │ -00045a80: 7265 6520 3220 2020 2020 2020 2020 2020 ree 2 │ │ │ │ -00045a90: 2020 2020 2020 7c0a 7c20 2020 2020 736f |.| so │ │ │ │ -00045aa0: 7572 6365 2076 6172 6965 7479 3a20 5050 urce variety: PP │ │ │ │ -00045ab0: 5e34 2020 2020 2020 2020 2020 2020 2020 ^4 │ │ │ │ +00045a50: 2020 2020 2020 2020 2020 7c0a 7c6f 3320 |.|o3 │ │ │ │ +00045a60: 3d20 7261 7469 6f6e 616c 206d 6170 2064 = rational map d │ │ │ │ +00045a70: 6566 696e 6564 2062 7920 666f 726d 7320 efined by forms │ │ │ │ +00045a80: 6f66 2064 6567 7265 6520 3220 2020 2020 of degree 2 │ │ │ │ +00045a90: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00045aa0: 2020 2020 736f 7572 6365 2076 6172 6965 source varie │ │ │ │ +00045ab0: 7479 3a20 5050 5e34 2020 2020 2020 2020 ty: PP^4 │ │ │ │ 00045ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00045ae0: 7461 7267 6574 2076 6172 6965 7479 3a20 target variety: │ │ │ │ -00045af0: 5050 5e35 2020 2020 2020 2020 2020 2020 PP^5 │ │ │ │ +00045ad0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00045ae0: 7c20 2020 2020 7461 7267 6574 2076 6172 | target var │ │ │ │ +00045af0: 6965 7479 3a20 5050 5e35 2020 2020 2020 iety: PP^5 │ │ │ │ 00045b00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b10: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00045b20: 2020 636f 6566 6669 6369 656e 7420 7269 coefficient ri │ │ │ │ -00045b30: 6e67 3a20 5a5a 2f33 3333 3331 2020 2020 ng: ZZ/33331 │ │ │ │ -00045b40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045b50: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00045b60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045b20: 7c0a 7c20 2020 2020 636f 6566 6669 6369 |.| coeffici │ │ │ │ +00045b30: 656e 7420 7269 6e67 3a20 5a5a 2f33 3333 ent ring: ZZ/333 │ │ │ │ +00045b40: 3331 2020 2020 2020 2020 2020 2020 2020 31 │ │ │ │ +00045b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045b60: 2020 7c0a 2b2d 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 2b0a --------------+. │ │ │ │ -00045ba0: 7c69 3420 3a20 4920 3d20 696d 6167 6520 |i4 : I = image │ │ │ │ -00045bb0: 7068 693b 2020 2020 2020 2020 2020 2020 phi; │ │ │ │ +00045b90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045ba0: 2d2d 2d2d 2b0a 7c69 3420 3a20 4920 3d20 ----+.|i4 : I = │ │ │ │ +00045bb0: 696d 6167 6520 7068 693b 2020 2020 2020 image phi; │ │ │ │ 00045bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045bd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045be0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00045be0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00045bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c20: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00045c30: 2020 2020 205a 5a20 2020 2020 2020 2020 ZZ │ │ │ │ +00045c20: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00045c30: 2020 2020 2020 2020 2020 205a 5a20 2020 ZZ │ │ │ │ 00045c40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045c50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045c60: 2020 2020 7c0a 7c6f 3420 3a20 4964 6561 |.|o4 : Idea │ │ │ │ -00045c70: 6c20 6f66 202d 2d2d 2d2d 5b78 202e 2e78 l of -----[x ..x │ │ │ │ -00045c80: 205d 2020 2020 2020 2020 2020 2020 2020 ] │ │ │ │ +00045c60: 2020 2020 2020 2020 2020 7c0a 7c6f 3420 |.|o4 │ │ │ │ +00045c70: 3a20 4964 6561 6c20 6f66 202d 2d2d 2d2d : Ideal of ----- │ │ │ │ +00045c80: 5b78 202e 2e78 205d 2020 2020 2020 2020 [x ..x ] │ │ │ │ 00045c90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045ca0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ -00045cb0: 2020 2020 2020 2033 3333 3331 2020 3020 33331 0 │ │ │ │ -00045cc0: 2020 3520 2020 2020 2020 2020 2020 2020 5 │ │ │ │ +00045ca0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00045cb0: 2020 2020 2020 2020 2020 2020 2033 3333 333 │ │ │ │ +00045cc0: 3331 2020 3020 2020 3520 2020 2020 2020 31 0 5 │ │ │ │ 00045cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045ce0: 2020 2020 2020 2020 7c0a 2b2d 2d2d 2d2d |.+----- │ │ │ │ -00045cf0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045ce0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00045cf0: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ 00045d00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00045d10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00045d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 3520 ----------+.|i5 │ │ │ │ -00045d30: 3a20 6465 7363 7269 6265 2070 6869 2020 : describe phi │ │ │ │ -00045d40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045d20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045d30: 2b0a 7c69 3520 3a20 6465 7363 7269 6265 +.|i5 : describe │ │ │ │ +00045d40: 2070 6869 2020 2020 2020 2020 2020 2020 phi │ │ │ │ 00045d50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045d60: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00045d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045d60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045d70: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00045d80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045d90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045da0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00045db0: 7c6f 3520 3d20 7261 7469 6f6e 616c 206d |o5 = rational m │ │ │ │ -00045dc0: 6170 2064 6566 696e 6564 2062 7920 666f ap defined by fo │ │ │ │ -00045dd0: 726d 7320 6f66 2064 6567 7265 6520 3220 rms of degree 2 │ │ │ │ -00045de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045df0: 7c0a 7c20 2020 2020 736f 7572 6365 2076 |.| source v │ │ │ │ -00045e00: 6172 6965 7479 3a20 5050 5e34 2020 2020 ariety: PP^4 │ │ │ │ -00045e10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045da0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045db0: 2020 2020 7c0a 7c6f 3520 3d20 7261 7469 |.|o5 = rati │ │ │ │ +00045dc0: 6f6e 616c 206d 6170 2064 6566 696e 6564 onal map defined │ │ │ │ +00045dd0: 2062 7920 666f 726d 7320 6f66 2064 6567 by forms of deg │ │ │ │ +00045de0: 7265 6520 3220 2020 2020 2020 2020 2020 ree 2 │ │ │ │ +00045df0: 2020 2020 2020 7c0a 7c20 2020 2020 736f |.| so │ │ │ │ +00045e00: 7572 6365 2076 6172 6965 7479 3a20 5050 urce variety: PP │ │ │ │ +00045e10: 5e34 2020 2020 2020 2020 2020 2020 2020 ^4 │ │ │ │ 00045e20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045e30: 2020 7c0a 7c20 2020 2020 7461 7267 6574 |.| target │ │ │ │ -00045e40: 2076 6172 6965 7479 3a20 5050 5e35 2020 variety: PP^5 │ │ │ │ -00045e50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045e30: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00045e40: 7461 7267 6574 2076 6172 6965 7479 3a20 target variety: │ │ │ │ +00045e50: 5050 5e35 2020 2020 2020 2020 2020 2020 PP^5 │ │ │ │ 00045e60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045e70: 2020 2020 7c0a 7c20 2020 2020 696d 6167 |.| imag │ │ │ │ -00045e80: 653a 2073 6d6f 6f74 6820 7175 6164 7269 e: smooth quadri │ │ │ │ -00045e90: 6320 6879 7065 7273 7572 6661 6365 2069 c hypersurface i │ │ │ │ -00045ea0: 6e20 5050 5e35 2020 2020 2020 2020 2020 n PP^5 │ │ │ │ -00045eb0: 2020 2020 2020 7c0a 7c20 2020 2020 646f |.| do │ │ │ │ -00045ec0: 6d69 6e61 6e63 653a 2066 616c 7365 2020 minance: false │ │ │ │ -00045ed0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045e70: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +00045e80: 2020 696d 6167 653a 2073 6d6f 6f74 6820 image: smooth │ │ │ │ +00045e90: 7175 6164 7269 6320 6879 7065 7273 7572 quadric hypersur │ │ │ │ +00045ea0: 6661 6365 2069 6e20 5050 5e35 2020 2020 face in PP^5 │ │ │ │ +00045eb0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +00045ec0: 2020 2020 646f 6d69 6e61 6e63 653a 2066 dominance: f │ │ │ │ +00045ed0: 616c 7365 2020 2020 2020 2020 2020 2020 alse │ │ │ │ 00045ee0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045ef0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00045f00: 6269 7261 7469 6f6e 616c 6974 793a 2066 birationality: f │ │ │ │ -00045f10: 616c 7365 2020 2020 2020 2020 2020 2020 alse │ │ │ │ +00045ef0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00045f00: 7c20 2020 2020 6269 7261 7469 6f6e 616c | birational │ │ │ │ +00045f10: 6974 793a 2066 616c 7365 2020 2020 2020 ity: false │ │ │ │ 00045f20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045f30: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00045f40: 2020 636f 6566 6669 6369 656e 7420 7269 coefficient ri │ │ │ │ -00045f50: 6e67 3a20 5a5a 2f33 3333 3331 2020 2020 ng: ZZ/33331 │ │ │ │ -00045f60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00045f70: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ -00045f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045f30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045f40: 7c0a 7c20 2020 2020 636f 6566 6669 6369 |.| coeffici │ │ │ │ +00045f50: 656e 7420 7269 6e67 3a20 5a5a 2f33 3333 ent ring: ZZ/333 │ │ │ │ +00045f60: 3331 2020 2020 2020 2020 2020 2020 2020 31 │ │ │ │ +00045f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00045f80: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 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 2b0a --------------+. │ │ │ │ -00045fc0: 7c69 3620 3a20 3f20 4920 2020 2020 2020 |i6 : ? I │ │ │ │ +00045fb0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00045fc0: 2d2d 2d2d 2b0a 7c69 3620 3a20 3f20 4920 ----+.|i6 : ? I │ │ │ │ 00045fd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045fe0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00045ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046000: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00046000: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00046010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046030: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046040: 2020 7c0a 7c6f 3620 3d20 736d 6f6f 7468 |.|o6 = smooth │ │ │ │ -00046050: 2071 7561 6472 6963 2068 7970 6572 7375 quadric hypersu │ │ │ │ -00046060: 7266 6163 6520 696e 2050 505e 3520 2020 rface in PP^5 │ │ │ │ -00046070: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046080: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ +00046040: 2020 2020 2020 2020 7c0a 7c6f 3620 3d20 |.|o6 = │ │ │ │ +00046050: 736d 6f6f 7468 2071 7561 6472 6963 2068 smooth quadric h │ │ │ │ +00046060: 7970 6572 7375 7266 6163 6520 696e 2050 ypersurface in P │ │ │ │ +00046070: 505e 3520 2020 2020 2020 2020 2020 2020 P^5 │ │ │ │ +00046080: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ 00046090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000460a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000460b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000460c0: 2d2d 2d2d 2d2d 2b0a 7c69 3720 3a20 7068 ------+.|i7 : ph │ │ │ │ -000460d0: 6921 3b20 2020 2020 2020 2020 2020 2020 i!; │ │ │ │ +000460c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +000460d0: 3720 3a20 7068 6921 3b20 2020 2020 2020 7 : phi!; │ │ │ │ 000460e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000460f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046100: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00046110: 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 | │ │ │ │ 00046120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046140: 2020 2020 2020 2020 2020 7c0a 7c6f 3720 |.|o7 │ │ │ │ -00046150: 3a20 5261 7469 6f6e 616c 4d61 7020 2871 : RationalMap (q │ │ │ │ -00046160: 7561 6472 6174 6963 2072 6174 696f 6e61 uadratic rationa │ │ │ │ -00046170: 6c20 6d61 7020 6672 6f6d 2050 505e 3420 l map from PP^4 │ │ │ │ -00046180: 746f 2050 505e 3529 2020 2020 7c0a 2b2d to PP^5) |.+- │ │ │ │ -00046190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046140: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046150: 7c0a 7c6f 3720 3a20 5261 7469 6f6e 616c |.|o7 : Rational │ │ │ │ +00046160: 4d61 7020 2871 7561 6472 6174 6963 2072 Map (quadratic r │ │ │ │ +00046170: 6174 696f 6e61 6c20 6d61 7020 6672 6f6d ational map from │ │ │ │ +00046180: 2050 505e 3420 746f 2050 505e 3529 2020 PP^4 to PP^5) │ │ │ │ +00046190: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 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 2b0a --------------+. │ │ │ │ -000461d0: 7c69 3820 3a20 6465 7363 7269 6265 2070 |i8 : describe p │ │ │ │ -000461e0: 6869 2020 2020 2020 2020 2020 2020 2020 hi │ │ │ │ +000461c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000461d0: 2d2d 2d2d 2b0a 7c69 3820 3a20 6465 7363 ----+.|i8 : desc │ │ │ │ +000461e0: 7269 6265 2070 6869 2020 2020 2020 2020 ribe phi │ │ │ │ 000461f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046210: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00046210: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00046220: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046240: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046250: 2020 7c0a 7c6f 3820 3d20 7261 7469 6f6e |.|o8 = ration │ │ │ │ -00046260: 616c 206d 6170 2064 6566 696e 6564 2062 al map defined b │ │ │ │ -00046270: 7920 666f 726d 7320 6f66 2064 6567 7265 y forms of degre │ │ │ │ -00046280: 6520 3220 2020 2020 2020 2020 2020 2020 e 2 │ │ │ │ -00046290: 2020 2020 7c0a 7c20 2020 2020 736f 7572 |.| sour │ │ │ │ -000462a0: 6365 2076 6172 6965 7479 3a20 5050 5e34 ce variety: PP^4 │ │ │ │ -000462b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046250: 2020 2020 2020 2020 7c0a 7c6f 3820 3d20 |.|o8 = │ │ │ │ +00046260: 7261 7469 6f6e 616c 206d 6170 2064 6566 rational map def │ │ │ │ +00046270: 696e 6564 2062 7920 666f 726d 7320 6f66 ined by forms of │ │ │ │ +00046280: 2064 6567 7265 6520 3220 2020 2020 2020 degree 2 │ │ │ │ +00046290: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000462a0: 2020 736f 7572 6365 2076 6172 6965 7479 source variety │ │ │ │ +000462b0: 3a20 5050 5e34 2020 2020 2020 2020 2020 : PP^4 │ │ │ │ 000462c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000462d0: 2020 2020 2020 7c0a 7c20 2020 2020 7461 |.| ta │ │ │ │ -000462e0: 7267 6574 2076 6172 6965 7479 3a20 5050 rget variety: PP │ │ │ │ -000462f0: 5e35 2020 2020 2020 2020 2020 2020 2020 ^5 │ │ │ │ +000462d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000462e0: 2020 2020 7461 7267 6574 2076 6172 6965 target varie │ │ │ │ +000462f0: 7479 3a20 5050 5e35 2020 2020 2020 2020 ty: PP^5 │ │ │ │ 00046300: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046310: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00046320: 696d 6167 653a 2073 6d6f 6f74 6820 7175 image: smooth qu │ │ │ │ -00046330: 6164 7269 6320 6879 7065 7273 7572 6661 adric hypersurfa │ │ │ │ -00046340: 6365 2069 6e20 5050 5e35 2020 2020 2020 ce in PP^5 │ │ │ │ -00046350: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ -00046360: 2020 646f 6d69 6e61 6e63 653a 2066 616c dominance: fal │ │ │ │ -00046370: 7365 2020 2020 2020 2020 2020 2020 2020 se │ │ │ │ +00046310: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00046320: 7c20 2020 2020 696d 6167 653a 2073 6d6f | image: smo │ │ │ │ +00046330: 6f74 6820 7175 6164 7269 6320 6879 7065 oth quadric hype │ │ │ │ +00046340: 7273 7572 6661 6365 2069 6e20 5050 5e35 rsurface in PP^5 │ │ │ │ +00046350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046360: 7c0a 7c20 2020 2020 646f 6d69 6e61 6e63 |.| dominanc │ │ │ │ +00046370: 653a 2066 616c 7365 2020 2020 2020 2020 e: false │ │ │ │ 00046380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046390: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -000463a0: 2020 2020 6269 7261 7469 6f6e 616c 6974 birationalit │ │ │ │ -000463b0: 793a 2066 616c 7365 2020 2020 2020 2020 y: false │ │ │ │ +00046390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000463a0: 2020 7c0a 7c20 2020 2020 6269 7261 7469 |.| birati │ │ │ │ +000463b0: 6f6e 616c 6974 793a 2066 616c 7365 2020 onality: false │ │ │ │ 000463c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000463d0: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -000463e0: 7c20 2020 2020 6465 6772 6565 206f 6620 | degree of │ │ │ │ -000463f0: 6d61 703a 2031 2020 2020 2020 2020 2020 map: 1 │ │ │ │ +000463d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000463e0: 2020 2020 7c0a 7c20 2020 2020 6465 6772 |.| degr │ │ │ │ +000463f0: 6565 206f 6620 6d61 703a 2031 2020 2020 ee of map: 1 │ │ │ │ 00046400: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00046410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046420: 7c0a 7c20 2020 2020 7072 6f6a 6563 7469 |.| projecti │ │ │ │ -00046430: 7665 2064 6567 7265 6573 3a20 7b31 2c20 ve degrees: {1, │ │ │ │ -00046440: 322c 2034 2c20 342c 2032 7d20 2020 2020 2, 4, 4, 2} │ │ │ │ -00046450: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046460: 2020 7c0a 7c20 2020 2020 6e75 6d62 6572 |.| number │ │ │ │ -00046470: 206f 6620 6d69 6e69 6d61 6c20 7265 7072 of minimal repr │ │ │ │ -00046480: 6573 656e 7461 7469 7665 733a 2031 2020 esentatives: 1 │ │ │ │ -00046490: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000464a0: 2020 2020 7c0a 7c20 2020 2020 6469 6d65 |.| dime │ │ │ │ -000464b0: 6e73 696f 6e20 6261 7365 206c 6f63 7573 nsion base locus │ │ │ │ -000464c0: 3a20 3120 2020 2020 2020 2020 2020 2020 : 1 │ │ │ │ +00046420: 2020 2020 2020 7c0a 7c20 2020 2020 7072 |.| pr │ │ │ │ +00046430: 6f6a 6563 7469 7665 2064 6567 7265 6573 ojective degrees │ │ │ │ +00046440: 3a20 7b31 2c20 322c 2034 2c20 342c 2032 : {1, 2, 4, 4, 2 │ │ │ │ +00046450: 7d20 2020 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00046460: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00046470: 6e75 6d62 6572 206f 6620 6d69 6e69 6d61 number of minima │ │ │ │ +00046480: 6c20 7265 7072 6573 656e 7461 7469 7665 l representative │ │ │ │ +00046490: 733a 2031 2020 2020 2020 2020 2020 2020 s: 1 │ │ │ │ +000464a0: 2020 2020 2020 2020 2020 7c0a 7c20 2020 |.| │ │ │ │ +000464b0: 2020 6469 6d65 6e73 696f 6e20 6261 7365 dimension base │ │ │ │ +000464c0: 206c 6f63 7573 3a20 3120 2020 2020 2020 locus: 1 │ │ │ │ 000464d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000464e0: 2020 2020 2020 7c0a 7c20 2020 2020 6465 |.| de │ │ │ │ -000464f0: 6772 6565 2062 6173 6520 6c6f 6375 733a gree base locus: │ │ │ │ -00046500: 2034 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +000464e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +000464f0: 2020 2020 6465 6772 6565 2062 6173 6520 degree base │ │ │ │ +00046500: 6c6f 6375 733a 2034 2020 2020 2020 2020 locus: 4 │ │ │ │ 00046510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046520: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ -00046530: 636f 6566 6669 6369 656e 7420 7269 6e67 coefficient ring │ │ │ │ -00046540: 3a20 5a5a 2f33 3333 3331 2020 2020 2020 : ZZ/33331 │ │ │ │ +00046520: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ +00046530: 7c20 2020 2020 636f 6566 6669 6369 656e | coefficien │ │ │ │ +00046540: 7420 7269 6e67 3a20 5a5a 2f33 3333 3331 t ring: ZZ/33331 │ │ │ │ 00046550: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00046560: 2020 2020 2020 2020 2020 7c0a 2b2d 2d2d |.+--- │ │ │ │ -00046570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00046560: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00046570: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ 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 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 │ │ │ │ -000465d0: 6f6e 616c 4d61 7020 213a 2052 6174 696f onalMap !: Ratio │ │ │ │ -000465e0: 6e61 6c4d 6170 2021 2c20 2d2d 2063 616c nalMap !, -- cal │ │ │ │ -000465f0: 6375 6c61 7465 7320 6576 6572 7920 706f culates every po │ │ │ │ -00046600: 7373 6962 6c65 2074 6869 6e67 0a0a 5761 ssible thing..Wa │ │ │ │ -00046610: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ -00046620: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ -00046630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -00046640: 0a20 202a 202a 6e6f 7465 2064 6573 6372 . * *note descr │ │ │ │ -00046650: 6962 6528 5261 7469 6f6e 616c 4d61 7029 ibe(RationalMap) │ │ │ │ -00046660: 3a20 6465 7363 7269 6265 5f6c 7052 6174 : describe_lpRat │ │ │ │ -00046670: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20 ionalMap_rp, -- │ │ │ │ -00046680: 6465 7363 7269 6265 2061 0a20 2020 2072 describe a. r │ │ │ │ -00046690: 6174 696f 6e61 6c20 6d61 700a 2d2d 2d2d ational map.---- │ │ │ │ -000466a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000465a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000465b0: 2d2d 2b0a 0a53 6565 2061 6c73 6f0a 3d3d --+..See also.== │ │ │ │ +000465c0: 3d3d 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 ======.. * *not │ │ │ │ +000465d0: 6520 5261 7469 6f6e 616c 4d61 7020 213a e RationalMap !: │ │ │ │ +000465e0: 2052 6174 696f 6e61 6c4d 6170 2021 2c20 RationalMap !, │ │ │ │ +000465f0: 2d2d 2063 616c 6375 6c61 7465 7320 6576 -- calculates ev │ │ │ │ +00046600: 6572 7920 706f 7373 6962 6c65 2074 6869 ery possible thi │ │ │ │ +00046610: 6e67 0a0a 5761 7973 2074 6f20 7573 6520 ng..Ways to use │ │ │ │ +00046620: 7468 6973 206d 6574 686f 643a 0a3d 3d3d this method:.=== │ │ │ │ +00046630: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00046640: 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f 7465 =====.. * *note │ │ │ │ +00046650: 2064 6573 6372 6962 6528 5261 7469 6f6e describe(Ration │ │ │ │ +00046660: 616c 4d61 7029 3a20 6465 7363 7269 6265 alMap): describe │ │ │ │ +00046670: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72 _lpRationalMap_r │ │ │ │ +00046680: 702c 202d 2d20 6465 7363 7269 6265 2061 p, -- describe a │ │ │ │ +00046690: 0a20 2020 2072 6174 696f 6e61 6c20 6d61 . rational ma │ │ │ │ +000466a0: 700a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d p.-------------- │ │ │ │ 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 2d0a 0a54 6865 -----------..The │ │ │ │ -000466f0: 2073 6f75 7263 6520 6f66 2074 6869 7320 source of this │ │ │ │ -00046700: 646f 6375 6d65 6e74 2069 7320 696e 0a2f document is in./ │ │ │ │ -00046710: 6275 696c 642f 7265 7072 6f64 7563 6962 build/reproducib │ │ │ │ -00046720: 6c65 2d70 6174 682f 6d61 6361 756c 6179 le-path/macaulay │ │ │ │ -00046730: 322d 312e 3236 2e30 362b 6473 2f4d 322f 2-1.26.06+ds/M2/ │ │ │ │ -00046740: 4d61 6361 756c 6179 322f 7061 636b 6167 Macaulay2/packag │ │ │ │ -00046750: 6573 2f43 7265 6d6f 6e61 2f0a 646f 6375 es/Cremona/.docu │ │ │ │ -00046760: 6d65 6e74 6174 696f 6e2e 6d32 3a39 3833 mentation.m2:983 │ │ │ │ -00046770: 3a30 2e0a 1f0a 4669 6c65 3a20 4372 656d :0....File: Crem │ │ │ │ -00046780: 6f6e 612e 696e 666f 2c20 4e6f 6465 3a20 ona.info, Node: │ │ │ │ -00046790: 446f 6d69 6e61 6e74 2c20 4e65 7874 3a20 Dominant, Next: │ │ │ │ -000467a0: 656e 7472 6965 735f 6c70 5261 7469 6f6e entries_lpRation │ │ │ │ -000467b0: 616c 4d61 705f 7270 2c20 5072 6576 3a20 alMap_rp, Prev: │ │ │ │ -000467c0: 6465 7363 7269 6265 5f6c 7052 6174 696f describe_lpRatio │ │ │ │ -000467d0: 6e61 6c4d 6170 5f72 702c 2055 703a 2054 nalMap_rp, Up: T │ │ │ │ -000467e0: 6f70 0a0a 446f 6d69 6e61 6e74 0a2a 2a2a op..Dominant.*** │ │ │ │ -000467f0: 2a2a 2a2a 2a0a 0a44 6573 6372 6970 7469 *****..Descripti │ │ │ │ -00046800: 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a on.===========.. │ │ │ │ -00046810: 5468 6973 2069 7320 616e 206f 7074 696f This is an optio │ │ │ │ -00046820: 6e61 6c20 6172 6775 6d65 6e74 2066 6f72 nal argument for │ │ │ │ -00046830: 202a 6e6f 7465 2074 6f4d 6170 3a20 746f *note toMap: to │ │ │ │ -00046840: 4d61 702c 2e20 5768 656e 202a 6e6f 7465 Map,. When *note │ │ │ │ -00046850: 2074 7275 653a 0a28 4d61 6361 756c 6179 true:.(Macaulay │ │ │ │ -00046860: 3244 6f63 2974 7275 652c 206f 7220 2a6e 2Doc)true, or *n │ │ │ │ -00046870: 6f74 6520 696e 6669 6e69 7479 3a20 284d ote infinity: (M │ │ │ │ -00046880: 6163 6175 6c61 7932 446f 6329 696e 6669 acaulay2Doc)infi │ │ │ │ -00046890: 6e69 7479 2c20 6973 2070 6173 7365 6420 nity, is passed │ │ │ │ -000468a0: 746f 0a74 6869 7320 6f70 7469 6f6e 2c20 to.this option, │ │ │ │ -000468b0: 7468 6520 6b65 726e 656c 206f 6620 7468 the kernel of th │ │ │ │ -000468c0: 6520 7265 7475 726e 6564 2072 696e 6720 e returned ring │ │ │ │ -000468d0: 6d61 7020 7769 6c6c 2062 6520 7a65 726f map will be zero │ │ │ │ -000468e0: 2e0a 0a46 756e 6374 696f 6e73 2077 6974 ...Functions wit │ │ │ │ -000468f0: 6820 6f70 7469 6f6e 616c 2061 7267 756d h optional argum │ │ │ │ -00046900: 656e 7420 6e61 6d65 6420 446f 6d69 6e61 ent named Domina │ │ │ │ -00046910: 6e74 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d nt:.============ │ │ │ │ +000466e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000466f0: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ +00046700: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ +00046710: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ +00046720: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ +00046730: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ +00046740: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ +00046750: 7061 636b 6167 6573 2f43 7265 6d6f 6e61 packages/Cremona │ │ │ │ +00046760: 2f0a 646f 6375 6d65 6e74 6174 696f 6e2e /.documentation. │ │ │ │ +00046770: 6d32 3a39 3833 3a30 2e0a 1f0a 4669 6c65 m2:983:0....File │ │ │ │ +00046780: 3a20 4372 656d 6f6e 612e 696e 666f 2c20 : Cremona.info, │ │ │ │ +00046790: 4e6f 6465 3a20 446f 6d69 6e61 6e74 2c20 Node: Dominant, │ │ │ │ +000467a0: 4e65 7874 3a20 656e 7472 6965 735f 6c70 Next: entries_lp │ │ │ │ +000467b0: 5261 7469 6f6e 616c 4d61 705f 7270 2c20 RationalMap_rp, │ │ │ │ +000467c0: 5072 6576 3a20 6465 7363 7269 6265 5f6c Prev: describe_l │ │ │ │ +000467d0: 7052 6174 696f 6e61 6c4d 6170 5f72 702c pRationalMap_rp, │ │ │ │ +000467e0: 2055 703a 2054 6f70 0a0a 446f 6d69 6e61 Up: Top..Domina │ │ │ │ +000467f0: 6e74 0a2a 2a2a 2a2a 2a2a 2a0a 0a44 6573 nt.********..Des │ │ │ │ +00046800: 6372 6970 7469 6f6e 0a3d 3d3d 3d3d 3d3d cription.======= │ │ │ │ +00046810: 3d3d 3d3d 0a0a 5468 6973 2069 7320 616e ====..This is an │ │ │ │ +00046820: 206f 7074 696f 6e61 6c20 6172 6775 6d65 optional argume │ │ │ │ +00046830: 6e74 2066 6f72 202a 6e6f 7465 2074 6f4d nt for *note toM │ │ │ │ +00046840: 6170 3a20 746f 4d61 702c 2e20 5768 656e ap: toMap,. When │ │ │ │ +00046850: 202a 6e6f 7465 2074 7275 653a 0a28 4d61 *note true:.(Ma │ │ │ │ +00046860: 6361 756c 6179 3244 6f63 2974 7275 652c caulay2Doc)true, │ │ │ │ +00046870: 206f 7220 2a6e 6f74 6520 696e 6669 6e69 or *note infini │ │ │ │ +00046880: 7479 3a20 284d 6163 6175 6c61 7932 446f ty: (Macaulay2Do │ │ │ │ +00046890: 6329 696e 6669 6e69 7479 2c20 6973 2070 c)infinity, is p │ │ │ │ +000468a0: 6173 7365 6420 746f 0a74 6869 7320 6f70 assed to.this op │ │ │ │ +000468b0: 7469 6f6e 2c20 7468 6520 6b65 726e 656c tion, the kernel │ │ │ │ +000468c0: 206f 6620 7468 6520 7265 7475 726e 6564 of the returned │ │ │ │ +000468d0: 2072 696e 6720 6d61 7020 7769 6c6c 2062 ring map will b │ │ │ │ +000468e0: 6520 7a65 726f 2e0a 0a46 756e 6374 696f e zero...Functio │ │ │ │ +000468f0: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00046900: 2061 7267 756d 656e 7420 6e61 6d65 6420 argument named │ │ │ │ +00046910: 446f 6d69 6e61 6e74 3a0a 3d3d 3d3d 3d3d Dominant:.====== │ │ │ │ 00046920: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ 00046930: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00046940: 3d3d 3d3d 0a0a 2020 2a20 2272 6174 696f ====.. * "ratio │ │ │ │ -00046950: 6e61 6c4d 6170 282e 2e2e 2c44 6f6d 696e nalMap(...,Domin │ │ │ │ -00046960: 616e 743d 3e2e 2e2e 2922 0a20 202a 2022 ant=>...)". * " │ │ │ │ -00046970: 746f 4d61 7028 2e2e 2e2c 446f 6d69 6e61 toMap(...,Domina │ │ │ │ -00046980: 6e74 3d3e 2e2e 2e29 220a 0a46 6f72 2074 nt=>...)"..For t │ │ │ │ -00046990: 6865 2070 726f 6772 616d 6d65 720a 3d3d he programmer.== │ │ │ │ -000469a0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -000469b0: 0a0a 5468 6520 6f62 6a65 6374 202a 6e6f ..The object *no │ │ │ │ -000469c0: 7465 2044 6f6d 696e 616e 743a 2044 6f6d te Dominant: Dom │ │ │ │ -000469d0: 696e 616e 742c 2069 7320 6120 2a6e 6f74 inant, is a *not │ │ │ │ -000469e0: 6520 7379 6d62 6f6c 3a20 284d 6163 6175 e symbol: (Macau │ │ │ │ -000469f0: 6c61 7932 446f 6329 5379 6d62 6f6c 2c2e lay2Doc)Symbol,. │ │ │ │ -00046a00: 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..-------------- │ │ │ │ +00046940: 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 ==========.. * │ │ │ │ +00046950: 2272 6174 696f 6e61 6c4d 6170 282e 2e2e "rationalMap(... │ │ │ │ +00046960: 2c44 6f6d 696e 616e 743d 3e2e 2e2e 2922 ,Dominant=>...)" │ │ │ │ +00046970: 0a20 202a 2022 746f 4d61 7028 2e2e 2e2c . * "toMap(..., │ │ │ │ +00046980: 446f 6d69 6e61 6e74 3d3e 2e2e 2e29 220a Dominant=>...)". │ │ │ │ +00046990: 0a46 6f72 2074 6865 2070 726f 6772 616d .For the program │ │ │ │ +000469a0: 6d65 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d mer.============ │ │ │ │ +000469b0: 3d3d 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 ======..The obje │ │ │ │ +000469c0: 6374 202a 6e6f 7465 2044 6f6d 696e 616e ct *note Dominan │ │ │ │ +000469d0: 743a 2044 6f6d 696e 616e 742c 2069 7320 t: Dominant, is │ │ │ │ +000469e0: 6120 2a6e 6f74 6520 7379 6d62 6f6c 3a20 a *note symbol: │ │ │ │ +000469f0: 284d 6163 6175 6c61 7932 446f 6329 5379 (Macaulay2Doc)Sy │ │ │ │ +00046a00: 6d62 6f6c 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d mbol,...-------- │ │ │ │ 00046a10: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00046a20: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00046a30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00046a40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00046a50: 2d0a 0a54 6865 2073 6f75 7263 6520 6f66 -..The source of │ │ │ │ -00046a60: 2074 6869 7320 646f 6375 6d65 6e74 2069 this document i │ │ │ │ -00046a70: 7320 696e 0a2f 6275 696c 642f 7265 7072 s in./build/repr │ │ │ │ -00046a80: 6f64 7563 6962 6c65 2d70 6174 682f 6d61 oducible-path/ma │ │ │ │ -00046a90: 6361 756c 6179 322d 312e 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -00046aa0: 6473 2f4d 322f 4d61 6361 756c 6179 322f ds/M2/Macaulay2/ │ │ │ │ -00046ab0: 7061 636b 6167 6573 2f43 7265 6d6f 6e61 packages/Cremona │ │ │ │ -00046ac0: 2f0a 646f 6375 6d65 6e74 6174 696f 6e2e /.documentation. │ │ │ │ -00046ad0: 6d32 3a38 383a 302e 0a1f 0a46 696c 653a m2:88:0....File: │ │ │ │ -00046ae0: 2043 7265 6d6f 6e61 2e69 6e66 6f2c 204e Cremona.info, N │ │ │ │ -00046af0: 6f64 653a 2065 6e74 7269 6573 5f6c 7052 ode: entries_lpR │ │ │ │ -00046b00: 6174 696f 6e61 6c4d 6170 5f72 702c 204e ationalMap_rp, N │ │ │ │ -00046b10: 6578 743a 2045 756c 6572 4368 6172 6163 ext: EulerCharac │ │ │ │ -00046b20: 7465 7269 7374 6963 2c20 5072 6576 3a20 teristic, Prev: │ │ │ │ -00046b30: 446f 6d69 6e61 6e74 2c20 5570 3a20 546f Dominant, Up: To │ │ │ │ -00046b40: 700a 0a65 6e74 7269 6573 2852 6174 696f p..entries(Ratio │ │ │ │ -00046b50: 6e61 6c4d 6170 2920 2d2d 2074 6865 2065 nalMap) -- the e │ │ │ │ -00046b60: 6e74 7269 6573 206f 6620 7468 6520 6d61 ntries of the ma │ │ │ │ -00046b70: 7472 6978 2061 7373 6f63 6961 7465 6420 trix associated │ │ │ │ -00046b80: 746f 2061 2072 6174 696f 6e61 6c20 6d61 to a rational ma │ │ │ │ -00046b90: 700a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a p.************** │ │ │ │ +00046a50: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00046a60: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00046a70: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +00046a80: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ +00046a90: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ +00046aa0: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ +00046ab0: 756c 6179 322f 7061 636b 6167 6573 2f43 ulay2/packages/C │ │ │ │ +00046ac0: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74 remona/.document │ │ │ │ +00046ad0: 6174 696f 6e2e 6d32 3a38 383a 302e 0a1f ation.m2:88:0... │ │ │ │ +00046ae0: 0a46 696c 653a 2043 7265 6d6f 6e61 2e69 .File: Cremona.i │ │ │ │ +00046af0: 6e66 6f2c 204e 6f64 653a 2065 6e74 7269 nfo, Node: entri │ │ │ │ +00046b00: 6573 5f6c 7052 6174 696f 6e61 6c4d 6170 es_lpRationalMap │ │ │ │ +00046b10: 5f72 702c 204e 6578 743a 2045 756c 6572 _rp, Next: Euler │ │ │ │ +00046b20: 4368 6172 6163 7465 7269 7374 6963 2c20 Characteristic, │ │ │ │ +00046b30: 5072 6576 3a20 446f 6d69 6e61 6e74 2c20 Prev: Dominant, │ │ │ │ +00046b40: 5570 3a20 546f 700a 0a65 6e74 7269 6573 Up: Top..entries │ │ │ │ +00046b50: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d (RationalMap) -- │ │ │ │ +00046b60: 2074 6865 2065 6e74 7269 6573 206f 6620 the entries of │ │ │ │ +00046b70: 7468 6520 6d61 7472 6978 2061 7373 6f63 the matrix assoc │ │ │ │ +00046b80: 6961 7465 6420 746f 2061 2072 6174 696f iated to a ratio │ │ │ │ +00046b90: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a nal map.******** │ │ │ │ 00046ba0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00046bb0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00046bc0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00046bd0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00046be0: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20 .. * Function: │ │ │ │ -00046bf0: 2a6e 6f74 6520 656e 7472 6965 733a 2028 *note entries: ( │ │ │ │ -00046c00: 4d61 6361 756c 6179 3244 6f63 2965 6e74 Macaulay2Doc)ent │ │ │ │ -00046c10: 7269 6573 2c0a 2020 2a20 5573 6167 653a ries,. * Usage: │ │ │ │ -00046c20: 200a 2020 2020 2020 2020 656e 7472 6965 . entrie │ │ │ │ -00046c30: 7320 5068 690a 2020 2a20 496e 7075 7473 s Phi. * Inputs │ │ │ │ -00046c40: 3a0a 2020 2020 2020 2a20 5068 692c 2061 :. * Phi, a │ │ │ │ -00046c50: 202a 6e6f 7465 2072 6174 696f 6e61 6c20 *note rational │ │ │ │ -00046c60: 6d61 703a 2052 6174 696f 6e61 6c4d 6170 map: RationalMap │ │ │ │ -00046c70: 2c0a 2020 2a20 4f75 7470 7574 733a 0a20 ,. * Outputs:. │ │ │ │ -00046c80: 2020 2020 202a 2061 202a 6e6f 7465 206c * a *note l │ │ │ │ -00046c90: 6973 743a 2028 4d61 6361 756c 6179 3244 ist: (Macaulay2D │ │ │ │ -00046ca0: 6f63 294c 6973 742c 2c20 7468 6520 656e oc)List,, the en │ │ │ │ -00046cb0: 7472 6965 7320 6f66 2074 6865 206d 6174 tries of the mat │ │ │ │ -00046cc0: 7269 7820 6173 736f 6369 6174 6564 0a20 rix associated. │ │ │ │ -00046cd0: 2020 2020 2020 2074 6f20 7468 6520 7269 to the ri │ │ │ │ -00046ce0: 6e67 206d 6170 2064 6566 696e 696e 6720 ng map defining │ │ │ │ -00046cf0: 7468 6520 7261 7469 6f6e 616c 206d 6170 the rational map │ │ │ │ -00046d00: 2050 6869 0a0a 4465 7363 7269 7074 696f Phi..Descriptio │ │ │ │ -00046d10: 6e0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 n.===========..T │ │ │ │ -00046d20: 6869 7320 6973 2065 7175 6976 616c 656e his is equivalen │ │ │ │ -00046d30: 7420 746f 2066 6c61 7474 656e 2065 6e74 t to flatten ent │ │ │ │ -00046d40: 7269 6573 206d 6174 7269 7820 5068 692e ries matrix Phi. │ │ │ │ -00046d50: 0a0a 5365 6520 616c 736f 0a3d 3d3d 3d3d ..See also.===== │ │ │ │ -00046d60: 3d3d 3d0a 0a20 202a 202a 6e6f 7465 206d ===.. * *note m │ │ │ │ -00046d70: 6174 7269 7828 5261 7469 6f6e 616c 4d61 atrix(RationalMa │ │ │ │ -00046d80: 7029 3a20 6d61 7472 6978 5f6c 7052 6174 p): matrix_lpRat │ │ │ │ -00046d90: 696f 6e61 6c4d 6170 5f72 702c 202d 2d20 ionalMap_rp, -- │ │ │ │ -00046da0: 7468 6520 6d61 7472 6978 0a20 2020 2061 the matrix. a │ │ │ │ -00046db0: 7373 6f63 6961 7465 6420 746f 2061 2072 ssociated to a r │ │ │ │ -00046dc0: 6174 696f 6e61 6c20 6d61 700a 0a57 6179 ational map..Way │ │ │ │ -00046dd0: 7320 746f 2075 7365 2074 6869 7320 6d65 s to use this me │ │ │ │ -00046de0: 7468 6f64 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d thod:.========== │ │ │ │ -00046df0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00046e00: 2020 2a20 2a6e 6f74 6520 656e 7472 6965 * *note entrie │ │ │ │ -00046e10: 7328 5261 7469 6f6e 616c 4d61 7029 3a20 s(RationalMap): │ │ │ │ -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. │ │ │ │ -00046e50: 2020 2020 6d61 7472 6978 2061 7373 6f63 matrix assoc │ │ │ │ -00046e60: 6961 7465 6420 746f 2061 2072 6174 696f iated to a ratio │ │ │ │ -00046e70: 6e61 6c20 6d61 700a 2d2d 2d2d 2d2d 2d2d nal map.-------- │ │ │ │ +00046be0: 2a2a 2a2a 2a2a 0a0a 2020 2a20 4675 6e63 ******.. * Func │ │ │ │ +00046bf0: 7469 6f6e 3a20 2a6e 6f74 6520 656e 7472 tion: *note entr │ │ │ │ +00046c00: 6965 733a 2028 4d61 6361 756c 6179 3244 ies: (Macaulay2D │ │ │ │ +00046c10: 6f63 2965 6e74 7269 6573 2c0a 2020 2a20 oc)entries,. * │ │ │ │ +00046c20: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +00046c30: 656e 7472 6965 7320 5068 690a 2020 2a20 entries Phi. * │ │ │ │ +00046c40: 496e 7075 7473 3a0a 2020 2020 2020 2a20 Inputs:. * │ │ │ │ +00046c50: 5068 692c 2061 202a 6e6f 7465 2072 6174 Phi, a *note rat │ │ │ │ +00046c60: 696f 6e61 6c20 6d61 703a 2052 6174 696f ional map: Ratio │ │ │ │ +00046c70: 6e61 6c4d 6170 2c0a 2020 2a20 4f75 7470 nalMap,. * Outp │ │ │ │ +00046c80: 7574 733a 0a20 2020 2020 202a 2061 202a uts:. * a * │ │ │ │ +00046c90: 6e6f 7465 206c 6973 743a 2028 4d61 6361 note list: (Maca │ │ │ │ +00046ca0: 756c 6179 3244 6f63 294c 6973 742c 2c20 ulay2Doc)List,, │ │ │ │ +00046cb0: 7468 6520 656e 7472 6965 7320 6f66 2074 the entries of t │ │ │ │ +00046cc0: 6865 206d 6174 7269 7820 6173 736f 6369 he matrix associ │ │ │ │ +00046cd0: 6174 6564 0a20 2020 2020 2020 2074 6f20 ated. to │ │ │ │ +00046ce0: 7468 6520 7269 6e67 206d 6170 2064 6566 the ring map def │ │ │ │ +00046cf0: 696e 696e 6720 7468 6520 7261 7469 6f6e ining the ration │ │ │ │ +00046d00: 616c 206d 6170 2050 6869 0a0a 4465 7363 al map Phi..Desc │ │ │ │ +00046d10: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ +00046d20: 3d3d 3d0a 0a54 6869 7320 6973 2065 7175 ===..This is equ │ │ │ │ +00046d30: 6976 616c 656e 7420 746f 2066 6c61 7474 ivalent to flatt │ │ │ │ +00046d40: 656e 2065 6e74 7269 6573 206d 6174 7269 en entries matri │ │ │ │ +00046d50: 7820 5068 692e 0a0a 5365 6520 616c 736f x Phi...See also │ │ │ │ +00046d60: 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a .========.. * * │ │ │ │ +00046d70: 6e6f 7465 206d 6174 7269 7828 5261 7469 note matrix(Rati │ │ │ │ +00046d80: 6f6e 616c 4d61 7029 3a20 6d61 7472 6978 onalMap): matrix │ │ │ │ +00046d90: 5f6c 7052 6174 696f 6e61 6c4d 6170 5f72 _lpRationalMap_r │ │ │ │ +00046da0: 702c 202d 2d20 7468 6520 6d61 7472 6978 p, -- the matrix │ │ │ │ +00046db0: 0a20 2020 2061 7373 6f63 6961 7465 6420 . associated │ │ │ │ +00046dc0: 746f 2061 2072 6174 696f 6e61 6c20 6d61 to a rational ma │ │ │ │ +00046dd0: 700a 0a57 6179 7320 746f 2075 7365 2074 p..Ways to use t │ │ │ │ +00046de0: 6869 7320 6d65 7468 6f64 3a0a 3d3d 3d3d his method:.==== │ │ │ │ +00046df0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ +00046e00: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ +00046e10: 656e 7472 6965 7328 5261 7469 6f6e 616c entries(Rational │ │ │ │ +00046e20: 4d61 7029 3a20 656e 7472 6965 735f 6c70 Map): entries_lp │ │ │ │ +00046e30: 5261 7469 6f6e 616c 4d61 705f 7270 2c20 RationalMap_rp, │ │ │ │ +00046e40: 2d2d 2074 6865 2065 6e74 7269 6573 206f -- the entries o │ │ │ │ +00046e50: 6620 7468 650a 2020 2020 6d61 7472 6978 f the. matrix │ │ │ │ +00046e60: 2061 7373 6f63 6961 7465 6420 746f 2061 associated to a │ │ │ │ +00046e70: 2072 6174 696f 6e61 6c20 6d61 700a 2d2d rational map.-- │ │ │ │ 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 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -00046ed0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -00046ee0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -00046ef0: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -00046f00: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -00046f10: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -00046f20: 756c 6179 322f 7061 636b 6167 6573 2f43 ulay2/packages/C │ │ │ │ -00046f30: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74 remona/.document │ │ │ │ -00046f40: 6174 696f 6e2e 6d32 3a37 3430 3a30 2e0a ation.m2:740:0.. │ │ │ │ -00046f50: 1f0a 4669 6c65 3a20 4372 656d 6f6e 612e ..File: Cremona. │ │ │ │ -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 │ │ │ │ -00046ff0: 6374 6572 6973 7469 6320 6f66 2061 2028 cteristic of a ( │ │ │ │ -00047000: 736d 6f6f 7468 2920 7072 6f6a 6563 7469 smooth) projecti │ │ │ │ -00047010: 7665 2076 6172 6965 7479 0a2a 2a2a 2a2a ve variety.***** │ │ │ │ -00047020: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00046ec0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +00046ed0: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +00046ee0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +00046ef0: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +00046f00: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +00046f10: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +00046f20: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +00046f30: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f ages/Cremona/.do │ │ │ │ +00046f40: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a37 cumentation.m2:7 │ │ │ │ +00046f50: 3430 3a30 2e0a 1f0a 4669 6c65 3a20 4372 40:0....File: Cr │ │ │ │ +00046f60: 656d 6f6e 612e 696e 666f 2c20 4e6f 6465 emona.info, Node │ │ │ │ +00046f70: 3a20 4575 6c65 7243 6861 7261 6374 6572 : EulerCharacter │ │ │ │ +00046f80: 6973 7469 632c 204e 6578 743a 2065 7863 istic, Next: exc │ │ │ │ +00046f90: 6570 7469 6f6e 616c 4c6f 6375 732c 2050 eptionalLocus, P │ │ │ │ +00046fa0: 7265 763a 2065 6e74 7269 6573 5f6c 7052 rev: entries_lpR │ │ │ │ +00046fb0: 6174 696f 6e61 6c4d 6170 5f72 702c 2055 ationalMap_rp, U │ │ │ │ +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 .*************** │ │ │ │ 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 2a0a 0a20 202a 2055 7361 6765 3a20 ***.. * Usage: │ │ │ │ -00047080: 0a20 2020 2020 2020 2045 756c 6572 4368 . EulerCh │ │ │ │ -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 │ │ │ │ -000470c0: 6465 616c 3a20 284d 6163 6175 6c61 7932 deal: (Macaulay2 │ │ │ │ -000470d0: 446f 6329 4964 6561 6c2c 2c20 6120 686f Doc)Ideal,, a ho │ │ │ │ -000470e0: 6d6f 6765 6e65 6f75 7320 6964 6561 6c20 mogeneous ideal │ │ │ │ -000470f0: 6465 6669 6e69 6e67 2061 0a20 2020 2020 defining a. │ │ │ │ -00047100: 2020 2073 6d6f 6f74 6820 7072 6f6a 6563 smooth projec │ │ │ │ -00047110: 7469 7665 2076 6172 6965 7479 2024 585c tive variety $X\ │ │ │ │ -00047120: 7375 6273 6574 5c6d 6174 6862 627b 507d subset\mathbb{P} │ │ │ │ -00047130: 5e6e 240a 2020 2a20 2a6e 6f74 6520 4f70 ^n$. * *note Op │ │ │ │ -00047140: 7469 6f6e 616c 2069 6e70 7574 733a 2028 tional inputs: ( │ │ │ │ -00047150: 4d61 6361 756c 6179 3244 6f63 2975 7369 Macaulay2Doc)usi │ │ │ │ -00047160: 6e67 2066 756e 6374 696f 6e73 2077 6974 ng functions wit │ │ │ │ -00047170: 6820 6f70 7469 6f6e 616c 2069 6e70 7574 h optional input │ │ │ │ -00047180: 732c 3a0a 2020 2020 2020 2a20 2a6e 6f74 s,:. * *not │ │ │ │ -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 │ │ │ │ -00047220: 7220 746f 2065 6e73 7572 650a 2020 2020 r to ensure. │ │ │ │ -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 │ │ │ │ -00047380: 2c2e 2053 6565 2061 6c73 6f0a 7468 6520 ,. See also.the │ │ │ │ -00047390: 636f 7272 6573 706f 6e64 696e 6720 6d65 corresponding me │ │ │ │ -000473a0: 7468 6f64 7320 696e 2074 6865 2070 6163 thods in the pac │ │ │ │ -000473b0: 6b61 6765 7320 4353 4d2d 4120 2873 6565 kages CSM-A (see │ │ │ │ -000473c0: 0a68 7474 703a 2f2f 7777 772e 6d61 7468 .http://www.math │ │ │ │ -000473d0: 2e66 7375 2e65 6475 2f7e 616c 7566 6669 .fsu.edu/~aluffi │ │ │ │ -000473e0: 2f43 534d 2f43 534d 2e68 746d 6c20 292c /CSM/CSM.html ), │ │ │ │ -000473f0: 2062 7920 502e 2041 6c75 6666 692c 2061 by P. Aluffi, a │ │ │ │ -00047400: 6e64 202a 6e6f 7465 0a43 6861 7261 6374 nd *note.Charact │ │ │ │ -00047410: 6572 6973 7469 6343 6c61 7373 6573 3a20 eristicClasses: │ │ │ │ -00047420: 2843 6861 7261 6374 6572 6973 7469 6343 (CharacteristicC │ │ │ │ -00047430: 6c61 7373 6573 2954 6f70 2c2c 2062 7920 lasses)Top,, by │ │ │ │ -00047440: 4d2e 2048 656c 6d65 7220 616e 6420 432e M. Helmer and C. │ │ │ │ -00047450: 204a 6f73 742e 0a0a 496e 2067 656e 6572 Jost...In gener │ │ │ │ -00047460: 616c 2c20 6576 656e 2069 6620 7468 6520 al, even if the │ │ │ │ -00047470: 696e 7075 7420 6964 6561 6c20 6465 6669 input ideal defi │ │ │ │ -00047480: 6e65 7320 6120 7369 6e67 756c 6172 2076 nes a singular v │ │ │ │ -00047490: 6172 6965 7479 2024 5824 2c20 7468 650a ariety $X$, the. │ │ │ │ -000474a0: 7265 7475 726e 6564 2076 616c 7565 2065 returned value e │ │ │ │ -000474b0: 7175 616c 7320 7468 6520 6465 6772 6565 quals the degree │ │ │ │ -000474c0: 206f 6620 7468 6520 636f 6d70 6f6e 656e of the componen │ │ │ │ -000474d0: 7420 6f66 2064 696d 656e 7369 6f6e 2030 t of dimension 0 │ │ │ │ -000474e0: 206f 6620 7468 650a 4368 6572 6e2d 4675 of the.Chern-Fu │ │ │ │ -000474f0: 6c74 6f6e 2063 6c61 7373 206f 6620 2458 lton class of $X │ │ │ │ -00047500: 242e 2054 6865 2045 756c 6572 2063 6861 $. The Euler cha │ │ │ │ -00047510: 7261 6374 6572 6973 7469 6320 6f66 2061 racteristic of a │ │ │ │ -00047520: 2073 696e 6775 6c61 7220 7661 7269 6574 singular variet │ │ │ │ -00047530: 7920 6361 6e0a 6265 2063 6f6d 7075 7465 y can.be compute │ │ │ │ -00047540: 6420 7669 6120 7468 6520 6d65 7468 6f64 d via the method │ │ │ │ -00047550: 202a 6e6f 7465 2043 6865 726e 5363 6877 *note ChernSchw │ │ │ │ -00047560: 6172 747a 4d61 6350 6865 7273 6f6e 3a0a artzMacPherson:. │ │ │ │ -00047570: 4368 6572 6e53 6368 7761 7274 7a4d 6163 ChernSchwartzMac │ │ │ │ -00047580: 5068 6572 736f 6e2c 2e0a 0a49 6e20 7468 Pherson,...In th │ │ │ │ -00047590: 6520 6578 616d 706c 6520 6265 6c6f 772c e example below, │ │ │ │ -000475a0: 2077 6520 636f 6d70 7574 6520 7468 6520 we compute the │ │ │ │ -000475b0: 4575 6c65 7220 6368 6172 6163 7465 7269 Euler characteri │ │ │ │ -000475c0: 7374 6963 206f 660a 245c 6d61 7468 6262 stic of.$\mathbb │ │ │ │ -000475d0: 7b47 7d28 312c 3429 5c73 7562 7365 745c {G}(1,4)\subset\ │ │ │ │ -000475e0: 6d61 7468 6262 7b50 7d5e 7b39 7d24 2c20 mathbb{P}^{9}$, │ │ │ │ -000475f0: 7573 696e 6720 626f 7468 2061 2070 726f using both a pro │ │ │ │ -00047600: 6261 6269 6c69 7374 6963 2061 6e64 2061 babilistic and a │ │ │ │ -00047610: 0a6e 6f6e 2d70 726f 6261 6269 6c69 7374 .non-probabilist │ │ │ │ -00047620: 6963 2061 7070 726f 6163 682e 0a0a 2b2d ic approach...+- │ │ │ │ -00047630: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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 │ │ │ │ +000470d0: 6175 6c61 7932 446f 6329 4964 6561 6c2c aulay2Doc)Ideal, │ │ │ │ +000470e0: 2c20 6120 686f 6d6f 6765 6e65 6f75 7320 , a homogeneous │ │ │ │ +000470f0: 6964 6561 6c20 6465 6669 6e69 6e67 2061 ideal defining a │ │ │ │ +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 │ │ │ │ +00047150: 7574 733a 2028 4d61 6361 756c 6179 3244 uts: (Macaulay2D │ │ │ │ +00047160: 6f63 2975 7369 6e67 2066 756e 6374 696f oc)using functio │ │ │ │ +00047170: 6e73 2077 6974 6820 6f70 7469 6f6e 616c ns with optional │ │ │ │ +00047180: 2069 6e70 7574 732c 3a0a 2020 2020 2020 inputs,:. │ │ │ │ +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 │ │ │ │ +00047390: 6f0a 7468 6520 636f 7272 6573 706f 6e64 o.the correspond │ │ │ │ +000473a0: 696e 6720 6d65 7468 6f64 7320 696e 2074 ing methods in t │ │ │ │ +000473b0: 6865 2070 6163 6b61 6765 7320 4353 4d2d he packages CSM- │ │ │ │ +000473c0: 4120 2873 6565 0a68 7474 703a 2f2f 7777 A (see.http://ww │ │ │ │ +000473d0: 772e 6d61 7468 2e66 7375 2e65 6475 2f7e w.math.fsu.edu/~ │ │ │ │ +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 │ │ │ │ +00047470: 6620 7468 6520 696e 7075 7420 6964 6561 f the input idea │ │ │ │ +00047480: 6c20 6465 6669 6e65 7320 6120 7369 6e67 l defines a sing │ │ │ │ +00047490: 756c 6172 2076 6172 6965 7479 2024 5824 ular variety $X$ │ │ │ │ +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 │ │ │ │ +000474d0: 6d70 6f6e 656e 7420 6f66 2064 696d 656e mponent of dimen │ │ │ │ +000474e0: 7369 6f6e 2030 206f 6620 7468 650a 4368 sion 0 of the.Ch │ │ │ │ +000474f0: 6572 6e2d 4675 6c74 6f6e 2063 6c61 7373 ern-Fulton class │ │ │ │ +00047500: 206f 6620 2458 242e 2054 6865 2045 756c of $X$. 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3930 190 │ │ │ │ -000477d0: 3138 3120 2030 2c31 2020 2030 2c32 2020 181 0,1 0,2 │ │ │ │ -000477e0: 2031 2c32 2020 2030 2c33 2020 2031 2c33 1,2 0,3 1,3 │ │ │ │ -000477f0: 2020 2032 2c33 2020 2030 2c34 2020 2031 2,3 0,4 1 │ │ │ │ -00047800: 2c34 2020 2032 2c34 2020 2020 7c0a 7c2d ,4 2,4 |.|- │ │ │ │ -00047810: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000477b0: 2020 2c20 7020 2020 2c20 7020 2020 2c20 , p , p , │ │ │ │ +000477c0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +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 │ │ │ │ +000477f0: 2020 2031 2c33 2020 2032 2c33 2020 2030 1,3 2,3 0 │ │ │ │ +00047800: 2c34 2020 2031 2c34 2020 2032 2c34 2020 ,4 1,4 2,4 │ │ │ │ +00047810: 2020 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|----------- │ │ │ │ 00047820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00047830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2020 │ │ │ │ +000479f0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00047a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047a30: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00047a40: 3220 3d20 3130 2020 2020 2020 2020 2020 2 = 10 │ │ │ │ +00047a30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047a40: 2020 7c0a 7c6f 3220 3d20 3130 2020 2020 |.|o2 = 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 7c0a 2b2d |.+- │ │ │ │ -00047a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047a90: 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+----------- │ │ │ │ 00047aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 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2c43 6572 7469 6679 3d3e 7472 ic(I,Certify=>tr │ │ │ │ +00047b10: 7565 2920 2020 2020 2020 2020 2020 2020 ue) │ │ │ │ +00047b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047b30: 2020 7c0a 7c43 6572 7469 6679 3a20 6f75 |.|Certify: ou │ │ │ │ +00047b40: 7470 7574 2063 6572 7469 6669 6564 2120 tput certified! │ │ │ │ 00047b50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047b60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047b70: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ -00047b80: 2d2d 2075 7365 6420 302e 3031 3237 3437 -- used 0.012747 │ │ │ │ -00047b90: 3373 2028 6370 7529 3b20 302e 3031 3232 3s (cpu); 0.0122 │ │ │ │ -00047ba0: 3936 3773 2028 7468 7265 6164 293b 2030 967s (thread); 0 │ │ │ │ -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 │ │ │ │ +00047b70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047b80: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ +00047b90: 3036 3730 3833 3473 2028 6370 7529 3b20 0670834s (cpu); │ │ │ │ +00047ba0: 302e 3031 3434 3534 3773 2028 7468 7265 0.0144547s (thre │ │ │ │ +00047bb0: 6164 293b 2030 7320 2867 6329 2020 2020 ad); 0s (gc) │ │ │ │ +00047bc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047bd0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ 00047be0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047bf0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00047c00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00047c10: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ -00047c20: 3320 3d20 3130 2020 2020 2020 2020 2020 3 = 10 │ │ │ │ +00047c10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047c20: 2020 7c0a 7c6f 3320 3d20 3130 2020 2020 |.|o3 = 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 7c0a 2b2d |.+- │ │ │ │ -00047c70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00047c60: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00047c70: 2020 7c0a 2b2d 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 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:.============= │ │ │ │ +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 │ │ │ │ +00047d40: 7269 6574 7929 3a20 2856 6172 6965 7469 riety): (Varieti │ │ │ │ +00047d50: 6573 2965 756c 6572 5f6c 7050 726f 6a65 es)euler_lpProje │ │ │ │ +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 │ │ │ │ +00047db0: 6520 7661 7269 6574 790a 2020 2a20 2a6e e variety. * *n │ │ │ │ +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:.======= │ │ │ │ 00047ec0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -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 ---------------- │ │ │ │ +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 ...------------- │ │ │ │ 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 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 362e 3036 2b64 732f 4d32 y2-1.26.06+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 .*************** │ │ │ │ +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 362e 3036 acaulay2-1.26.06 │ │ │ │ +00048050: 2b64 732f 4d32 2f4d 6163 6175 6c61 7932 +ds/M2/Macaulay2 │ │ │ │ +00048060: 2f70 6163 6b61 6765 732f 4372 656d 6f6e /packages/Cremon │ │ │ │ +00048070: 612f 0a64 6f63 756d 656e 7461 7469 6f6e a/.documentation │ │ │ │ +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.********* │ │ │ │ 00048140: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00048150: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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 ---------------- │ │ │ │ +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 .+-------------- │ │ │ │ 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 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 │ │ │ │ +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 ; │ │ │ │ 00048580: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048590: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -000485a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048590: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000485a0: 0a2b 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 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 │ │ │ │ +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 .| │ │ │ │ 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 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 ---------------- │ │ │ │ +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 .|-------------- │ │ │ │ 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 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); │ │ │ │ +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); │ │ │ │ 00048750: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048760: 2020 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 │ │ │ │ +00048770: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00048780: 0a7c 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 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 │ │ │ │ +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) │ │ │ │ 000487f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048810: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00048820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048810: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00048820: 0a2b 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 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; │ │ │ │ +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; │ │ │ │ 00048890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000488a0: 2020 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 │ │ │ │ +000488b0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +000488c0: 0a7c 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 207c 0a7c 2020 2020 |.| │ │ │ │ -00048910: 2020 2020 2020 2020 2020 2020 5a5a 2020 ZZ │ │ │ │ -00048920: 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 │ │ │ │ 00048930: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048940: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 ] │ │ │ │ +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 ] │ │ │ │ 00048980: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00048990: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 000489d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000489e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000489f0: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00048a00: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +000489f0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00048a00: 0a2b 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 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 │ │ │ │ +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) │ │ │ │ 00048a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00048a90: 2020 2020 2020 2020 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00048aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048a90: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00048aa0: 0a2b 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 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 ---------------- │ │ │ │ +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 .|-------------- │ │ │ │ 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 2d7c 0a7c 2c78 5f34 ---------|.|,x_4 │ │ │ │ -00048b90: 2c78 5f35 7d7d 2929 2020 2020 2020 2020 ,x_5}})) │ │ │ │ +00048b80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d7c ---------------| │ │ │ │ +00048b90: 0a7c 2c78 5f34 2c78 5f35 7d7d 2929 2020 .|,x_4,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 207c 0a2b 2d2d 2d2d |.+---- │ │ │ │ -00048be0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00048bd0: 2020 2020 2020 2020 2020 2020 2020 207c | │ │ │ │ +00048be0: 0a2b 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 2d2b 0a0a 5365 6520 ---------+..See │ │ │ │ -00048c30: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ -00048c40: 202a 202a 6e6f 7465 2069 6465 616c 2852 * *note ideal(R │ │ │ │ -00048c50: 6174 696f 6e61 6c4d 6170 293a 2069 6465 ationalMap): ide │ │ │ │ -00048c60: 616c 5f6c 7052 6174 696f 6e61 6c4d 6170 al_lpRationalMap │ │ │ │ -00048c70: 5f72 702c 202d 2d20 6261 7365 206c 6f63 _rp, -- base loc │ │ │ │ -00048c80: 7573 206f 6620 610a 2020 2020 7261 7469 us of a. rati │ │ │ │ -00048c90: 6f6e 616c 206d 6170 0a20 202a 202a 6e6f onal map. * *no │ │ │ │ -00048ca0: 7465 2069 6e76 6572 7365 4d61 7028 5261 te inverseMap(Ra │ │ │ │ -00048cb0: 7469 6f6e 616c 4d61 7029 3a20 696e 7665 tionalMap): inve │ │ │ │ -00048cc0: 7273 654d 6170 2c20 2d2d 2069 6e76 6572 rseMap, -- inver │ │ │ │ -00048cd0: 7365 206f 6620 6120 6269 7261 7469 6f6e se of a biration │ │ │ │ -00048ce0: 616c 206d 6170 0a20 202a 202a 6e6f 7465 al map. * *note │ │ │ │ -00048cf0: 2052 6174 696f 6e61 6c4d 6170 205e 2a2a RationalMap ^** │ │ │ │ -00048d00: 2049 6465 616c 3a20 5261 7469 6f6e 616c Ideal: Rational │ │ │ │ -00048d10: 4d61 7020 5e5f 7374 5f73 7420 4964 6561 Map ^_st_st Idea │ │ │ │ -00048d20: 6c2c 202d 2d20 696e 7665 7273 6520 696d l, -- inverse im │ │ │ │ -00048d30: 6167 650a 2020 2020 7669 6120 6120 7261 age. via a ra │ │ │ │ -00048d40: 7469 6f6e 616c 206d 6170 0a20 202a 202a tional map. * * │ │ │ │ -00048d50: 6e6f 7465 2069 7349 736f 6d6f 7270 6869 note isIsomorphi │ │ │ │ -00048d60: 736d 2852 6174 696f 6e61 6c4d 6170 293a sm(RationalMap): │ │ │ │ -00048d70: 2069 7349 736f 6d6f 7270 6869 736d 5f6c isIsomorphism_l │ │ │ │ -00048d80: 7052 6174 696f 6e61 6c4d 6170 5f72 702c pRationalMap_rp, │ │ │ │ -00048d90: 202d 2d0a 2020 2020 7768 6574 6865 7220 --. whether │ │ │ │ -00048da0: 6120 6269 7261 7469 6f6e 616c 206d 6170 a birational map │ │ │ │ -00048db0: 2069 7320 616e 2069 736f 6d6f 7270 6869 is an isomorphi │ │ │ │ -00048dc0: 736d 0a20 202a 202a 6e6f 7465 2066 6f72 sm. * *note for │ │ │ │ -00048dd0: 6365 496e 7665 7273 654d 6170 3a20 666f ceInverseMap: fo │ │ │ │ -00048de0: 7263 6549 6e76 6572 7365 4d61 702c 202d rceInverseMap, - │ │ │ │ -00048df0: 2d20 6465 636c 6172 6520 7468 6174 2074 - declare that t │ │ │ │ -00048e00: 776f 2072 6174 696f 6e61 6c20 6d61 7073 wo rational maps │ │ │ │ -00048e10: 0a20 2020 2061 7265 206f 6e65 2074 6865 . are one the │ │ │ │ -00048e20: 2069 6e76 6572 7365 206f 6620 7468 6520 inverse of the │ │ │ │ -00048e30: 6f74 6865 720a 0a57 6179 7320 746f 2075 other..Ways to u │ │ │ │ -00048e40: 7365 2065 7863 6570 7469 6f6e 616c 4c6f se exceptionalLo │ │ │ │ -00048e50: 6375 733a 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d cus:.=========== │ │ │ │ +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 │ │ │ │ +00048d30: 7273 6520 696d 6167 650a 2020 2020 7669 rse image. vi │ │ │ │ +00048d40: 6120 6120 7261 7469 6f6e 616c 206d 6170 a a rational map │ │ │ │ +00048d50: 0a20 202a 202a 6e6f 7465 2069 7349 736f . * *note isIso │ │ │ │ +00048d60: 6d6f 7270 6869 736d 2852 6174 696f 6e61 morphism(Rationa │ │ │ │ +00048d70: 6c4d 6170 293a 2069 7349 736f 6d6f 7270 lMap): isIsomorp │ │ │ │ +00048d80: 6869 736d 5f6c 7052 6174 696f 6e61 6c4d hism_lpRationalM │ │ │ │ +00048d90: 6170 5f72 702c 202d 2d0a 2020 2020 7768 ap_rp, --. wh │ │ │ │ +00048da0: 6574 6865 7220 6120 6269 7261 7469 6f6e ether a biration │ │ │ │ +00048db0: 616c 206d 6170 2069 7320 616e 2069 736f al map is an iso │ │ │ │ +00048dc0: 6d6f 7270 6869 736d 0a20 202a 202a 6e6f morphism. * *no │ │ │ │ +00048dd0: 7465 2066 6f72 6365 496e 7665 7273 654d te forceInverseM │ │ │ │ +00048de0: 6170 3a20 666f 7263 6549 6e76 6572 7365 ap: forceInverse │ │ │ │ +00048df0: 4d61 702c 202d 2d20 6465 636c 6172 6520 Map, -- declare │ │ │ │ +00048e00: 7468 6174 2074 776f 2072 6174 696f 6e61 that two rationa │ │ │ │ +00048e10: 6c20 6d61 7073 0a20 2020 2061 7265 206f l maps. are o │ │ │ │ +00048e20: 6e65 2074 6865 2069 6e76 6572 7365 206f ne the inverse o │ │ │ │ +00048e30: 6620 7468 6520 6f74 6865 720a 0a57 6179 f the other..Way │ │ │ │ +00048e40: 7320 746f 2075 7365 2065 7863 6570 7469 s to use excepti │ │ │ │ +00048e50: 6f6e 616c 4c6f 6375 733a 0a3d 3d3d 3d3d onalLocus:.===== │ │ │ │ 00048e60: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00048e70: 3d3d 0a0a 2020 2a20 2265 7863 6570 7469 ==.. * "excepti │ │ │ │ -00048e80: 6f6e 616c 4c6f 6375 7328 5261 7469 6f6e onalLocus(Ration │ │ │ │ -00048e90: 616c 4d61 7029 220a 0a46 6f72 2074 6865 alMap)"..For the │ │ │ │ -00048ea0: 2070 726f 6772 616d 6d65 720a 3d3d 3d3d programmer.==== │ │ │ │ -00048eb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ -00048ec0: 5468 6520 6f62 6a65 6374 202a 6e6f 7465 The object *note │ │ │ │ -00048ed0: 2065 7863 6570 7469 6f6e 616c 4c6f 6375 exceptionalLocu │ │ │ │ -00048ee0: 733a 2065 7863 6570 7469 6f6e 616c 4c6f s: exceptionalLo │ │ │ │ -00048ef0: 6375 732c 2069 7320 6120 2a6e 6f74 6520 cus, is a *note │ │ │ │ -00048f00: 6d65 7468 6f64 2066 756e 6374 696f 6e0a method function. │ │ │ │ -00048f10: 7769 7468 206f 7074 696f 6e73 3a20 284d with options: (M │ │ │ │ -00048f20: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ -00048f30: 6f64 4675 6e63 7469 6f6e 5769 7468 4f70 odFunctionWithOp │ │ │ │ -00048f40: 7469 6f6e 732c 2e0a 0a2d 2d2d 2d2d 2d2d tions,...------- │ │ │ │ +00048e70: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2265 ========.. * "e │ │ │ │ +00048e80: 7863 6570 7469 6f6e 616c 4c6f 6375 7328 xceptionalLocus( │ │ │ │ +00048e90: 5261 7469 6f6e 616c 4d61 7029 220a 0a46 RationalMap)"..F │ │ │ │ +00048ea0: 6f72 2074 6865 2070 726f 6772 616d 6d65 or the programme │ │ │ │ +00048eb0: 720a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d r.============== │ │ │ │ +00048ec0: 3d3d 3d3d 0a0a 5468 6520 6f62 6a65 6374 ====..The object │ │ │ │ +00048ed0: 202a 6e6f 7465 2065 7863 6570 7469 6f6e *note exception │ │ │ │ +00048ee0: 616c 4c6f 6375 733a 2065 7863 6570 7469 alLocus: excepti │ │ │ │ +00048ef0: 6f6e 616c 4c6f 6375 732c 2069 7320 6120 onalLocus, is a │ │ │ │ +00048f00: 2a6e 6f74 6520 6d65 7468 6f64 2066 756e *note method fun │ │ │ │ +00048f10: 6374 696f 6e0a 7769 7468 206f 7074 696f ction.with optio │ │ │ │ +00048f20: 6e73 3a20 284d 6163 6175 6c61 7932 446f ns: (Macaulay2Do │ │ │ │ +00048f30: 6329 4d65 7468 6f64 4675 6e63 7469 6f6e c)MethodFunction │ │ │ │ +00048f40: 5769 7468 4f70 7469 6f6e 732c 2e0a 0a2d WithOptions,...- │ │ │ │ 00048f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00048f60: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00048f70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00048f80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00048f90: 2d2d 2d2d 2d2d 2d2d 0a0a 5468 6520 736f --------..The so │ │ │ │ -00048fa0: 7572 6365 206f 6620 7468 6973 2064 6f63 urce of this doc │ │ │ │ -00048fb0: 756d 656e 7420 6973 2069 6e0a 2f62 7569 ument is in./bui │ │ │ │ -00048fc0: 6c64 2f72 6570 726f 6475 6369 626c 652d ld/reproducible- │ │ │ │ -00048fd0: 7061 7468 2f6d 6163 6175 6c61 7932 2d31 path/macaulay2-1 │ │ │ │ -00048fe0: 2e32 362e 3036 2b64 732f 4d32 2f4d 6163 .26.06+ds/M2/Mac │ │ │ │ -00048ff0: 6175 6c61 7932 2f70 6163 6b61 6765 732f aulay2/packages/ │ │ │ │ -00049000: 4372 656d 6f6e 612f 0a64 6f63 756d 656e Cremona/.documen │ │ │ │ -00049010: 7461 7469 6f6e 2e6d 323a 3435 303a 302e tation.m2:450:0. │ │ │ │ -00049020: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61 ...File: Cremona │ │ │ │ -00049030: 2e69 6e66 6f2c 204e 6f64 653a 2066 6c61 .info, Node: fla │ │ │ │ -00049040: 7474 656e 5f6c 7052 6174 696f 6e61 6c4d tten_lpRationalM │ │ │ │ -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) - │ │ │ │ -000490a0: 2d20 7772 6974 6520 736f 7572 6365 2061 - write source a │ │ │ │ -000490b0: 6e64 2074 6172 6765 7420 6173 206e 6f6e nd target as non │ │ │ │ -000490c0: 6465 6765 6e65 7261 7465 2076 6172 6965 degenerate varie │ │ │ │ -000490d0: 7469 6573 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a ties.*********** │ │ │ │ +00048f90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ +00048fa0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ +00048fb0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ +00048fc0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ +00048fd0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ +00048fe0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ +00048ff0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ +00049000: 6b61 6765 732f 4372 656d 6f6e 612f 0a64 kages/Cremona/.d │ │ │ │ +00049010: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a ocumentation.m2: │ │ │ │ +00049020: 3435 303a 302e 0a1f 0a46 696c 653a 2043 450:0....File: C │ │ │ │ +00049030: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 remona.info, Nod │ │ │ │ +00049040: 653a 2066 6c61 7474 656e 5f6c 7052 6174 e: flatten_lpRat │ │ │ │ +00049050: 696f 6e61 6c4d 6170 5f72 702c 204e 6578 ionalMap_rp, Nex │ │ │ │ +00049060: 743a 2066 6f72 6365 496d 6167 652c 2050 t: forceImage, P │ │ │ │ +00049070: 7265 763a 2065 7863 6570 7469 6f6e 616c rev: exceptional │ │ │ │ +00049080: 4c6f 6375 732c 2055 703a 2054 6f70 0a0a Locus, Up: Top.. │ │ │ │ +00049090: 666c 6174 7465 6e28 5261 7469 6f6e 616c flatten(Rational │ │ │ │ +000490a0: 4d61 7029 202d 2d20 7772 6974 6520 736f Map) -- write so │ │ │ │ +000490b0: 7572 6365 2061 6e64 2074 6172 6765 7420 urce and target │ │ │ │ +000490c0: 6173 206e 6f6e 6465 6765 6e65 7261 7465 as nondegenerate │ │ │ │ +000490d0: 2076 6172 6965 7469 6573 0a2a 2a2a 2a2a varieties.***** │ │ │ │ 000490e0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 000490f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00049100: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -00049110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a0a ***************. │ │ │ │ -00049120: 0a20 202a 2046 756e 6374 696f 6e3a 202a . * Function: * │ │ │ │ -00049130: 6e6f 7465 2066 6c61 7474 656e 3a20 284d note flatten: (M │ │ │ │ -00049140: 6163 6175 6c61 7932 446f 6329 666c 6174 acaulay2Doc)flat │ │ │ │ -00049150: 7465 6e2c 0a20 202a 2055 7361 6765 3a20 ten,. * Usage: │ │ │ │ -00049160: 0a20 2020 2020 2020 2066 6c61 7474 656e . flatten │ │ │ │ -00049170: 2070 6869 0a20 202a 2049 6e70 7574 733a phi. * Inputs: │ │ │ │ -00049180: 0a20 2020 2020 202a 2070 6869 2c20 6120 . * phi, a │ │ │ │ -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 │ │ │ │ -00049230: 6520 7365 6e73 6520 7468 6174 2074 6865 e sense that the │ │ │ │ -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 =====..+-------- │ │ │ │ +00049110: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00049120: 2a2a 2a2a 2a0a 0a20 202a 2046 756e 6374 *****.. * Funct │ │ │ │ +00049130: 696f 6e3a 202a 6e6f 7465 2066 6c61 7474 ion: *note flatt │ │ │ │ +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 │ │ │ │ +000491c0: 7473 3a0a 2020 2020 2020 2a20 6120 2a6e ts:. * a *n │ │ │ │ +000491d0: 6f74 6520 7261 7469 6f6e 616c 206d 6170 ote rational map │ │ │ │ +000491e0: 3a20 5261 7469 6f6e 616c 4d61 702c 2c20 : RationalMap,, │ │ │ │ +000491f0: 6120 7261 7469 6f6e 616c 206d 6170 2069 a rational map i │ │ │ │ +00049200: 736f 6d6f 7270 6869 6320 746f 2074 6865 somorphic to the │ │ │ │ +00049210: 0a20 2020 2020 2020 206f 7269 6769 6e61 . origina │ │ │ │ +00049220: 6c20 6d61 702c 2066 6c61 7474 656e 6564 l map, flattened │ │ │ │ +00049230: 2069 6e20 7468 6520 7365 6e73 6520 7468 in the sense th │ │ │ │ +00049240: 6174 2074 6865 2069 6465 616c 7320 6f66 at the ideals of │ │ │ │ +00049250: 2073 6f75 7263 6520 616e 640a 2020 2020 source and. │ │ │ │ +00049260: 2020 2020 7461 7267 6574 2063 6f6e 7461 target conta │ │ │ │ +00049270: 696e 206e 6f20 6c69 6e65 6172 2066 6f72 in no linear for │ │ │ │ +00049280: 6d73 0a0a 4465 7363 7269 7074 696f 6e0a ms..Description. │ │ │ │ +00049290: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a2b 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 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 |.| │ │ │ │ +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 |.| │ │ │ │ 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 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 |.|-------- │ │ │ │ +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 |.|-- │ │ │ │ 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 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|.| │ │ │ │ +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|.| │ │ │ │ 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 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 │ │ │ │ +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) │ │ │ │ 000494f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00049500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049510: 2020 2020 207c 0a7c 2d2d 2d2d 2d2d 2d2d |.|-------- │ │ │ │ +00049510: 2020 2020 2020 2020 2020 207c 0a7c 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 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 |.|-------- │ │ │ │ +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- |.|-- │ │ │ │ 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 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|.|-------- │ │ │ │ +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|.|-- │ │ │ │ 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 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 |.|-------- │ │ │ │ +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, |.|-- │ │ │ │ 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 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 |.+-------- │ │ │ │ +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 |.+-- │ │ │ │ 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 2d2b 0a7c 6933 203a 2064 6573 -----+.|i3 : des │ │ │ │ -000497f0: 6372 6962 6520 7068 6920 2020 2020 2020 cribe phi │ │ │ │ +000497e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6933 -----------+.|i3 │ │ │ │ +000497f0: 203a 2064 6573 6372 6962 6520 7068 6920 : describe 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00049830: 2020 2020 2020 2020 2020 207c 0a7c 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 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 │ │ │ │ +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 │ │ │ │ 000498c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00049960: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 000499a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000499b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000499c0: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +000499c0: 2020 2020 2020 2020 2020 207c 0a2b 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 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 │ │ │ │ +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; │ │ │ │ 00049a40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00049a50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049a60: 2020 2020 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00049a60: 2020 2020 2020 2020 2020 207c 0a7c 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 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 |.+-------- │ │ │ │ +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) |.+-- │ │ │ │ 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 2d2b 0a7c 6935 203a 2064 6573 -----+.|i5 : des │ │ │ │ -00049b60: 6372 6962 6520 7073 6920 2020 2020 2020 cribe psi │ │ │ │ +00049b50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a7c 6935 -----------+.|i5 │ │ │ │ +00049b60: 203a 2064 6573 6372 6962 6520 7073 6920 : describe 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 207c 0a7c 2020 2020 2020 2020 |.| │ │ │ │ +00049ba0: 2020 2020 2020 2020 2020 207c 0a7c 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 207c 0a7c 6f35 203d 2072 6174 |.|o5 = rat │ │ │ │ -00049c00: 696f 6e61 6c20 6d61 7020 6465 6669 6e65 ional map define │ │ │ │ -00049c10: 6420 6279 2066 6f72 6d73 206f 6620 6465 d by forms of de │ │ │ │ -00049c20: 6772 6565 2031 2020 2020 2020 2020 2020 gree 1 │ │ │ │ +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 │ │ │ │ 00049c30: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049c40: 2020 2020 207c 0a7c 2020 2020 2073 6f75 |.| sou │ │ │ │ -00049c50: 7263 6520 7661 7269 6574 793a 2073 6d6f rce variety: smo │ │ │ │ -00049c60: 6f74 6820 7175 6164 7269 6320 6879 7065 oth quadric hype │ │ │ │ -00049c70: 7273 7572 6661 6365 2069 6e20 5050 5e34 rsurface in PP^4 │ │ │ │ -00049c80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049c90: 2020 2020 207c 0a7c 2020 2020 2074 6172 |.| tar │ │ │ │ -00049ca0: 6765 7420 7661 7269 6574 793a 2050 505e get variety: PP^ │ │ │ │ -00049cb0: 3420 2020 2020 2020 2020 2020 2020 2020 4 │ │ │ │ +00049c40: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00049c50: 2020 2073 6f75 7263 6520 7661 7269 6574 source variet │ │ │ │ +00049c60: 793a 2073 6d6f 6f74 6820 7175 6164 7269 y: smooth quadri │ │ │ │ +00049c70: 6320 6879 7065 7273 7572 6661 6365 2069 c hypersurface i │ │ │ │ +00049c80: 6e20 5050 5e34 2020 2020 2020 2020 2020 n PP^4 │ │ │ │ +00049c90: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00049ca0: 2020 2074 6172 6765 7420 7661 7269 6574 target variet │ │ │ │ +00049cb0: 793a 2050 505e 3420 2020 2020 2020 2020 y: PP^4 │ │ │ │ 00049cc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00049cd0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049ce0: 2020 2020 207c 0a7c 2020 2020 2063 6f65 |.| coe │ │ │ │ -00049cf0: 6666 6963 6965 6e74 2072 696e 673a 2051 fficient ring: Q │ │ │ │ -00049d00: 5120 2020 2020 2020 2020 2020 2020 2020 Q │ │ │ │ +00049ce0: 2020 2020 2020 2020 2020 207c 0a7c 2020 |.| │ │ │ │ +00049cf0: 2020 2063 6f65 6666 6963 6965 6e74 2072 coefficient r │ │ │ │ +00049d00: 696e 673a 2051 5120 2020 2020 2020 2020 ing: QQ │ │ │ │ 00049d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00049d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00049d30: 2020 2020 207c 0a2b 2d2d 2d2d 2d2d 2d2d |.+-------- │ │ │ │ +00049d30: 2020 2020 2020 2020 2020 207c 0a2b 2d2d |.+-- │ │ │ │ 00049d40: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -00049d80: 2d2d 2d2d 2d2b 0a0a 5761 7973 2074 6f20 -----+..Ways to │ │ │ │ -00049d90: 7573 6520 7468 6973 206d 6574 686f 643a use this method: │ │ │ │ -00049da0: 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d .=============== │ │ │ │ -00049db0: 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a =========.. * * │ │ │ │ -00049dc0: 6e6f 7465 2066 6c61 7474 656e 2852 6174 note flatten(Rat │ │ │ │ -00049dd0: 696f 6e61 6c4d 6170 293a 2066 6c61 7474 ionalMap): flatt │ │ │ │ -00049de0: 656e 5f6c 7052 6174 696f 6e61 6c4d 6170 en_lpRationalMap │ │ │ │ -00049df0: 5f72 702c 202d 2d20 7772 6974 6520 736f _rp, -- write so │ │ │ │ -00049e00: 7572 6365 2061 6e64 0a20 2020 2074 6172 urce and. tar │ │ │ │ -00049e10: 6765 7420 6173 206e 6f6e 6465 6765 6e65 get as nondegene │ │ │ │ -00049e20: 7261 7465 2076 6172 6965 7469 6573 0a2d rate varieties.- │ │ │ │ -00049e30: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049d80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2b 0a0a 5761 -----------+..Wa │ │ │ │ +00049d90: 7973 2074 6f20 7573 6520 7468 6973 206d ys to use this m │ │ │ │ +00049da0: 6574 686f 643a 0a3d 3d3d 3d3d 3d3d 3d3d ethod:.========= │ │ │ │ +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 │ │ │ │ +00049df0: 6e61 6c4d 6170 5f72 702c 202d 2d20 7772 nalMap_rp, -- wr │ │ │ │ +00049e00: 6974 6520 736f 7572 6365 2061 6e64 0a20 ite source and. │ │ │ │ +00049e10: 2020 2074 6172 6765 7420 6173 206e 6f6e target as non │ │ │ │ +00049e20: 6465 6765 6e65 7261 7465 2076 6172 6965 degenerate varie │ │ │ │ +00049e30: 7469 6573 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d ties.----------- │ │ │ │ 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 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 │ │ │ │ -00049eb0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00049ec0: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00049ed0: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00049ee0: 6b61 6765 732f 4372 656d 6f6e 612f 0a64 kages/Cremona/.d │ │ │ │ -00049ef0: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a ocumentation.m2: │ │ │ │ -00049f00: 3535 323a 302e 0a1f 0a46 696c 653a 2043 552:0....File: C │ │ │ │ -00049f10: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 remona.info, Nod │ │ │ │ -00049f20: 653a 2066 6f72 6365 496d 6167 652c 204e e: forceImage, N │ │ │ │ -00049f30: 6578 743a 2066 6f72 6365 496e 7665 7273 ext: forceInvers │ │ │ │ -00049f40: 654d 6170 2c20 5072 6576 3a20 666c 6174 eMap, Prev: flat │ │ │ │ -00049f50: 7465 6e5f 6c70 5261 7469 6f6e 616c 4d61 ten_lpRationalMa │ │ │ │ -00049f60: 705f 7270 2c20 5570 3a20 546f 700a 0a66 p_rp, Up: Top..f │ │ │ │ -00049f70: 6f72 6365 496d 6167 6520 2d2d 2064 6563 orceImage -- dec │ │ │ │ -00049f80: 6c61 7265 2077 6869 6368 2069 7320 7468 lare which is th │ │ │ │ -00049f90: 6520 696d 6167 6520 6f66 2061 2072 6174 e image of a rat │ │ │ │ -00049fa0: 696f 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a ional map.****** │ │ │ │ +00049e70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00049e80: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +00049e90: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +00049ea0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +00049eb0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +00049ec0: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +00049ed0: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +00049ee0: 7932 2f70 6163 6b61 6765 732f 4372 656d y2/packages/Crem │ │ │ │ +00049ef0: 6f6e 612f 0a64 6f63 756d 656e 7461 7469 ona/.documentati │ │ │ │ +00049f00: 6f6e 2e6d 323a 3535 323a 302e 0a1f 0a46 on.m2:552:0....F │ │ │ │ +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 │ │ │ │ +00049f90: 2069 7320 7468 6520 696d 6167 6520 6f66 is the image of │ │ │ │ +00049fa0: 2061 2072 6174 696f 6e61 6c20 6d61 700a a rational 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 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 │ │ │ │ -0004a040: 6e61 6c4d 6170 2c0a 2020 2020 2020 2a20 nalMap,. * │ │ │ │ -0004a050: 492c 2061 6e20 2a6e 6f74 6520 6964 6561 I, an *note idea │ │ │ │ -0004a060: 6c3a 2028 4d61 6361 756c 6179 3244 6f63 l: (Macaulay2Doc │ │ │ │ -0004a070: 2949 6465 616c 2c0a 2020 2a20 4f75 7470 )Ideal,. * Outp │ │ │ │ -0004a080: 7574 733a 0a20 2020 2020 202a 202a 6e6f uts:. * *no │ │ │ │ -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 │ │ │ │ -0004a0d0: 6f64 2061 6c6c 6f77 7320 746f 2069 6e66 od allows to inf │ │ │ │ -0004a0e0: 6f72 6d20 7468 6520 7379 7374 656d 2061 orm the system a │ │ │ │ -0004a0f0: 626f 7574 2074 6865 2069 6d61 6765 206f bout the image o │ │ │ │ -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 │ │ │ │ -0004a120: 6572 666f 726d 696e 6720 616e 7920 636f erforming any co │ │ │ │ -0004a130: 6d70 7574 6174 696f 6e2e 2049 6e20 7061 mputation. In pa │ │ │ │ -0004a140: 7274 6963 756c 6172 2c20 7468 6973 2063 rticular, this c │ │ │ │ -0004a150: 616e 2062 6520 7573 6564 2074 6f20 6465 an be used to de │ │ │ │ -0004a160: 636c 6172 650a 7468 6174 2061 2072 6174 clare.that a rat │ │ │ │ -0004a170: 696f 6e61 6c20 6d61 7020 6973 2064 6f6d ional map is dom │ │ │ │ -0004a180: 696e 616e 742e 0a0a 2b2d 2d2d 2d2d 2d2d inant...+------- │ │ │ │ +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 │ │ │ │ +0004a060: 6520 6964 6561 6c3a 2028 4d61 6361 756c e ideal: (Macaul │ │ │ │ +0004a070: 6179 3244 6f63 2949 6465 616c 2c0a 2020 ay2Doc)Ideal,. │ │ │ │ +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 │ │ │ │ +0004a0e0: 746f 2069 6e66 6f72 6d20 7468 6520 7379 to inform the sy │ │ │ │ +0004a0f0: 7374 656d 2061 626f 7574 2074 6865 2069 stem about the i │ │ │ │ +0004a100: 6d61 6765 206f 6620 6120 6769 7665 6e20 mage of a given │ │ │ │ +0004a110: 7261 7469 6f6e 616c 206d 6170 0a77 6974 rational map.wit │ │ │ │ +0004a120: 686f 7574 2070 6572 666f 726d 696e 6720 hout performing │ │ │ │ +0004a130: 616e 7920 636f 6d70 7574 6174 696f 6e2e any computation. │ │ │ │ +0004a140: 2049 6e20 7061 7274 6963 756c 6172 2c20 In particular, │ │ │ │ +0004a150: 7468 6973 2063 616e 2062 6520 7573 6564 this can be used │ │ │ │ +0004a160: 2074 6f20 6465 636c 6172 650a 7468 6174 to declare.that │ │ │ │ +0004a170: 2061 2072 6174 696f 6e61 6c20 6d61 7020 a rational map │ │ │ │ +0004a180: 6973 2064 6f6d 696e 616e 742e 0a0a 2b2d is dominant...+- │ │ │ │ 0004a190: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -0004a1d0: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 5036 ------+.|i1 : P6 │ │ │ │ -0004a1e0: 203d 2051 515b 745f 302e 2e74 5f36 5d3b = QQ[t_0..t_6]; │ │ │ │ -0004a1f0: 2058 203d 2020 2020 2020 2020 2020 2020 X = │ │ │ │ +0004a1d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0004a1e0: 3120 3a20 5036 203d 2051 515b 745f 302e 1 : P6 = QQ[t_0. │ │ │ │ +0004a1f0: 2e74 5f36 5d3b 2058 203d 2020 2020 2020 .t_6]; X = │ │ │ │ 0004a200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a210: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a220: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0004a220: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0004a230: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 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 7c0a 7c6f 3220 3a20 4964 |.|o2 : Id │ │ │ │ -0004a280: 6561 6c20 6f66 2050 3620 2020 2020 2020 eal of P6 │ │ │ │ +0004a270: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004a280: 3220 3a20 4964 6561 6c20 6f66 2050 3620 2 : Ideal of P6 │ │ │ │ 0004a290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a2a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a2b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a2c0: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ +0004a2c0: 2020 2020 2020 2020 2020 2020 7c0a 7c2d |.|- │ │ │ │ 0004a2d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a2e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a2f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a300: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a310: 2d2d 2d2d 2d2d 7c0a 7c6d 696e 6f72 7328 ------|.|minors( │ │ │ │ -0004a320: 332c 6d61 7472 6978 7b7b 745f 302e 2e74 3,matrix{{t_0..t │ │ │ │ -0004a330: 5f34 7d2c 7b74 5f31 2e2e 745f 357d 2c7b _4},{t_1..t_5},{ │ │ │ │ -0004a340: 745f 322e 2e74 5f36 7d7d 293b 2020 2020 t_2..t_6}}); │ │ │ │ -0004a350: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a360: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004a310: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c6d ------------|.|m │ │ │ │ +0004a320: 696e 6f72 7328 332c 6d61 7472 6978 7b7b inors(3,matrix{{ │ │ │ │ +0004a330: 745f 302e 2e74 5f34 7d2c 7b74 5f31 2e2e t_0..t_4},{t_1.. │ │ │ │ +0004a340: 745f 357d 2c7b 745f 322e 2e74 5f36 7d7d t_5},{t_2..t_6}} │ │ │ │ +0004a350: 293b 2020 2020 2020 2020 2020 2020 2020 ); │ │ │ │ +0004a360: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004a370: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 2b0a 7c69 3320 3a20 5068 ------+.|i3 : Ph │ │ │ │ -0004a3c0: 6920 3d20 7261 7469 6f6e 616c 4d61 7028 i = rationalMap( │ │ │ │ -0004a3d0: 582c 446f 6d69 6e61 6e74 3d3e 3229 3b20 X,Dominant=>2); │ │ │ │ -0004a3e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a3b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0004a3c0: 3320 3a20 5068 6920 3d20 7261 7469 6f6e 3 : Phi = ration │ │ │ │ +0004a3d0: 616c 4d61 7028 582c 446f 6d69 6e61 6e74 alMap(X,Dominant │ │ │ │ +0004a3e0: 3d3e 3229 3b20 2020 2020 2020 2020 2020 =>2); │ │ │ │ 0004a3f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a400: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0004a400: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 0004a410: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a420: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a440: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a450: 2020 2020 2020 7c0a 7c6f 3320 3a20 5261 |.|o3 : Ra │ │ │ │ -0004a460: 7469 6f6e 616c 4d61 7020 2863 7562 6963 tionalMap (cubic │ │ │ │ -0004a470: 2072 6174 696f 6e61 6c20 6d61 7020 6672 rational map fr │ │ │ │ -0004a480: 6f6d 2050 505e 3620 746f 2036 2d64 696d om PP^6 to 6-dim │ │ │ │ -0004a490: 656e 7369 6f6e 616c 2073 7562 7661 7269 ensional subvari │ │ │ │ -0004a4a0: 6574 7920 6f66 7c0a 7c2d 2d2d 2d2d 2d2d ety of|.|------- │ │ │ │ +0004a450: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004a460: 3320 3a20 5261 7469 6f6e 616c 4d61 7020 3 : RationalMap │ │ │ │ +0004a470: 2863 7562 6963 2072 6174 696f 6e61 6c20 (cubic rational │ │ │ │ +0004a480: 6d61 7020 6672 6f6d 2050 505e 3620 746f map from PP^6 to │ │ │ │ +0004a490: 2036 2d64 696d 656e 7369 6f6e 616c 2073 6-dimensional s │ │ │ │ +0004a4a0: 7562 7661 7269 6574 7920 6f66 7c0a 7c2d ubvariety of|.|- │ │ │ │ 0004a4b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a4c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a4d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a4e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a4f0: 2d2d 2d2d 2d2d 7c0a 7c50 505e 3929 2020 ------|.|PP^9) │ │ │ │ -0004a500: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a4f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c50 ------------|.|P │ │ │ │ +0004a500: 505e 3929 2020 2020 2020 2020 2020 2020 P^9) │ │ │ │ 0004a510: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a520: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a540: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004a540: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004a550: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a560: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a570: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a580: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a590: 2d2d 2d2d 2d2d 2b0a 7c69 3420 3a20 7469 ------+.|i4 : ti │ │ │ │ -0004a5a0: 6d65 2066 6f72 6365 496d 6167 6528 5068 me forceImage(Ph │ │ │ │ -0004a5b0: 692c 6964 6561 6c20 305f 2874 6172 6765 i,ideal 0_(targe │ │ │ │ -0004a5c0: 7420 5068 6929 2920 2020 2020 2020 2020 t Phi)) │ │ │ │ +0004a590: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0004a5a0: 3420 3a20 7469 6d65 2066 6f72 6365 496d 4 : time forceIm │ │ │ │ +0004a5b0: 6167 6528 5068 692c 6964 6561 6c20 305f age(Phi,ideal 0_ │ │ │ │ +0004a5c0: 2874 6172 6765 7420 5068 6929 2920 2020 (target Phi)) │ │ │ │ 0004a5d0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a5e0: 2020 2020 2020 7c0a 7c20 2d2d 2075 7365 |.| -- use │ │ │ │ -0004a5f0: 6420 302e 3030 3036 3637 3736 3273 2028 d 0.000667762s ( │ │ │ │ -0004a600: 6370 7529 3b20 302e 3030 3036 3536 3037 cpu); 0.00065607 │ │ │ │ -0004a610: 7320 2874 6872 6561 6429 3b20 3073 2028 s (thread); 0s ( │ │ │ │ -0004a620: 6763 2920 2020 2020 2020 2020 2020 2020 gc) │ │ │ │ -0004a630: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004a5e0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ +0004a5f0: 2d2d 2075 7365 6420 302e 3030 3037 3635 -- used 0.000765 │ │ │ │ +0004a600: 3638 3873 2028 6370 7529 3b20 302e 3030 688s (cpu); 0.00 │ │ │ │ +0004a610: 3037 3630 3835 3173 2028 7468 7265 6164 0760851s (thread │ │ │ │ +0004a620: 293b 2030 7320 2867 6329 2020 2020 2020 ); 0s (gc) │ │ │ │ +0004a630: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004a640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a660: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a670: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a680: 2d2d 2d2d 2d2d 2b0a 7c69 3520 3a20 5068 ------+.|i5 : Ph │ │ │ │ -0004a690: 693b 2020 2020 2020 2020 2020 2020 2020 i; │ │ │ │ +0004a680: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 7c69 ------------+.|i │ │ │ │ +0004a690: 3520 3a20 5068 693b 2020 2020 2020 2020 5 : Phi; │ │ │ │ 0004a6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a6b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a6c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a6d0: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ +0004a6d0: 2020 2020 2020 2020 2020 2020 7c0a 7c20 |.| │ │ │ │ 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 7c0a 7c6f 3520 3a20 5261 |.|o5 : Ra │ │ │ │ -0004a730: 7469 6f6e 616c 4d61 7020 2863 7562 6963 tionalMap (cubic │ │ │ │ -0004a740: 2064 6f6d 696e 616e 7420 7261 7469 6f6e dominant ration │ │ │ │ -0004a750: 616c 206d 6170 2066 726f 6d20 5050 5e36 al map from PP^6 │ │ │ │ -0004a760: 2074 6f20 362d 6469 6d65 6e73 696f 6e61 to 6-dimensiona │ │ │ │ -0004a770: 6c20 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d l |.|------- │ │ │ │ +0004a720: 2020 2020 2020 2020 2020 2020 7c0a 7c6f |.|o │ │ │ │ +0004a730: 3520 3a20 5261 7469 6f6e 616c 4d61 7020 5 : RationalMap │ │ │ │ +0004a740: 2863 7562 6963 2064 6f6d 696e 616e 7420 (cubic dominant │ │ │ │ +0004a750: 7261 7469 6f6e 616c 206d 6170 2066 726f rational map fro │ │ │ │ +0004a760: 6d20 5050 5e36 2074 6f20 362d 6469 6d65 m PP^6 to 6-dime │ │ │ │ +0004a770: 6e73 696f 6e61 6c20 2020 2020 7c0a 7c2d nsional |.|- │ │ │ │ 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 ---------------- │ │ │ │ -0004a7c0: 2d2d 2d2d 2d2d 7c0a 7c73 7562 7661 7269 ------|.|subvari │ │ │ │ -0004a7d0: 6574 7920 6f66 2050 505e 3929 2020 2020 ety of PP^9) │ │ │ │ -0004a7e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +0004a7c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 7c0a 7c73 ------------|.|s │ │ │ │ +0004a7d0: 7562 7661 7269 6574 7920 6f66 2050 505e ubvariety of PP^ │ │ │ │ +0004a7e0: 3929 2020 2020 2020 2020 2020 2020 2020 9) │ │ │ │ 0004a7f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004a800: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004a810: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ +0004a810: 2020 2020 2020 2020 2020 2020 7c0a 2b2d |.+- │ │ │ │ 0004a820: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a830: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a840: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004a850: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004a860: 2d2d 2d2d 2d2d 2b0a 0a43 6176 6561 740a ------+..Caveat. │ │ │ │ -0004a870: 3d3d 3d3d 3d3d 0a0a 4966 2074 6865 2064 ======..If the d │ │ │ │ -0004a880: 6563 6c61 7261 7469 6f6e 2069 7320 6661 eclaration is fa │ │ │ │ -0004a890: 6c73 652c 206e 6f6e 7365 6e73 6963 616c lse, nonsensical │ │ │ │ -0004a8a0: 2061 6e73 7765 7273 206d 6179 2072 6573 answers may res │ │ │ │ -0004a8b0: 756c 742e 0a0a 5365 6520 616c 736f 0a3d ult...See also.= │ │ │ │ -0004a8c0: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ -0004a8d0: 7465 2069 6d61 6765 2852 6174 696f 6e61 te image(Rationa │ │ │ │ -0004a8e0: 6c4d 6170 293a 2069 6d61 6765 5f6c 7052 lMap): image_lpR │ │ │ │ -0004a8f0: 6174 696f 6e61 6c4d 6170 5f63 6d5a 5a5f ationalMap_cmZZ_ │ │ │ │ -0004a900: 7270 2c20 2d2d 2063 6c6f 7375 7265 206f rp, -- closure o │ │ │ │ -0004a910: 6620 7468 650a 2020 2020 696d 6167 6520 f the. image │ │ │ │ -0004a920: 6f66 2061 2072 6174 696f 6e61 6c20 6d61 of a rational ma │ │ │ │ -0004a930: 700a 2020 2a20 2a6e 6f74 6520 666f 7263 p. * *note forc │ │ │ │ -0004a940: 6549 6e76 6572 7365 4d61 703a 2066 6f72 eInverseMap: for │ │ │ │ -0004a950: 6365 496e 7665 7273 654d 6170 2c20 2d2d ceInverseMap, -- │ │ │ │ -0004a960: 2064 6563 6c61 7265 2074 6861 7420 7477 declare that tw │ │ │ │ -0004a970: 6f20 7261 7469 6f6e 616c 206d 6170 730a o rational maps. │ │ │ │ -0004a980: 2020 2020 6172 6520 6f6e 6520 7468 6520 are one the │ │ │ │ -0004a990: 696e 7665 7273 6520 6f66 2074 6865 206f inverse of the o │ │ │ │ -0004a9a0: 7468 6572 0a0a 5761 7973 2074 6f20 7573 ther..Ways to us │ │ │ │ -0004a9b0: 6520 666f 7263 6549 6d61 6765 3a0a 3d3d e forceImage:.== │ │ │ │ -0004a9c0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004a9d0: 3d3d 3d3d 3d0a 0a20 202a 2022 666f 7263 =====.. * "forc │ │ │ │ -0004a9e0: 6549 6d61 6765 2852 6174 696f 6e61 6c4d eImage(RationalM │ │ │ │ -0004a9f0: 6170 2c49 6465 616c 2922 0a0a 466f 7220 ap,Ideal)"..For │ │ │ │ -0004aa00: 7468 6520 7072 6f67 7261 6d6d 6572 0a3d the programmer.= │ │ │ │ -0004aa10: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004aa20: 3d0a 0a54 6865 206f 626a 6563 7420 2a6e =..The object *n │ │ │ │ -0004aa30: 6f74 6520 666f 7263 6549 6d61 6765 3a20 ote forceImage: │ │ │ │ -0004aa40: 666f 7263 6549 6d61 6765 2c20 6973 2061 forceImage, is a │ │ │ │ -0004aa50: 202a 6e6f 7465 206d 6574 686f 6420 6675 *note method fu │ │ │ │ -0004aa60: 6e63 7469 6f6e 3a0a 284d 6163 6175 6c61 nction:.(Macaula │ │ │ │ -0004aa70: 7932 446f 6329 4d65 7468 6f64 4675 6e63 y2Doc)MethodFunc │ │ │ │ -0004aa80: 7469 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d tion,...-------- │ │ │ │ +0004a860: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a 0a43 ------------+..C │ │ │ │ +0004a870: 6176 6561 740a 3d3d 3d3d 3d3d 0a0a 4966 aveat.======..If │ │ │ │ +0004a880: 2074 6865 2064 6563 6c61 7261 7469 6f6e the declaration │ │ │ │ +0004a890: 2069 7320 6661 6c73 652c 206e 6f6e 7365 is false, nonse │ │ │ │ +0004a8a0: 6e73 6963 616c 2061 6e73 7765 7273 206d nsical answers m │ │ │ │ +0004a8b0: 6179 2072 6573 756c 742e 0a0a 5365 6520 ay result...See │ │ │ │ +0004a8c0: 616c 736f 0a3d 3d3d 3d3d 3d3d 3d0a 0a20 also.========.. │ │ │ │ +0004a8d0: 202a 202a 6e6f 7465 2069 6d61 6765 2852 * *note image(R │ │ │ │ +0004a8e0: 6174 696f 6e61 6c4d 6170 293a 2069 6d61 ationalMap): ima │ │ │ │ +0004a8f0: 6765 5f6c 7052 6174 696f 6e61 6c4d 6170 ge_lpRationalMap │ │ │ │ +0004a900: 5f63 6d5a 5a5f 7270 2c20 2d2d 2063 6c6f _cmZZ_rp, -- clo │ │ │ │ +0004a910: 7375 7265 206f 6620 7468 650a 2020 2020 sure of the. │ │ │ │ +0004a920: 696d 6167 6520 6f66 2061 2072 6174 696f image of a ratio │ │ │ │ +0004a930: 6e61 6c20 6d61 700a 2020 2a20 2a6e 6f74 nal map. * *not │ │ │ │ +0004a940: 6520 666f 7263 6549 6e76 6572 7365 4d61 e forceInverseMa │ │ │ │ +0004a950: 703a 2066 6f72 6365 496e 7665 7273 654d p: forceInverseM │ │ │ │ +0004a960: 6170 2c20 2d2d 2064 6563 6c61 7265 2074 ap, -- declare t │ │ │ │ +0004a970: 6861 7420 7477 6f20 7261 7469 6f6e 616c hat two rational │ │ │ │ +0004a980: 206d 6170 730a 2020 2020 6172 6520 6f6e maps. are on │ │ │ │ +0004a990: 6520 7468 6520 696e 7665 7273 6520 6f66 e the inverse of │ │ │ │ +0004a9a0: 2074 6865 206f 7468 6572 0a0a 5761 7973 the other..Ways │ │ │ │ +0004a9b0: 2074 6f20 7573 6520 666f 7263 6549 6d61 to use forceIma │ │ │ │ +0004a9c0: 6765 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ge:.============ │ │ │ │ +0004a9d0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a 0a20 202a ===========.. * │ │ │ │ +0004a9e0: 2022 666f 7263 6549 6d61 6765 2852 6174 "forceImage(Rat │ │ │ │ +0004a9f0: 696f 6e61 6c4d 6170 2c49 6465 616c 2922 ionalMap,Ideal)" │ │ │ │ +0004aa00: 0a0a 466f 7220 7468 6520 7072 6f67 7261 ..For the progra │ │ │ │ +0004aa10: 6d6d 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d mmer.=========== │ │ │ │ +0004aa20: 3d3d 3d3d 3d3d 3d0a 0a54 6865 206f 626a =======..The obj │ │ │ │ +0004aa30: 6563 7420 2a6e 6f74 6520 666f 7263 6549 ect *note forceI │ │ │ │ +0004aa40: 6d61 6765 3a20 666f 7263 6549 6d61 6765 mage: forceImage │ │ │ │ +0004aa50: 2c20 6973 2061 202a 6e6f 7465 206d 6574 , is a *note met │ │ │ │ +0004aa60: 686f 6420 6675 6e63 7469 6f6e 3a0a 284d hod function:.(M │ │ │ │ +0004aa70: 6163 6175 6c61 7932 446f 6329 4d65 7468 acaulay2Doc)Meth │ │ │ │ +0004aa80: 6f64 4675 6e63 7469 6f6e 2c2e 0a0a 2d2d odFunction,...-- │ │ │ │ 0004aa90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004aaa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004aab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004aac0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004aad0: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ -0004aae0: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ -0004aaf0: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ -0004ab00: 642f 7265 7072 6f64 7563 6962 6c65 2d70 d/reproducible-p │ │ │ │ -0004ab10: 6174 682f 6d61 6361 756c 6179 322d 312e ath/macaulay2-1. │ │ │ │ -0004ab20: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+ds/M2/Maca │ │ │ │ -0004ab30: 756c 6179 322f 7061 636b 6167 6573 2f43 ulay2/packages/C │ │ │ │ -0004ab40: 7265 6d6f 6e61 2f0a 646f 6375 6d65 6e74 remona/.document │ │ │ │ -0004ab50: 6174 696f 6e2e 6d32 3a31 3031 323a 302e ation.m2:1012:0. │ │ │ │ -0004ab60: 0a1f 0a46 696c 653a 2043 7265 6d6f 6e61 ...File: Cremona │ │ │ │ -0004ab70: 2e69 6e66 6f2c 204e 6f64 653a 2066 6f72 .info, Node: for │ │ │ │ -0004ab80: 6365 496e 7665 7273 654d 6170 2c20 4e65 ceInverseMap, Ne │ │ │ │ -0004ab90: 7874 3a20 6772 6170 682c 2050 7265 763a xt: graph, Prev: │ │ │ │ -0004aba0: 2066 6f72 6365 496d 6167 652c 2055 703a forceImage, Up: │ │ │ │ -0004abb0: 2054 6f70 0a0a 666f 7263 6549 6e76 6572 Top..forceInver │ │ │ │ -0004abc0: 7365 4d61 7020 2d2d 2064 6563 6c61 7265 seMap -- declare │ │ │ │ -0004abd0: 2074 6861 7420 7477 6f20 7261 7469 6f6e that two ration │ │ │ │ -0004abe0: 616c 206d 6170 7320 6172 6520 6f6e 6520 al maps are one │ │ │ │ -0004abf0: 7468 6520 696e 7665 7273 6520 6f66 2074 the inverse of t │ │ │ │ -0004ac00: 6865 206f 7468 6572 0a2a 2a2a 2a2a 2a2a he other.******* │ │ │ │ +0004aad0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a 0a54 -------------..T │ │ │ │ +0004aae0: 6865 2073 6f75 7263 6520 6f66 2074 6869 he source of thi │ │ │ │ +0004aaf0: 7320 646f 6375 6d65 6e74 2069 7320 696e s document is in │ │ │ │ +0004ab00: 0a2f 6275 696c 642f 7265 7072 6f64 7563 ./build/reproduc │ │ │ │ +0004ab10: 6962 6c65 2d70 6174 682f 6d61 6361 756c ible-path/macaul │ │ │ │ +0004ab20: 6179 322d 312e 3236 2e30 362b 6473 2f4d ay2-1.26.06+ds/M │ │ │ │ +0004ab30: 322f 4d61 6361 756c 6179 322f 7061 636b 2/Macaulay2/pack │ │ │ │ +0004ab40: 6167 6573 2f43 7265 6d6f 6e61 2f0a 646f ages/Cremona/.do │ │ │ │ +0004ab50: 6375 6d65 6e74 6174 696f 6e2e 6d32 3a31 cumentation.m2:1 │ │ │ │ +0004ab60: 3031 323a 302e 0a1f 0a46 696c 653a 2043 012:0....File: C │ │ │ │ +0004ab70: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 remona.info, Nod │ │ │ │ +0004ab80: 653a 2066 6f72 6365 496e 7665 7273 654d e: forceInverseM │ │ │ │ +0004ab90: 6170 2c20 4e65 7874 3a20 6772 6170 682c ap, Next: graph, │ │ │ │ +0004aba0: 2050 7265 763a 2066 6f72 6365 496d 6167 Prev: forceImag │ │ │ │ +0004abb0: 652c 2055 703a 2054 6f70 0a0a 666f 7263 e, Up: Top..forc │ │ │ │ +0004abc0: 6549 6e76 6572 7365 4d61 7020 2d2d 2064 eInverseMap -- d │ │ │ │ +0004abd0: 6563 6c61 7265 2074 6861 7420 7477 6f20 eclare that two │ │ │ │ +0004abe0: 7261 7469 6f6e 616c 206d 6170 7320 6172 rational maps ar │ │ │ │ +0004abf0: 6520 6f6e 6520 7468 6520 696e 7665 7273 e one the invers │ │ │ │ +0004ac00: 6520 6f66 2074 6865 206f 7468 6572 0a2a e of the other.* │ │ │ │ 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 2a0a 0a20 202a ***********.. * │ │ │ │ -0004ac60: 2055 7361 6765 3a20 0a20 2020 2020 2020 Usage: . │ │ │ │ -0004ac70: 2066 6f72 6365 496e 7665 7273 654d 6170 forceInverseMap │ │ │ │ -0004ac80: 2850 6869 2c50 7369 290a 2020 2a20 496e (Phi,Psi). * In │ │ │ │ -0004ac90: 7075 7473 3a0a 2020 2020 2020 2a20 5068 puts:. * Ph │ │ │ │ -0004aca0: 692c 2061 202a 6e6f 7465 2072 6174 696f i, a *note ratio │ │ │ │ -0004acb0: 6e61 6c20 6d61 703a 2052 6174 696f 6e61 nal map: Rationa │ │ │ │ -0004acc0: 6c4d 6170 2c0a 2020 2020 2020 2a20 5073 lMap,. * Ps │ │ │ │ -0004acd0: 692c 2061 202a 6e6f 7465 2072 6174 696f i, a *note ratio │ │ │ │ -0004ace0: 6e61 6c20 6d61 703a 2052 6174 696f 6e61 nal map: Rationa │ │ │ │ -0004acf0: 6c4d 6170 2c0a 2020 2a20 4f75 7470 7574 lMap,. * Output │ │ │ │ -0004ad00: 733a 0a20 2020 2020 202a 202a 6e6f 7465 s:. * *note │ │ │ │ -0004ad10: 206e 756c 6c3a 2028 4d61 6361 756c 6179 null: (Macaulay │ │ │ │ -0004ad20: 3244 6f63 296e 756c 6c2c 0a0a 4465 7363 2Doc)null,..Desc │ │ │ │ -0004ad30: 7269 7074 696f 6e0a 3d3d 3d3d 3d3d 3d3d ription.======== │ │ │ │ -0004ad40: 3d3d 3d0a 0a54 6869 7320 6d65 7468 6f64 ===..This method │ │ │ │ -0004ad50: 2061 6c6c 6f77 7320 746f 2069 6e66 6f72 allows to infor │ │ │ │ -0004ad60: 6d20 7468 6520 7379 7374 656d 2074 6861 m the system tha │ │ │ │ -0004ad70: 7420 7477 6f20 6d61 7073 2061 7265 206f t two maps are o │ │ │ │ -0004ad80: 6e65 2074 6865 2069 6e76 6572 7365 206f ne the inverse o │ │ │ │ -0004ad90: 660a 7468 6520 6f74 6865 7220 7769 7468 f.the other with │ │ │ │ -0004ada0: 6f75 7420 7065 7266 6f72 6d69 6e67 2061 out performing a │ │ │ │ -0004adb0: 6e79 2063 6f6d 7075 7461 7469 6f6e 2e20 ny computation. │ │ │ │ -0004adc0: 5468 6973 2069 7320 7573 6566 756c 2069 This is useful i │ │ │ │ -0004add0: 6e20 7061 7274 6963 756c 6172 2069 660a n particular if. │ │ │ │ -0004ade0: 796f 7520 6361 6c63 756c 6174 6520 7468 you calculate th │ │ │ │ -0004adf0: 6520 696e 7665 7273 6520 6d61 7020 7573 e inverse map us │ │ │ │ -0004ae00: 696e 6720 796f 7572 206f 776e 206d 6574 ing your own met │ │ │ │ -0004ae10: 686f 642e 0a0a 4361 7665 6174 0a3d 3d3d hod...Caveat.=== │ │ │ │ -0004ae20: 3d3d 3d0a 0a49 6620 7468 6520 6465 636c ===..If the decl │ │ │ │ -0004ae30: 6172 6174 696f 6e20 6973 2066 616c 7365 aration is false │ │ │ │ -0004ae40: 2c20 6e6f 6e73 656e 7369 6361 6c20 616e , nonsensical an │ │ │ │ -0004ae50: 7377 6572 7320 6d61 7920 7265 7375 6c74 swers may result │ │ │ │ -0004ae60: 2e0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d ...See also.==== │ │ │ │ -0004ae70: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -0004ae80: 6973 496e 7665 7273 654d 6170 2852 6174 isInverseMap(Rat │ │ │ │ -0004ae90: 696f 6e61 6c4d 6170 2c52 6174 696f 6e61 ionalMap,Rationa │ │ │ │ -0004aea0: 6c4d 6170 293a 0a20 2020 2069 7349 6e76 lMap):. isInv │ │ │ │ -0004aeb0: 6572 7365 4d61 705f 6c70 5261 7469 6f6e erseMap_lpRation │ │ │ │ -0004aec0: 616c 4d61 705f 636d 5261 7469 6f6e 616c alMap_cmRational │ │ │ │ -0004aed0: 4d61 705f 7270 2c20 2d2d 2063 6865 636b Map_rp, -- check │ │ │ │ -0004aee0: 7320 7768 6574 6865 7220 7477 6f20 7261 s whether two ra │ │ │ │ -0004aef0: 7469 6f6e 616c 0a20 2020 206d 6170 7320 tional. maps │ │ │ │ -0004af00: 6172 6520 6f6e 6520 7468 6520 696e 7665 are one the inve │ │ │ │ -0004af10: 7273 6520 6f66 2074 6865 206f 7468 6572 rse of the other │ │ │ │ -0004af20: 0a20 202a 202a 6e6f 7465 2069 6e76 6572 . * *note inver │ │ │ │ -0004af30: 7365 2852 6174 696f 6e61 6c4d 6170 293a se(RationalMap): │ │ │ │ -0004af40: 2069 6e76 6572 7365 5f6c 7052 6174 696f inverse_lpRatio │ │ │ │ -0004af50: 6e61 6c4d 6170 5f72 702c 202d 2d20 696e nalMap_rp, -- in │ │ │ │ -0004af60: 7665 7273 6520 6f66 2061 0a20 2020 2062 verse of a. b │ │ │ │ -0004af70: 6972 6174 696f 6e61 6c20 6d61 700a 0a57 irational map..W │ │ │ │ -0004af80: 6179 7320 746f 2075 7365 2066 6f72 6365 ays to use force │ │ │ │ -0004af90: 496e 7665 7273 654d 6170 3a0a 3d3d 3d3d InverseMap:.==== │ │ │ │ -0004afa0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -0004afb0: 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 2266 ========.. * "f │ │ │ │ -0004afc0: 6f72 6365 496e 7665 7273 654d 6170 2852 orceInverseMap(R │ │ │ │ -0004afd0: 6174 696f 6e61 6c4d 6170 2c52 6174 696f ationalMap,Ratio │ │ │ │ -0004afe0: 6e61 6c4d 6170 2922 0a0a 466f 7220 7468 nalMap)"..For th │ │ │ │ -0004aff0: 6520 7072 6f67 7261 6d6d 6572 0a3d 3d3d e programmer.=== │ │ │ │ -0004b000: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d0a ===============. │ │ │ │ -0004b010: 0a54 6865 206f 626a 6563 7420 2a6e 6f74 .The object *not │ │ │ │ -0004b020: 6520 666f 7263 6549 6e76 6572 7365 4d61 e forceInverseMa │ │ │ │ -0004b030: 703a 2066 6f72 6365 496e 7665 7273 654d p: forceInverseM │ │ │ │ -0004b040: 6170 2c20 6973 2061 202a 6e6f 7465 206d ap, is a *note m │ │ │ │ -0004b050: 6574 686f 6420 6675 6e63 7469 6f6e 3a0a ethod function:. │ │ │ │ -0004b060: 284d 6163 6175 6c61 7932 446f 6329 4d65 (Macaulay2Doc)Me │ │ │ │ -0004b070: 7468 6f64 4675 6e63 7469 6f6e 2c2e 0a0a thodFunction,... │ │ │ │ -0004b080: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +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). │ │ │ │ +0004ac90: 2020 2a20 496e 7075 7473 3a0a 2020 2020 * Inputs:. │ │ │ │ +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:. * │ │ │ │ +0004ad10: 202a 6e6f 7465 206e 756c 6c3a 2028 4d61 *note null: (Ma │ │ │ │ +0004ad20: 6361 756c 6179 3244 6f63 296e 756c 6c2c caulay2Doc)null, │ │ │ │ +0004ad30: 0a0a 4465 7363 7269 7074 696f 6e0a 3d3d ..Description.== │ │ │ │ +0004ad40: 3d3d 3d3d 3d3d 3d3d 3d0a 0a54 6869 7320 =========..This │ │ │ │ +0004ad50: 6d65 7468 6f64 2061 6c6c 6f77 7320 746f method allows to │ │ │ │ +0004ad60: 2069 6e66 6f72 6d20 7468 6520 7379 7374 inform the syst │ │ │ │ +0004ad70: 656d 2074 6861 7420 7477 6f20 6d61 7073 em that two maps │ │ │ │ +0004ad80: 2061 7265 206f 6e65 2074 6865 2069 6e76 are one the inv │ │ │ │ +0004ad90: 6572 7365 206f 660a 7468 6520 6f74 6865 erse of.the othe │ │ │ │ +0004ada0: 7220 7769 7468 6f75 7420 7065 7266 6f72 r without perfor │ │ │ │ +0004adb0: 6d69 6e67 2061 6e79 2063 6f6d 7075 7461 ming any computa │ │ │ │ +0004adc0: 7469 6f6e 2e20 5468 6973 2069 7320 7573 tion. This is us │ │ │ │ +0004add0: 6566 756c 2069 6e20 7061 7274 6963 756c eful in particul │ │ │ │ +0004ade0: 6172 2069 660a 796f 7520 6361 6c63 756c ar if.you calcul │ │ │ │ +0004adf0: 6174 6520 7468 6520 696e 7665 7273 6520 ate the inverse │ │ │ │ +0004ae00: 6d61 7020 7573 696e 6720 796f 7572 206f map using your o │ │ │ │ +0004ae10: 776e 206d 6574 686f 642e 0a0a 4361 7665 wn method...Cave │ │ │ │ +0004ae20: 6174 0a3d 3d3d 3d3d 3d0a 0a49 6620 7468 at.======..If th │ │ │ │ +0004ae30: 6520 6465 636c 6172 6174 696f 6e20 6973 e declaration is │ │ │ │ +0004ae40: 2066 616c 7365 2c20 6e6f 6e73 656e 7369 false, nonsensi │ │ │ │ +0004ae50: 6361 6c20 616e 7377 6572 7320 6d61 7920 cal answers may │ │ │ │ +0004ae60: 7265 7375 6c74 2e0a 0a53 6565 2061 6c73 result...See als │ │ │ │ +0004ae70: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +0004ae80: 2a6e 6f74 6520 6973 496e 7665 7273 654d *note isInverseM │ │ │ │ +0004ae90: 6170 2852 6174 696f 6e61 6c4d 6170 2c52 ap(RationalMap,R │ │ │ │ +0004aea0: 6174 696f 6e61 6c4d 6170 293a 0a20 2020 ationalMap):. │ │ │ │ +0004aeb0: 2069 7349 6e76 6572 7365 4d61 705f 6c70 isInverseMap_lp │ │ │ │ +0004aec0: 5261 7469 6f6e 616c 4d61 705f 636d 5261 RationalMap_cmRa │ │ │ │ +0004aed0: 7469 6f6e 616c 4d61 705f 7270 2c20 2d2d tionalMap_rp, -- │ │ │ │ +0004aee0: 2063 6865 636b 7320 7768 6574 6865 7220 checks whether │ │ │ │ +0004aef0: 7477 6f20 7261 7469 6f6e 616c 0a20 2020 two rational. │ │ │ │ +0004af00: 206d 6170 7320 6172 6520 6f6e 6520 7468 maps are one th │ │ │ │ +0004af10: 6520 696e 7665 7273 6520 6f66 2074 6865 e inverse of the │ │ │ │ +0004af20: 206f 7468 6572 0a20 202a 202a 6e6f 7465 other. * *note │ │ │ │ +0004af30: 2069 6e76 6572 7365 2852 6174 696f 6e61 inverse(Rationa │ │ │ │ +0004af40: 6c4d 6170 293a 2069 6e76 6572 7365 5f6c lMap): inverse_l │ │ │ │ +0004af50: 7052 6174 696f 6e61 6c4d 6170 5f72 702c pRationalMap_rp, │ │ │ │ +0004af60: 202d 2d20 696e 7665 7273 6520 6f66 2061 -- inverse of a │ │ │ │ +0004af70: 0a20 2020 2062 6972 6174 696f 6e61 6c20 . birational │ │ │ │ +0004af80: 6d61 700a 0a57 6179 7320 746f 2075 7365 map..Ways to use │ │ │ │ +0004af90: 2066 6f72 6365 496e 7665 7273 654d 6170 forceInverseMap │ │ │ │ +0004afa0: 3a0a 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d :.============== │ │ │ │ +0004afb0: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 0a0a ==============.. │ │ │ │ +0004afc0: 2020 2a20 2266 6f72 6365 496e 7665 7273 * "forceInvers │ │ │ │ +0004afd0: 654d 6170 2852 6174 696f 6e61 6c4d 6170 eMap(RationalMap │ │ │ │ +0004afe0: 2c52 6174 696f 6e61 6c4d 6170 2922 0a0a ,RationalMap)".. │ │ │ │ +0004aff0: 466f 7220 7468 6520 7072 6f67 7261 6d6d For the programm │ │ │ │ +0004b000: 6572 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d er.============= │ │ │ │ +0004b010: 3d3d 3d3d 3d0a 0a54 6865 206f 626a 6563 =====..The objec │ │ │ │ +0004b020: 7420 2a6e 6f74 6520 666f 7263 6549 6e76 t *note forceInv │ │ │ │ +0004b030: 6572 7365 4d61 703a 2066 6f72 6365 496e erseMap: forceIn │ │ │ │ +0004b040: 7665 7273 654d 6170 2c20 6973 2061 202a verseMap, is a * │ │ │ │ +0004b050: 6e6f 7465 206d 6574 686f 6420 6675 6e63 note method func │ │ │ │ +0004b060: 7469 6f6e 3a0a 284d 6163 6175 6c61 7932 tion:.(Macaulay2 │ │ │ │ +0004b070: 446f 6329 4d65 7468 6f64 4675 6e63 7469 Doc)MethodFuncti │ │ │ │ +0004b080: 6f6e 2c2e 0a0a 2d2d 2d2d 2d2d 2d2d 2d2d on,...---------- │ │ │ │ 0004b090: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004b0a0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 0004b0b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -0004b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d0a ---------------. │ │ │ │ -0004b0d0: 0a54 6865 2073 6f75 7263 6520 6f66 2074 .The source of t │ │ │ │ -0004b0e0: 6869 7320 646f 6375 6d65 6e74 2069 7320 his document is │ │ │ │ -0004b0f0: 696e 0a2f 6275 696c 642f 7265 7072 6f64 in./build/reprod │ │ │ │ -0004b100: 7563 6962 6c65 2d70 6174 682f 6d61 6361 ucible-path/maca │ │ │ │ -0004b110: 756c 6179 322d 312e 3236 2e30 362b 6473 ulay2-1.26.06+ds │ │ │ │ -0004b120: 2f4d 322f 4d61 6361 756c 6179 322f 7061 /M2/Macaulay2/pa │ │ │ │ -0004b130: 636b 6167 6573 2f43 7265 6d6f 6e61 2f0a ckages/Cremona/. │ │ │ │ -0004b140: 646f 6375 6d65 6e74 6174 696f 6e2e 6d32 documentation.m2 │ │ │ │ -0004b150: 3a39 3935 3a30 2e0a 1f0a 4669 6c65 3a20 :995:0....File: │ │ │ │ -0004b160: 4372 656d 6f6e 612e 696e 666f 2c20 4e6f Cremona.info, No │ │ │ │ -0004b170: 6465 3a20 6772 6170 682c 204e 6578 743a de: graph, Next: │ │ │ │ -0004b180: 2067 7261 7068 5f6c 7052 696e 674d 6170 graph_lpRingMap │ │ │ │ -0004b190: 5f72 702c 2050 7265 763a 2066 6f72 6365 _rp, Prev: force │ │ │ │ -0004b1a0: 496e 7665 7273 654d 6170 2c20 5570 3a20 InverseMap, Up: │ │ │ │ -0004b1b0: 546f 700a 0a67 7261 7068 202d 2d20 636c Top..graph -- cl │ │ │ │ -0004b1c0: 6f73 7572 6520 6f66 2074 6865 2067 7261 osure of the gra │ │ │ │ -0004b1d0: 7068 206f 6620 6120 7261 7469 6f6e 616c ph of a rational │ │ │ │ -0004b1e0: 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a 2a2a map.*********** │ │ │ │ +0004b0c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +0004b0d0: 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 7263 -----..The sourc │ │ │ │ +0004b0e0: 6520 6f66 2074 6869 7320 646f 6375 6d65 e of this docume │ │ │ │ +0004b0f0: 6e74 2069 7320 696e 0a2f 6275 696c 642f nt is in./build/ │ │ │ │ +0004b100: 7265 7072 6f64 7563 6962 6c65 2d70 6174 reproducible-pat │ │ │ │ +0004b110: 682f 6d61 6361 756c 6179 322d 312e 3236 h/macaulay2-1.26 │ │ │ │ +0004b120: 2e30 362b 6473 2f4d 322f 4d61 6361 756c .06+ds/M2/Macaul │ │ │ │ +0004b130: 6179 322f 7061 636b 6167 6573 2f43 7265 ay2/packages/Cre │ │ │ │ +0004b140: 6d6f 6e61 2f0a 646f 6375 6d65 6e74 6174 mona/.documentat │ │ │ │ +0004b150: 696f 6e2e 6d32 3a39 3935 3a30 2e0a 1f0a ion.m2:995:0.... │ │ │ │ +0004b160: 4669 6c65 3a20 4372 656d 6f6e 612e 696e File: Cremona.in │ │ │ │ +0004b170: 666f 2c20 4e6f 6465 3a20 6772 6170 682c fo, Node: graph, │ │ │ │ +0004b180: 204e 6578 743a 2067 7261 7068 5f6c 7052 Next: graph_lpR │ │ │ │ +0004b190: 696e 674d 6170 5f72 702c 2050 7265 763a ingMap_rp, Prev: │ │ │ │ +0004b1a0: 2066 6f72 6365 496e 7665 7273 654d 6170 forceInverseMap │ │ │ │ +0004b1b0: 2c20 5570 3a20 546f 700a 0a67 7261 7068 , Up: Top..graph │ │ │ │ +0004b1c0: 202d 2d20 636c 6f73 7572 6520 6f66 2074 -- closure of t │ │ │ │ +0004b1d0: 6865 2067 7261 7068 206f 6620 6120 7261 he graph of a ra │ │ │ │ +0004b1e0: 7469 6f6e 616c 206d 6170 0a2a 2a2a 2a2a tional map.***** │ │ │ │ 0004b1f0: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 0004b200: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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 =..+------------ │ │ │ │ +0004b210: 2a2a 2a2a 2a2a 2a2a 2a2a 0a0a 2020 2a20 **********.. * │ │ │ │ +0004b220: 5573 6167 653a 200a 2020 2020 2020 2020 Usage: . │ │ │ │ +0004b230: 6772 6170 6820 7068 690a 2020 2a20 496e graph phi. * In │ │ │ │ +0004b240: 7075 7473 3a0a 2020 2020 2020 2a20 7068 puts:. * ph │ │ │ │ +0004b250: 692c 2061 202a 6e6f 7465 2072 6174 696f i, a *note ratio │ │ │ │ +0004b260: 6e61 6c20 6d61 703a 2052 6174 696f 6e61 nal map: Rationa │ │ │ │ +0004b270: 6c4d 6170 2c0a 2020 2a20 2a6e 6f74 6520 lMap,. * *note │ │ │ │ +0004b280: 4f70 7469 6f6e 616c 2069 6e70 7574 733a Optional inputs: │ │ │ │ +0004b290: 2028 4d61 6361 756c 6179 3244 6f63 2975 (Macaulay2Doc)u │ │ │ │ +0004b2a0: 7369 6e67 2066 756e 6374 696f 6e73 2077 sing functions w │ │ │ │ +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 =======..+------ │ │ │ │ 0004b3d0: 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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_ |.| │ │ │ │ 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: 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 │ │ │ │ +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 - │ │ │ │ 0004b4e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b4f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b500: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ 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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 │ │ │ │ +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 │ │ │ │ 0004b5e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b5f0: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +0004b5f0: 2020 2020 2020 207c 0a7c 2020 2020 2020 |.| │ │ │ │ 0004b600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b610: 2020 2020 2020 5a5a 2020 2020 2020 2020 ZZ │ │ │ │ +0004b610: 2020 2020 2020 2020 2020 2020 5a5a 2020 ZZ │ │ │ │ 0004b620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 0004b630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 |.| │ │ │ │ +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 |.| │ │ │ │ 0004b6a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -0004b6b0: 2020 2020 3139 3031 3831 2020 3020 2020 190181 0 │ │ │ │ -0004b6c0: 3120 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7266 6163 6520 696e hypersurface in │ │ │ │ -000514f0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +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 |.|------ │ │ │ │ 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: 2d7c 0a7c 5050 5e35 2920 2020 2020 2020 -|.|PP^5) │ │ │ │ +00051540: 2d2d 2d2d 2d2d 2d7c 0a7c 5050 5e35 2920 -------|.|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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00051590: 2020 2020 2020 207c 0a2b 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: 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) │ │ │ │ +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) │ │ │ │ 00051620: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00051630: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00051630: 2020 2020 2020 207c 0a2b 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: 2d2b 0a7c 6937 203a 2064 6573 6372 6962 -+.|i7 : describ │ │ │ │ -00051690: 6520 7032 2020 2020 2020 2020 2020 2020 e p2 │ │ │ │ +00051680: 2d2d 2d2d 2d2d 2d2b 0a7c 6937 203a 2064 -------+.|i7 : d │ │ │ │ +00051690: 6573 6372 6962 6520 7032 2020 2020 2020 escribe 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: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +000516d0: 2020 2020 2020 207c 0a7c 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: 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 │ │ │ │ +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 │ │ │ │ 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: 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 │ │ │ │ +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 │ │ │ │ 00051890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000518a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000518b0: 207c 0a7c 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------ │ │ │ │ +000518b0: 2020 2020 2020 207c 0a7c 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: 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 |.|------------ │ │ │ │ +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 , |.|------ │ │ │ │ 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: 2d7c 0a7c 317d 2c7b 312c 2031 7d2c 7b31 -|.|1},{1, 1},{1 │ │ │ │ -000519b0: 2c20 317d 2920 2020 2020 2020 2020 2020 , 1}) │ │ │ │ +000519a0: 2d2d 2d2d 2d2d 2d7c 0a7c 317d 2c7b 312c -------|.|1},{1, │ │ │ │ +000519b0: 2031 7d2c 7b31 2c20 317d 2920 2020 2020 1},{1, 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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +000519f0: 2020 2020 2020 207c 0a2b 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: 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 │ │ │ │ +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 │ │ │ │ 00051a70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051a80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00051a90: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00051a90: 2020 2020 2020 207c 0a7c 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: 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 │ │ │ │ +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 } │ │ │ │ 00051b10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051b20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00051b30: 207c 0a7c 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00051b30: 2020 2020 2020 207c 0a7c 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: 207c 0a7c 6f38 203a 204c 6973 7420 2020 |.|o8 : List │ │ │ │ -00051b90: 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 │ │ │ │ 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: 207c 0a2b 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------ │ │ │ │ +00051bd0: 2020 2020 2020 207c 0a2b 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: 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 +--------------- │ │ │ │ +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...+--------- │ │ │ │ 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 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; │ │ │ │ +00051cd0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00051ce0: 2d2d 2d2d 2b0a 7c69 3920 3a20 7469 6d65 ----+.|i9 : time │ │ │ │ +00051cf0: 2067 203d 2067 7261 7068 2070 323b 2020 g = graph p2; │ │ │ │ 00051d00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00051d10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00051d20: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00051d30: 7c20 2d2d 2075 7365 6420 302e 3033 3139 | -- used 0.0319 │ │ │ │ -00051d40: 3939 3773 2028 6370 7529 3b20 302e 3033 997s (cpu); 0.03 │ │ │ │ -00051d50: 3136 3731 3573 2028 7468 7265 6164 293b 16715s (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 +--------------- │ │ │ │ +00051d20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00051d30: 2020 2020 7c0a 7c20 2d2d 2075 7365 6420 |.| -- used │ │ │ │ +00051d40: 302e 3039 3432 3135 3573 2028 6370 7529 0.0942155s (cpu) │ │ │ │ +00051d50: 3b20 302e 3034 3135 3832 7320 2874 6872 ; 0.041582s (thr │ │ │ │ +00051d60: 6561 6429 3b20 3073 2028 6763 2920 2020 ead); 0s (gc) │ │ │ │ +00051d70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00051d80: 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d |.+--------- │ │ │ │ 00051d90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051da0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00051db0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00051dc0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2b0a --------------+. │ │ │ │ -00051dd0: 7c69 3130 203a 2067 5f30 3b20 2020 2020 |i10 : g_0; │ │ │ │ -00051de0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 ; │ │ │ │ 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 7c0a |. │ │ │ │ -00051e20: 7c20 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 |.| │ │ │ │ 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 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 |--------------- │ │ │ │ +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|.|--------- │ │ │ │ 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 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 │ │ │ │ +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) │ │ │ │ 00051f40: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00051f50: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00051f60: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00051f50: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00051f60: 2020 2020 7c0a 2b2d 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 2b0a --------------+. │ │ │ │ -00051fb0: 7c69 3131 203a 2067 5f31 3b20 2020 2020 |i11 : g_1; │ │ │ │ -00051fc0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 ; │ │ │ │ 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 7c0a |. │ │ │ │ -00052000: 7c20 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 |.| │ │ │ │ 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 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 |--------------- │ │ │ │ +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|.|--------- │ │ │ │ 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 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 │ │ │ │ +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) │ │ │ │ 00052120: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00052130: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00052140: 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d +--------------- │ │ │ │ +00052130: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00052140: 2020 2020 7c0a 2b2d 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 2b0a --------------+. │ │ │ │ -00052190: 7c69 3132 203a 2067 5f32 3b20 2020 2020 |i12 : g_2; │ │ │ │ -000521a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +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 ; │ │ │ │ 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 7c0a |. │ │ │ │ -000521e0: 7c20 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 |.| │ │ │ │ 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 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 |--------------- │ │ │ │ +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 |.|--------- │ │ │ │ 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 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 +--------------- │ │ │ │ +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 |.+--------- │ │ │ │ 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 2b0a --------------+. │ │ │ │ -00052370: 7c69 3133 203a 2064 6573 6372 6962 6520 |i13 : describe │ │ │ │ -00052380: 675f 3020 2020 2020 2020 2020 2020 2020 g_0 │ │ │ │ +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 │ │ │ │ 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 7c0a |. │ │ │ │ -000523c0: 7c20 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 |.| │ │ │ │ 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 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 │ │ │ │ +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 │ │ │ │ 000524e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00052530: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00052540: 2020 2020 2020 2020 2020 2020 2020 7c0a |. │ │ │ │ -00052550: 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |--------------- │ │ │ │ +00052540: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00052550: 2020 2020 7c0a 7c2d 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 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 |--------------- │ │ │ │ +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, |.|--------- │ │ │ │ 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 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 |--------------- │ │ │ │ +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 |.|--------- │ │ │ │ 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 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 |--------------- │ │ │ │ +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, |.|--------- │ │ │ │ 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 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 |--------------- │ │ │ │ +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 |.|--------- │ │ │ │ 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 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 +--------------- │ │ │ │ +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 |.+--------- │ │ │ │ 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 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 ..-------------- │ │ │ │ +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,...-------- │ │ │ │ 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: 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 3236 2e30 362b caulay2-1.26.06+ │ │ │ │ -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 │ │ │ │ -00052c20: 6170 682c 2055 703a 2054 6f70 0a0a 6772 aph, Up: Top..gr │ │ │ │ -00052c30: 6170 6828 5269 6e67 4d61 7029 202d 2d20 aph(RingMap) -- │ │ │ │ -00052c40: 636c 6f73 7572 6520 6f66 2074 6865 2067 closure of the g │ │ │ │ -00052c50: 7261 7068 206f 6620 6120 7261 7469 6f6e raph of a ration │ │ │ │ -00052c60: 616c 206d 6170 0a2a 2a2a 2a2a 2a2a 2a2a al map.********* │ │ │ │ +00052b40: 2d2d 2d2d 2d2d 2d0a 0a54 6865 2073 6f75 -------..The sou │ │ │ │ +00052b50: 7263 6520 6f66 2074 6869 7320 646f 6375 rce of this docu │ │ │ │ +00052b60: 6d65 6e74 2069 7320 696e 0a2f 6275 696c ment is in./buil │ │ │ │ +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: 3236 2e30 362b 6473 2f4d 322f 4d61 6361 26.06+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.*** │ │ │ │ 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 2a0a ***************. │ │ │ │ -00052ca0: 0a20 202a 2046 756e 6374 696f 6e3a 202a . * Function: * │ │ │ │ -00052cb0: 6e6f 7465 2067 7261 7068 3a20 6772 6170 note graph: grap │ │ │ │ -00052cc0: 682c 0a20 202a 2055 7361 6765 3a20 0a20 h,. * Usage: . │ │ │ │ -00052cd0: 2020 2020 2020 2067 7261 7068 2070 6869 graph phi │ │ │ │ -00052ce0: 0a20 202a 2049 6e70 7574 733a 0a20 2020 . * Inputs:. │ │ │ │ -00052cf0: 2020 202a 2070 6869 2c20 6120 2a6e 6f74 * phi, a *not │ │ │ │ -00052d00: 6520 7269 6e67 206d 6170 3a20 284d 6163 e ring map: (Mac │ │ │ │ -00052d10: 6175 6c61 7932 446f 6329 5269 6e67 4d61 aulay2Doc)RingMa │ │ │ │ -00052d20: 702c 2c20 7265 7072 6573 656e 7469 6e67 p,, representing │ │ │ │ -00052d30: 2061 2072 6174 696f 6e61 6c0a 2020 2020 a rational. │ │ │ │ -00052d40: 2020 2020 6d61 7020 245c 5068 693a 5820 map $\Phi:X │ │ │ │ -00052d50: 5c64 6173 6872 6967 6874 6172 726f 7720 \dashrightarrow │ │ │ │ -00052d60: 5924 0a20 202a 202a 6e6f 7465 204f 7074 Y$. * *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 │ │ │ │ -00052d90: 6720 6675 6e63 7469 6f6e 7320 7769 7468 g functions with │ │ │ │ -00052da0: 206f 7074 696f 6e61 6c20 696e 7075 7473 optional inputs │ │ │ │ -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 │ │ │ │ -00052e50: 7365 6e74 696e 6720 7468 6520 7072 6f6a senting the proj │ │ │ │ -00052e60: 6563 7469 6f6e 0a20 2020 2020 2020 206f ection. o │ │ │ │ -00052e70: 6e20 7468 6520 6669 7273 7420 6661 6374 n the first fact │ │ │ │ -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. │ │ │ │ -00052ef0: 2020 2020 2020 6f6e 2074 6865 2073 6563 on the sec │ │ │ │ -00052f00: 6f6e 6420 6661 6374 6f72 2024 5c6d 6174 ond factor $\mat │ │ │ │ -00052f10: 6862 667b 4772 6170 687d 285c 5068 6929 hbf{Graph}(\Phi) │ │ │ │ -00052f20: 205c 746f 2059 240a 0a44 6573 6372 6970 \to Y$..Descrip │ │ │ │ -00052f30: 7469 6f6e 0a3d 3d3d 3d3d 3d3d 3d3d 3d3d tion.=========== │ │ │ │ -00052f40: 0a0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ..+------------- │ │ │ │ +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 │ │ │ │ +00052cc0: 3a20 6772 6170 682c 0a20 202a 2055 7361 : graph,. * Usa │ │ │ │ +00052cd0: 6765 3a20 0a20 2020 2020 2020 2067 7261 ge: . gra │ │ │ │ +00052ce0: 7068 2070 6869 0a20 202a 2049 6e70 7574 ph phi. * Input │ │ │ │ +00052cf0: 733a 0a20 2020 2020 202a 2070 6869 2c20 s:. * phi, │ │ │ │ +00052d00: 6120 2a6e 6f74 6520 7269 6e67 206d 6170 a *note ring map │ │ │ │ +00052d10: 3a20 284d 6163 6175 6c61 7932 446f 6329 : (Macaulay2Doc) │ │ │ │ +00052d20: 5269 6e67 4d61 702c 2c20 7265 7072 6573 RingMap,, repres │ │ │ │ +00052d30: 656e 7469 6e67 2061 2072 6174 696f 6e61 enting a rationa │ │ │ │ +00052d40: 6c0a 2020 2020 2020 2020 6d61 7020 245c l. map $\ │ │ │ │ +00052d50: 5068 693a 5820 5c64 6173 6872 6967 6874 Phi:X \dashright │ │ │ │ +00052d60: 6172 726f 7720 5924 0a20 202a 202a 6e6f arrow Y$. * *no │ │ │ │ +00052d70: 7465 204f 7074 696f 6e61 6c20 696e 7075 te Optional inpu │ │ │ │ +00052d80: 7473 3a20 284d 6163 6175 6c61 7932 446f ts: (Macaulay2Do │ │ │ │ +00052d90: 6329 7573 696e 6720 6675 6e63 7469 6f6e c)using function │ │ │ │ +00052da0: 7320 7769 7468 206f 7074 696f 6e61 6c20 s with optional │ │ │ │ +00052db0: 696e 7075 7473 2c3a 0a20 2020 2020 202a inputs,:. * │ │ │ │ +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 $\mathb │ │ │ │ +00052e90: 667b 4772 6170 687d 285c 5068 6929 205c f{Graph}(\Phi) \ │ │ │ │ +00052ea0: 746f 2058 240a 2020 2020 2020 2a20 6120 to X$. * a │ │ │ │ +00052eb0: 2a6e 6f74 6520 7269 6e67 206d 6170 3a20 *note ring map: │ │ │ │ +00052ec0: 284d 6163 6175 6c61 7932 446f 6329 5269 (Macaulay2Doc)Ri │ │ │ │ +00052ed0: 6e67 4d61 702c 2c20 7265 7072 6573 656e ngMap,, represen │ │ │ │ +00052ee0: 7469 6e67 2074 6865 2070 726f 6a65 6374 ting the project │ │ │ │ +00052ef0: 696f 6e0a 2020 2020 2020 2020 6f6e 2074 ion. on t │ │ │ │ +00052f00: 6865 2073 6563 6f6e 6420 6661 6374 6f72 he second factor │ │ │ │ +00052f10: 2024 5c6d 6174 6862 667b 4772 6170 687d $\mathbf{Graph} │ │ │ │ +00052f20: 285c 5068 6929 205c 746f 2059 240a 0a44 (\Phi) \to Y$..D │ │ │ │ +00052f30: 6573 6372 6970 7469 6f6e 0a3d 3d3d 3d3d escription.===== │ │ │ │ +00052f40: 3d3d 3d3d 3d3d 0a0a 2b2d 2d2d 2d2d 2d2d ======..+------- │ │ │ │ 00052f50: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -00052f90: 2b0a 7c69 3120 3a20 7068 6920 3d20 6d61 +.|i1 : phi = ma │ │ │ │ -00052fa0: 7028 5151 5b78 5f30 2e2e 785f 335d 2c51 p(QQ[x_0..x_3],Q │ │ │ │ -00052fb0: 515b 795f 302e 2e79 5f32 5d2c 7b2d 785f Q[y_0..y_2],{-x_ │ │ │ │ -00052fc0: 315e 322b 785f 302a 785f 322c 2d78 5f31 1^2+x_0*x_2,-x_1 │ │ │ │ -00052fd0: 2a78 5f32 2b78 5f30 2a78 5f33 2c2d 785f *x_2+x_0*x_3,-x_ │ │ │ │ -00052fe0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00052f90: 2d2d 2d2d 2d2d 2b0a 7c69 3120 3a20 7068 ------+.|i1 : ph │ │ │ │ +00052fa0: 6920 3d20 6d61 7028 5151 5b78 5f30 2e2e i = map(QQ[x_0.. │ │ │ │ +00052fb0: 785f 335d 2c51 515b 795f 302e 2e79 5f32 x_3],QQ[y_0..y_2 │ │ │ │ +00052fc0: 5d2c 7b2d 785f 315e 322b 785f 302a 785f ],{-x_1^2+x_0*x_ │ │ │ │ +00052fd0: 322c 2d78 5f31 2a78 5f32 2b78 5f30 2a78 2,-x_1*x_2+x_0*x │ │ │ │ +00052fe0: 5f33 2c2d 785f 7c0a 7c20 2020 2020 2020 _3,-x_|.| │ │ │ │ 00052ff0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053000: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053010: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053020: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053030: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00053030: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00053040: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053050: 2020 2020 2020 2020 2032 2020 2020 2020 2 │ │ │ │ +00053050: 2020 2020 2020 2020 2020 2020 2020 2032 2 │ │ │ │ 00053060: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053070: 2020 2020 2032 2020 2020 2020 2020 2020 2 │ │ │ │ -00053080: 7c0a 7c6f 3120 3d20 6d61 7020 2851 515b |.|o1 = map (QQ[ │ │ │ │ -00053090: 7820 2e2e 7820 5d2c 2051 515b 7920 2e2e x ..x ], QQ[y .. │ │ │ │ -000530a0: 7920 5d2c 207b 2d20 7820 202b 2078 2078 y ], {- x + x x │ │ │ │ -000530b0: 202c 202d 2078 2078 2020 2b20 7820 7820 , - x x + x x │ │ │ │ -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 |.| │ │ │ │ +00053070: 2020 2020 2020 2020 2020 2032 2020 2020 2 │ │ │ │ +00053080: 2020 2020 2020 7c0a 7c6f 3120 3d20 6d61 |.|o1 = ma │ │ │ │ +00053090: 7020 2851 515b 7820 2e2e 7820 5d2c 2051 p (QQ[x ..x ], Q │ │ │ │ +000530a0: 515b 7920 2e2e 7920 5d2c 207b 2d20 7820 Q[y ..y ], {- x │ │ │ │ +000530b0: 202b 2078 2078 202c 202d 2078 2078 2020 + x x , - x x │ │ │ │ +000530c0: 2b20 7820 7820 2c20 2d20 7820 202b 2078 + x x , - x + x │ │ │ │ +000530d0: 2078 207d 2920 7c0a 7c20 2020 2020 2020 x }) |.| │ │ │ │ +000530e0: 2020 2020 2020 2030 2020 2033 2020 2020 0 3 │ │ │ │ +000530f0: 2020 2030 2020 2032 2020 2020 2020 2031 0 2 1 │ │ │ │ +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 |.| │ │ │ │ 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: 7c0a 7c6f 3120 3a20 5269 6e67 4d61 7020 |.|o1 : RingMap │ │ │ │ -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 ] │ │ │ │ +00053170: 2020 2020 2020 7c0a 7c6f 3120 3a20 5269 |.|o1 : Ri │ │ │ │ +00053180: 6e67 4d61 7020 5151 5b78 202e 2e78 205d ngMap QQ[x ..x ] │ │ │ │ +00053190: 203c 2d2d 2051 515b 7920 2e2e 7920 5d20 <-- QQ[y ..y ] │ │ │ │ 000531a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000531b0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 000531f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053210: 7c0a 7c2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.|------------- │ │ │ │ +00053210: 2020 2020 2020 7c0a 7c2d 2d2d 2d2d 2d2d |.|------- │ │ │ │ 00053220: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 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 ---------------- │ │ │ │ -00053260: 7c0a 7c32 5e32 2b78 5f31 2a78 5f33 7d29 |.|2^2+x_1*x_3}) │ │ │ │ -00053270: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00053260: 2d2d 2d2d 2d2d 7c0a 7c32 5e32 2b78 5f31 ------|.|2^2+x_1 │ │ │ │ +00053270: 2a78 5f33 7d29 2020 2020 2020 2020 2020 *x_3}) │ │ │ │ 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: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000532b0: 2020 2020 2020 7c0a 2b2d 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 ---------------- │ │ │ │ 000532f0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053300: 2b0a 7c69 3220 3a20 6772 6170 6820 7068 +.|i2 : graph ph │ │ │ │ -00053310: 6920 2020 2020 2020 2020 2020 2020 2020 i │ │ │ │ +00053300: 2d2d 2d2d 2d2d 2b0a 7c69 3220 3a20 6772 ------+.|i2 : gr │ │ │ │ +00053310: 6170 6820 7068 6920 2020 2020 2020 2020 aph phi │ │ │ │ 00053320: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053330: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053340: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053350: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ +00053350: 2020 2020 2020 7c0a 7c20 2020 2020 2020 |.| │ │ │ │ 00053360: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053370: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053380: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053390: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000533a0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000533b0: 2020 2020 2020 2020 2051 515b 7820 2e2e QQ[x .. │ │ │ │ -000533c0: 7820 2c20 7920 2e2e 7920 5d20 2020 2020 x , y ..y ] │ │ │ │ -000533d0: 2020 2020 2020 2020 2020 2020 2020 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2020 7c0a 7c6f 3220 3d20 286d |.|o2 = (m │ │ │ │ +00053450: 6170 2028 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ap (------------ │ │ │ │ 00053460: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053470: 2d2d 2d2d 2d2d 2c20 5151 5b78 202e 2e78 ------, QQ[x ..x │ │ │ │ -00053480: 205d 2c20 7b78 202c 2078 202c 2078 202c ], {x , x , x , │ │ │ │ -00053490: 7c0a 7c20 2020 2020 2020 2020 2020 2878 |.| (x │ │ │ │ -000534a0: 2079 2020 2d20 7820 7920 202b 2078 2079 y - x y + x y │ │ │ │ -000534b0: 202c 2078 2079 2020 2d20 7820 7920 202b , x y - x y + │ │ │ │ -000534c0: 2078 2079 2029 2020 2020 2020 3020 2020 x y ) 0 │ │ │ │ -000534d0: 3320 2020 2020 3020 2020 3120 2020 3220 3 0 1 2 │ │ │ │ -000534e0: 7c0a 7c20 2020 2020 2020 2020 2020 2020 |.| │ │ │ │ -000534f0: 3320 3020 2020 2032 2031 2020 2020 3120 3 0 2 1 1 │ │ │ │ -00053500: 3220 2020 3220 3020 2020 2031 2031 2020 2 2 0 1 1 │ │ │ │ -00053510: 2020 3020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ +00053470: 2d2d 2d2d 2d2d 2d2d 2d2d 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-000535f0: 2020 3020 2020 3320 2020 3020 2020 3220 0 3 0 2 │ │ │ │ -00053600: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +000535f0: 2020 2020 2020 2020 3020 2020 3320 2020 0 3 │ │ │ │ +00053600: 3020 2020 3220 2020 2020 2020 2020 2020 0 2 │ │ │ │ 00053610: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053620: 7c0a 7c20 2020 2020 7820 7d29 2c20 6d61 |.| x }), ma │ │ │ │ -00053630: 7020 282d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d p (------------- │ │ │ │ +00053620: 2020 2020 2020 7c0a 7c20 2020 2020 7820 |.| x │ │ │ │ +00053630: 7d29 2c20 6d61 7020 282d 2d2d 2d2d 2d2d }), map (------- │ │ │ │ 00053640: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053650: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2c 2051 515b -----------, QQ[ │ │ │ │ -00053660: 7920 2e2e 7920 5d2c 207b 7920 2c20 2020 y ..y ], {y , │ │ │ │ -00053670: 7c0a 7c20 2020 2020 2033 2020 2020 2020 |.| 3 │ │ │ │ -00053680: 2020 2028 7820 7920 202d 2078 2079 2020 (x y - x y │ │ │ │ -00053690: 2b20 7820 7920 2c20 7820 7920 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+000536d0: 2020 2020 2020 2020 2020 2033 2030 2020 3 0 │ │ │ │ +000536e0: 2020 3220 3120 2020 2031 2032 2020 2032 2 1 1 2 2 │ │ │ │ +000536f0: 2030 2020 2020 3120 3120 2020 2030 2032 0 1 1 0 2 │ │ │ │ 00053700: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00053710: 7c0a 7c20 2020 2020 2d2d 2d2d 2d2d 2d2d |.| -------- │ │ │ │ +00053710: 2020 2020 2020 7c0a 7c20 2020 2020 2d2d |.| -- │ │ │ │ 00053720: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053730: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053740: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053750: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053760: 7c0a 7c20 2020 2020 7920 2c20 7920 7d29 |.| y , y }) │ │ │ │ -00053770: 2920 2020 2020 2020 2020 2020 2020 2020 ) │ │ │ │ +00053760: 2d2d 2d2d 2d2d 7c0a 7c20 2020 2020 7920 ------|.| y │ │ │ │ +00053770: 2c20 7920 7d29 2920 2020 2020 2020 2020 , y })) │ │ │ │ 00053780: 2020 2020 2020 2020 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│ │ -00053860: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ +00053850: 2020 2020 2020 7c0a 7c6f 3220 3a20 5365 |.|o2 : Se │ │ │ │ +00053860: 7175 656e 6365 2020 2020 2020 2020 2020 quence │ │ │ │ 00053870: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053880: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053890: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000538a0: 7c0a 2b2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d |.+------------- │ │ │ │ +000538a0: 2020 2020 2020 7c0a 2b2d 2d2d 2d2d 2d2d |.+------- │ │ │ │ 000538b0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000538c0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000538d0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 000538e0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -000538f0: 2b0a 0a53 6565 2061 6c73 6f0a 3d3d 3d3d +..See also.==== │ │ │ │ -00053900: 3d3d 3d3d 0a0a 2020 2a20 2a6e 6f74 6520 ====.. * *note │ │ │ │ -00053910: 6772 6170 6828 5261 7469 6f6e 616c 4d61 graph(RationalMa │ │ │ │ -00053920: 7029 3a20 6772 6170 682c 202d 2d20 636c p): graph, -- cl │ │ │ │ -00053930: 6f73 7572 6520 6f66 2074 6865 2067 7261 osure of the gra │ │ │ │ -00053940: 7068 206f 6620 6120 7261 7469 6f6e 616c ph of a rational │ │ │ │ -00053950: 206d 6170 0a20 202a 202a 6e6f 7465 2067 map. * *note g │ │ │ │ -00053960: 7261 7068 4964 6561 6c3a 2028 4d61 6361 raphIdeal: (Maca │ │ │ │ -00053970: 756c 6179 3244 6f63 2967 7261 7068 4964 ulay2Doc)graphId │ │ │ │ -00053980: 6561 6c5f 6c70 5269 6e67 4d61 705f 7270 eal_lpRingMap_rp │ │ │ │ -00053990: 2c20 2d2d 2074 6865 2069 6465 616c 206f , -- the ideal o │ │ │ │ -000539a0: 660a 2020 2020 7468 6520 6772 6170 6820 f. the graph │ │ │ │ -000539b0: 6f66 2074 6865 2072 6567 756c 6172 206d of the regular m │ │ │ │ -000539c0: 6170 2063 6f72 7265 7370 6f6e 6469 6e67 ap corresponding │ │ │ │ -000539d0: 2074 6f20 6120 7269 6e67 206d 6170 0a0a to a ring map.. │ │ │ │ -000539e0: 5761 7973 2074 6f20 7573 6520 7468 6973 Ways to use this │ │ │ │ -000539f0: 206d 6574 686f 643a 0a3d 3d3d 3d3d 3d3d method:.======= │ │ │ │ +000538f0: 2d2d 2d2d 2d2d 2b0a 0a53 6565 2061 6c73 ------+..See als │ │ │ │ +00053900: 6f0a 3d3d 3d3d 3d3d 3d3d 0a0a 2020 2a20 o.========.. * │ │ │ │ +00053910: 2a6e 6f74 6520 6772 6170 6828 5261 7469 *note graph(Rati │ │ │ │ +00053920: 6f6e 616c 4d61 7029 3a20 6772 6170 682c onalMap): graph, │ │ │ │ +00053930: 202d 2d20 636c 6f73 7572 6520 6f66 2074 -- closure of t │ │ │ │ +00053940: 6865 2067 7261 7068 206f 6620 6120 7261 he graph of a ra │ │ │ │ +00053950: 7469 6f6e 616c 206d 6170 0a20 202a 202a tional map. * * │ │ │ │ +00053960: 6e6f 7465 2067 7261 7068 4964 6561 6c3a note graphIdeal: │ │ │ │ +00053970: 2028 4d61 6361 756c 6179 3244 6f63 2967 (Macaulay2Doc)g │ │ │ │ +00053980: 7261 7068 4964 6561 6c5f 6c70 5269 6e67 raphIdeal_lpRing │ │ │ │ +00053990: 4d61 705f 7270 2c20 2d2d 2074 6865 2069 Map_rp, -- the i │ │ │ │ +000539a0: 6465 616c 206f 660a 2020 2020 7468 6520 deal of. the │ │ │ │ +000539b0: 6772 6170 6820 6f66 2074 6865 2072 6567 graph of the reg │ │ │ │ +000539c0: 756c 6172 206d 6170 2063 6f72 7265 7370 ular map corresp │ │ │ │ +000539d0: 6f6e 6469 6e67 2074 6f20 6120 7269 6e67 onding to a ring │ │ │ │ +000539e0: 206d 6170 0a0a 5761 7973 2074 6f20 7573 map..Ways to us │ │ │ │ +000539f0: 6520 7468 6973 206d 6574 686f 643a 0a3d e this method:.= │ │ │ │ 00053a00: 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d 3d3d ================ │ │ │ │ -00053a10: 3d0a 0a20 202a 202a 6e6f 7465 2067 7261 =.. * *note gra │ │ │ │ -00053a20: 7068 2852 696e 674d 6170 293a 2067 7261 ph(RingMap): gra │ │ │ │ -00053a30: 7068 5f6c 7052 696e 674d 6170 5f72 702c ph_lpRingMap_rp, │ │ │ │ -00053a40: 202d 2d20 636c 6f73 7572 6520 6f66 2074 -- closure of t │ │ │ │ -00053a50: 6865 2067 7261 7068 206f 6620 610a 2020 he graph of a. │ │ │ │ -00053a60: 2020 7261 7469 6f6e 616c 206d 6170 0a2d rational map.- │ │ │ │ -00053a70: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00053a10: 3d3d 3d3d 3d3d 3d0a 0a20 202a 202a 6e6f =======.. * *no │ │ │ │ +00053a20: 7465 2067 7261 7068 2852 696e 674d 6170 te graph(RingMap │ │ │ │ +00053a30: 293a 2067 7261 7068 5f6c 7052 696e 674d ): graph_lpRingM │ │ │ │ +00053a40: 6170 5f72 702c 202d 2d20 636c 6f73 7572 ap_rp, -- closur │ │ │ │ +00053a50: 6520 6f66 2074 6865 2067 7261 7068 206f e of the graph o │ │ │ │ +00053a60: 6620 610a 2020 2020 7261 7469 6f6e 616c f a. rational │ │ │ │ +00053a70: 206d 6170 0a2d 2d2d 2d2d 2d2d 2d2d 2d2d map.----------- │ │ │ │ 00053a80: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053a90: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ 00053aa0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ -00053ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 0a0a --------------.. │ │ │ │ -00053ac0: 5468 6520 736f 7572 6365 206f 6620 7468 The source of th │ │ │ │ -00053ad0: 6973 2064 6f63 756d 656e 7420 6973 2069 is document is i │ │ │ │ -00053ae0: 6e0a 2f62 7569 6c64 2f72 6570 726f 6475 n./build/reprodu │ │ │ │ -00053af0: 6369 626c 652d 7061 7468 2f6d 6163 6175 cible-path/macau │ │ │ │ -00053b00: 6c61 7932 2d31 2e32 362e 3036 2b64 732f lay2-1.26.06+ds/ │ │ │ │ -00053b10: 4d32 2f4d 6163 6175 6c61 7932 2f70 6163 M2/Macaulay2/pac │ │ │ │ -00053b20: 6b61 6765 732f 4372 656d 6f6e 612f 0a64 kages/Cremona/.d │ │ │ │ -00053b30: 6f63 756d 656e 7461 7469 6f6e 2e6d 323a ocumentation.m2: │ │ │ │ -00053b40: 3236 373a 302e 0a1f 0a46 696c 653a 2043 267:0....File: C │ │ │ │ -00053b50: 7265 6d6f 6e61 2e69 6e66 6f2c 204e 6f64 remona.info, Nod │ │ │ │ -00053b60: 653a 2069 6465 616c 5f6c 7052 6174 696f e: ideal_lpRatio │ │ │ │ -00053b70: 6e61 6c4d 6170 5f72 702c 204e 6578 743a nalMap_rp, Next: │ │ │ │ -00053b80: 2069 6d61 6765 5f6c 7052 6174 696f 6e61 image_lpRationa │ │ │ │ -00053b90: 6c4d 6170 5f63 6d53 7472 696e 675f 7270 lMap_cmString_rp │ │ │ │ -00053ba0: 2c20 5072 6576 3a20 6772 6170 685f 6c70 , Prev: graph_lp │ │ │ │ -00053bb0: 5269 6e67 4d61 705f 7270 2c20 5570 3a20 RingMap_rp, Up: │ │ │ │ -00053bc0: 546f 700a 0a69 6465 616c 2852 6174 696f Top..ideal(Ratio │ │ │ │ -00053bd0: 6e61 6c4d 6170 2920 2d2d 2062 6173 6520 nalMap) -- base │ │ │ │ -00053be0: 6c6f 6375 7320 6f66 2061 2072 6174 696f locus of a ratio │ │ │ │ -00053bf0: 6e61 6c20 6d61 700a 2a2a 2a2a 2a2a 2a2a nal map.******** │ │ │ │ +00053ab0: 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d 2d2d ---------------- │ │ │ │ +00053ac0: 2d2d 2d2d 0a0a 5468 6520 736f 7572 6365 ----..The source │ │ │ │ +00053ad0: 206f 6620 7468 6973 2064 6f63 756d 656e of this documen │ │ │ │ +00053ae0: 7420 6973 2069 6e0a 2f62 7569 6c64 2f72 t is in./build/r │ │ │ │ +00053af0: 6570 726f 6475 6369 626c 652d 7061 7468 eproducible-path │ │ │ │ +00053b00: 2f6d 6163 6175 6c61 7932 2d31 2e32 362e /macaulay2-1.26. │ │ │ │ +00053b10: 3036 2b64 732f 4d32 2f4d 6163 6175 6c61 06+ds/M2/Macaula │ │ │ │ +00053b20: 7932 2f70 6163 6b61 6765 732f 4372 656d y2/packages/Crem │ │ │ │ +00053b30: 6f6e 612f 0a64 6f63 756d 656e 7461 7469 ona/.documentati │ │ │ │ +00053b40: 6f6e 2e6d 323a 3236 373a 302e 0a1f 0a46 on.m2:267:0....F │ │ │ │ +00053b50: 696c 653a 2043 7265 6d6f 6e61 2e69 6e66 ile: Cremona.inf │ │ │ │ +00053b60: 6f2c 204e 6f64 653a 2069 6465 616c 5f6c o, Node: ideal_l │ │ │ │ +00053b70: 7052 6174 696f 6e61 6c4d 6170 5f72 702c pRationalMap_rp, │ │ │ │ +00053b80: 204e 6578 743a 2069 6d61 6765 5f6c 7052 Next: image_lpR │ │ │ │ +00053b90: 6174 696f 6e61 6c4d 6170 5f63 6d53 7472 ationalMap_cmStr │ │ │ │ +00053ba0: 696e 675f 7270 2c20 5072 6576 3a20 6772 ing_rp, Prev: gr │ │ │ │ +00053bb0: 6170 685f 6c70 5269 6e67 4d61 705f 7270 aph_lpRingMap_rp │ │ │ │ +00053bc0: 2c20 5570 3a20 546f 700a 0a69 6465 616c , Up: Top..ideal │ │ │ │ +00053bd0: 2852 6174 696f 6e61 6c4d 6170 2920 2d2d (RationalMap) -- │ │ │ │ +00053be0: 2062 6173 6520 6c6f 6375 7320 6f66 2061 base locus of a │ │ │ │ +00053bf0: 2072 6174 696f 6e61 6c20 6d61 700a 2a2a rational map.** │ │ │ │ 00053c00: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ 00053c10: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ -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...+----------- │ │ │ │ +00053c20: 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a 2a2a **************** │ │ │ │ +00053c30: 0a0a 2020 2a20 4675 6e63 7469 6f6e 3a20 .. * Function: │ │ │ │ +00053c40: 2a6e 6f74 6520 6964 6561 6c3a 2028 4d61 *note ideal: (Ma │ │ │ │ +00053c50: 6361 756c 6179 3244 6f63 2969 6465 616c caulay2Doc)ideal │ │ │ │ +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 │ │ │ │ +00053ca0: 2072 6174 696f 6e61 6c20 6d61 703a 2052 rational map: R │ │ │ │ +00053cb0: 6174 696f 6e61 6c4d 6170 2c0a 2020 2a20 ationalMap,. * │ │ │ │ +00053cc0: 4f75 7470 7574 733a 0a20 2020 2020 202a Outputs:. * │ │ │ │ +00053cd0: 2061 6e20 2a6e 6f74 6520 6964 6561 6c3a an *note ideal: │ │ │ │ +00053ce0: 2028 4d61 6361 756c 6179 3244 6f63 2949 (Macaulay2Doc)I │ │ │ │ +00053cf0: 6465 616c 2c2c 2074 6865 2069 6465 616c deal,, the ideal │ │ │ │ +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...+----- │ │ │ │ 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 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 |.+----------- │ │ │ │ +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 |.+----- │ │ │ │ 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 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^|.| │ │ │ │ +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^|.| │ │ │ │ 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 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 │ │ │ │ +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 -- │ │ │ │ 00053f70: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00053f80: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 |.| │ │ │ │ +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 |.| │ │ │ │ 00053ff0: 2020 2020 2020 2020 2020 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 { │ │ │ │ +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 { │ │ │ │ 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 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00054090: 2020 2020 3220 2020 2020 2020 2020 2020 2 │ │ │ │ +00054080: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00054090: 2020 2020 2020 2020 2020 3220 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 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 │ │ │ │ +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 │ │ │ │ 00054100: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054110: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00054150: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054160: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054170: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00054180: 2020 7d20 2020 2020 2020 2020 2020 2020 } │ │ │ │ +00054170: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00054180: 2020 2020 2020 2020 7d20 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 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 │ │ │ │ +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 ]) │ │ │ │ 00054200: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 00054250: 2020 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 │ │ │ │ +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 { │ │ │ │ 00054290: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000542a0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 │ │ │ │ +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 │ │ │ │ 000542e0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 000542f0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054300: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00054310: 2020 2020 2020 2020 2020 2078 2020 2d20 x - │ │ │ │ -00054320: 7820 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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 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 │ │ │ │ +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 , │ │ │ │ 00054430: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -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 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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 │ │ │ │ 000545c0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -000545d0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ +000545d0: 2020 2020 2020 2020 7c0a 7c20 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 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 │ │ │ │ +00054620: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00054630: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ 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ideal phi │ │ │ │ 00054a00: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a10: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054a20: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054a30: 2020 7c0a 7c20 2d2d 2075 7365 6420 302e |.| -- used 0. │ │ │ │ -00054a40: 3030 3335 3237 3832 7320 2863 7075 293b 00352782s (cpu); │ │ │ │ -00054a50: 2030 2e30 3033 3532 3237 3673 2028 7468 0.00352276s (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 |.| │ │ │ │ +00054a30: 2020 2020 2020 2020 7c0a 7c20 2d2d 2075 |.| -- u │ │ │ │ +00054a40: 7365 6420 302e 3030 3335 3139 3936 7320 sed 0.00351996s │ │ │ │ +00054a50: 2863 7075 293b 2030 2e30 3033 3531 3538 (cpu); 0.0035158 │ │ │ │ +00054a60: 3473 2028 7468 7265 6164 293b 2030 7320 4s (thread); 0s │ │ │ │ +00054a70: 2867 6329 2020 2020 2020 2020 2020 2020 (gc) │ │ │ │ +00054a80: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ 00054a90: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054aa0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ab0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ 00054ac0: 2020 2020 2020 2020 2020 2020 2020 2020 │ │ │ │ -00054ad0: 2020 7c0a 7c20 2020 2020 2020 2020 2020 |.| │ │ │ │ -00054ae0: 2020 3220 2020 2020 2020 2020 2020 2020 2 │ │ │ │ +00054ad0: 2020 2020 2020 2020 7c0a 7c20 2020 2020 |.| │ │ │ │ +00054ae0: 2020 2020 2020 202 TRUNCATED DUE TO SIZE LIMIT: 10485760 bytes